VERSJTY RESEARCH STUDIES IN EDUCATION NUMBER 6
WHMETIC IN GRADES I AND II
I Critical Summary of New and Prcviooas.y Reported
WILLIAM A. BROWNELL
WITH THE ASSISTANCE OF
ROY A. DOTY AND WILLIAM C. REIN
DUKE UNIVERSITY PRESS
Durham, North Carolina
"" : 'i
DUKE UNIVERSITY RESEARCH STUDIES IN EDUCATION
ARITHMETIC IN GRADES I AND II
A Critical Summary of New and Previously Reported
ARITHMETIC IN GRADES I AND II
A Critical Summary of New and Previously Reported
WILLIAM A. BROWNELL
WITH THE ASSISTANCE OF
ROY A. DOTY AND WILLIAM C. REIN
DUKE UNIVERSITY PRESS
Durham, North Carolina
COPYRIGHT, 1941, BY THE
DUKE UNIVERSITY PRESS
I I -
PRINTED IN THE UNITED STATES OF AMERICA BY
THE SEEMAN PRINTER, INC., DURHAM, N. C.
I am very glad indeed to acknowledge my indebtedness to the fol-
lowing agencies and persons for their assistance:
1. To Duke University, for research funds to prosecute this study
and for publishing the report.
2. To Ginn and Company of Boston, Massachusetts, for supply-
ing free of charge a special printing of the tests used in the investi-
gation reported in Chapter III and for permission to reproduce copies
of tests in this monograph.
3. To the administrators and primary teachers of the schools listed
below, in which the data for two original studies were obtained:
California Davidson Township
San Andreas Dushore
Florida Erie (Lawrence Park)
Massachusetts Forest City
Lexington (Hancock) Jenkins Township
Waltham (Plympton, Whitemore) New Brighton
North Carolina Smethport
Durham (North Durham, Watts) Summerhill Township
Durham County (Bragtown, Glenn, Susquehanna (Thompson)
Hope Valley, Oak Grove) Williamsport
Raleigh (Boylan Heights, Hayes- Virginia
Winston-Salem (Granville) Petersburg (Brown,
Ohio Jackson, Lee)
Marysville Prince George
Canton Township Wisconsin
Charleston Green Bay
(For one reason or another, the data from some of these schools
could not be used, but this fact in no way lessens my feeling of
obligation for the attempted co-operation.)
4. To Miss Hulda Kilmer of the Dushore, Pennsylvania, schools,
who was largely instrumental in securing data from rural schools in
- <, -L
5. To the graduate students in my courses, Experimental Educa-
tion and Investigations in Arithmetic, who helped materially in group
testing and in interviewing.
6. To the junior authors, Mr. Roy A. Doty and Mr. W. C. Rein,
for assisting in the tabulations reported in Chapters II and III and in
the bibliographical research in Chapters II and IV, for preparing
several sections of Chapter IV, and for reading and criticizing the
manuscript as a whole.
W. A. B.
I. THE PRESENT STATUS OF NUMBER IN THE PRIMARY GRADES 3
Reasons for Present Confusion........................... 3
The Commoner Programs of Primary Number Instruction... 6
The Crucial Questions.................... ............. 8
Purpose of This Monograph ............. ............... 9
II. ARITHMETIC IN THE POSSESSION OF SCHOOL BEGINNERS 11
Part I. Original Investigation, with Related Research
(Topics 1-7) .............. ............ .............. 12
1. Rote Counting.............. ...................... 14
2. Enum eration ................... ................... 18
3. Identification ...................... ................. 20
4. R production ........................ .............. 24
5. Crude Quantitative Comparison ....................... 28
Concrete objects or pictures................:.......... 28
Abstract numbers .................................... 32
6. Exact Quantitative Comparison, or Matching........... 33
7. Number Combinations or Facts, and Their Use in Problems 34
The number combinations when presented concretely.. 35
The number combinations when presented in verbal
problem s ...................................... 37
The number combinations when presented in abstract
form ......................................... 42
Conclusions with respect to the number combinations..... 44
Part II. Topics 8-17............... ...................... 44
8. F actions ........................................... 44
9. Ordinals ......................................... 47
10. Reading and Writing Numbers ....................... 48
Reading numbers............. .................. 48
Writing numbers ............................... 49
11. Recognition of Geometric Forms ..................... 49
12. Time, U. S. Money, and Measures ..................... 50
13. Sex Differences....................................... 51
14. City versus Rural Children............................ 52
15. Differences in Levels of Intelligence................... 53
16. Effect of Kindergarten Instruction .................... 54
[ vii ]
17. M miscellaneous Studies................................. 55
Preschool studies.................................. 55
Readiness testing .............................. 57
Part III. Appraisal of Research Findings ................... 58
Brief Summary of Findings.......................... 58
Limitations of Research............................... -59
Cautions in Applications.............................. 61
Significance of Research Findings...................... 62
III. RESULTS OF A PARTICULAR PROGRAM OF SYSTEMATIC
ARITHMETIC INSTRUCTION IN GRADES I AND II 64
The Plan of Instruction............. ..................... 65
Outcomes ............................................. 63
Methods of teaching................ .................. 66
M materials of instruction.................................. 67
Experimental subjects................... .............. 69
Tests ................................................. 69
Results from the Group Tests ............ .................. 70
Gross results............... .......................... 70
Results in Grade IB.................................. 72
Results in Grade IA.................................. 76
Results in Grade IIB.................................... 80
Results in Grade IIA................... .............. 86
Results of the Individual Tests.............................. 90
Procedure ........................ .... ............... 90
Summary of interview data, by grades ................... 92
Thought processes or procedures on "known" combinations.. 92
The process of learning the combinations ................. 96
The subtraction compared with the addition combinations.... 99
Concluding Statement................................... 102
IV. PREVIOUSLY REPORTED RESEARCH ON THE EFFECTS OF
INITIATING OR DEFERRING ARITHMETIC INSTRUCTION
IN GRADES I AND II 104
Evidence on the Values of Systematic Instruction Beginning in
Grade I ........................... .. ............. 105
Evidence on the Values of "Social" Arithmetic in Grade I (or in
Grades I and II) and of Deferring Systematic Instruction... 107
Evidence on the Values of Abandoning Systematic Instruction
Entirely, or at Least in the First Grades................... 112
Teaching the Simple Combinations or Facts.................. 116
Comparative Difficulty of the Number Combinations or Facts... 122
Reading Numbers.............. ......................... 128
Psychological Development of Number Concepts and Skills.... 129
Counting (Enumeration) ............................... 129
N um ber concepts......................... ............. 130
Transfer of Training in Learning Arithmetic................ 135
Transfer with number combinations ...................... 135
Learning simple addition and subtraction operations........ 138
Conclusions with regard to transfer....................... 141
Grade Placement of Topics................................. 141
Children's Uses of, and Needs for, Number in the Primary
Grades ...................... ......... .............. 144
Specific Techniques; Minor Problems....................... 147
D rill .................. ... ............. ............... 147
Use of number patterns and other perceptual aids.......... 149
Adding upward versus adding downward .................. 150
Subtractive versus additive subtraction .................... 151
Teaching the related combinations together or separately.... 152
Games and devices.................................... 154
Permanence of learning .................................. 155
Conclusion .............................. ............... 156
V. A POSITION WITH RESPECT TO ARITHMETIC IN
GRADES I AND II 160
The Three Crucial Questions................. .............. 160
Meaning and Significance in Arithmetic .................... 162
Evaluating Programs of Primary Arithmetic................. 163
Arguments against and for Systematic Instruction............. 165
Effects on personality................................. 165
The policy of postponement............................. 166
CODE OF TOPIcs 170
LIST OF TABLES
NUMBER TITLE PAGE
1. Grade I Pupils Tested...................... .............. 13
2. Present Study: Results of the Tests of Rote Counting and
Enumeration .................... ........................ 15
3. Summary of Findings with Respect to Rote Counting by l's.... 16
4. Summary of Findings with Respect to Enumeration............ 19
5. Present Study: Results on Group Test of Identification-Four,
Seven, and Ten Objects .................................... 22
6. Summary of Findings with Respect to Identification ............ 23
7. Present Study: Results on Group Test of Number Reproduction
-5, 6, and 9............................................. 25
8. Summary of Findings with Respect to Number Reproduction.... 26
9. Present Study: Results on Group Test of Crude Comparison-
Concrete Numbers............... .......................... 28
10. Summary of Data with Respect to Crude Comparison, Concrete
Numbers .............................................. 30
11. Present Study: Results on Individual Tests of Crude Comparison
Abstract Numbers.................. .. ... ............... 33
12. Present Study: Results on Group Test of Exact Comparison-
Concrete Numbers 4, 5, 7..................... ............ 34
13. Summary of Findings with Respect to Number Combinations
When Concretely Presented ................................. 36
14. Present Study: Results on Number Combinations in Verbal
Problems ................... ........................... 38
15. Summary of Findings with Respect to Number Combinations
When Presented in Verbal Problems......................... 39
16. Present Study: Results on the Number Combinations in
A abstract Form ....................... ........ ........ .. .. .. 43
17. Per Cents of 1,897 Grade IA Pupils Without Instruction on Num-
ber, Who Had Various Concepts of Time and U. S. Money
(After W oody) ............................................. 50
18. Scores on Term Group Tests, Grade IB Through Grade IIA.... 70
19. Classification of Grade IB Test Items by Outcomes, and Scores
by Test Parts.................... ..................... 73
20. Comparison of Total Grade IB Group and the Selected Sample
of 100 Pupils........................ ... ............... 74
21. Number of Errors and Per Cents of Success on Items of Grade
IB Test; Samples of 100 Cases.............................. 75
22. Classification of Grade IA Test Items by Outcomes, and Scores
by Test Parts.......................................... 78
List of Tables
NUMBER TITLE PAGE
23. Number of Errors and Per Cents of Success on Items of the
Grade IA Test; Sample of 100 Cases........................ 79
24. Classification of Grade IIB Test Items by Outcomes, and Scores
by Test Parts................. .......... ................ 83
25. Number of Errors and Per Cents of Success on Items of the
Grade IIB Test; Sample of 100 Cases ....................... 84
26. Classification of Grade IIA Test Items by Outcomes, and Scores
by Test Parts............. ..... ....... ... .............. 88
27. Number of Errors and Per Cents of Success on Items of the
Grade IIA Test; Sample of 100 Cases ....................... 89
28. Results of Interviews in Grade I, First and Second Terms; Forty
Children Selected from Six Schools.......................... 93
29. Results of Interviews in Grade II, First and Second Terms; Sixty
Children Selected from Seven Schools....................... 94
30. How Children Think About the Combinations They Are Sup-
posed to "Know"........................ ................. 95
31. Nature of Learning as Shown by Changes in Procedures Used
with Number Combinations from Term to Term................ 97
32. Comparison of Procedures Used with A (Addition) and S (Sub-
traction) Combinations in Grades I and II.................... 100
33. Extent to Which Various Types of Solution Were Used on A-
and S-Combinations in Grades I and II ..................... 101
34. A Comparison of the Rank Difficulty of Twelve Addition
Combinations .......................... ................. 125
35. Percentage Scores on First and Last Tests, Together with Per
Cents of Possible Gain Actually Made (Adapted from Overman) 139
36. Relative Frequency with Which Situations Involving Number
Occurred (From Smith) ..................................... 145
37. Results of the Analysis of Ten Series of Primary Readers
(A after Gunderson) .......................... ............. 146
Chart I. Outcomes by Half-Grades.............................. 66
ARITHMETIC IN GRADES I AND II
THE PRESENT STATUS OF NUMBER IN THE
Few problems relating to the elementary school curriculum are
as troublesome as are those associated with the kind and amount of
arithmetic to be taught in the primary grades. These problems are
essentially new problems; they did not exist, or at least they were
not generally recognized, a quarter century ago when practices with
regard to primary number were relatively more uniform. The last
two decades have witnessed a decided break from tradition, but as
yet no satisfactory solution has been found. Instead, there is such
variety of practice as to amount almost to confusion.
The extent of this confusion is readily noted if one but compares
different course of study offerings in primary arithmetic. For ex-
ample, one school expects children to learn the addition combinations
with sums to 10 in Grade I; another defers systematic instruction on
these facts to Grade II; still another postpones such instruction to
Grade III. Needless to say, were other arithmetical topics included
in these comparisons, the variations suggested in the case of a single
topic would be greatly enhanced.
REASONS FOR PRESENT CONFUSION
Reasons for confusion with respect to primary grade number are
not hard to find. In the first place, evidence from school surveys,
from local testing programs, from investigations of the learning of
individual children, from research on adult usage-evidence from all
of these and other sources made it incontestably clear some years
ago that the anticipated results of arithmetical competence were not
being attained. Children proved to be inaccurate in computation and
unintelligent in problem solving; adults were found to employ neither
frequently nor effectively the arithmetic they had been taught, even
when it was obviously useful.
While most of these surveys and most of the experimentation
dealt with arithmetic in the intermediate and higher grades, search
for an explanation of the unsatisfactory conditions inevitably led to
an examination of the state of affairs in the primary grades. Still,
it should be observed that while attention was directed to arithmetic
4 Arithmetic in. Grades I and II
in the first grades (a tendency greatly accelerated by other influences
discussed below), the resulting studies did not in themselves provide
an answer for the questions they raised. TruLe. instruction, as meas-
ured by its results, was ineffective, but what was to be done? It
was possible from the data collected to argue in favor of widely
A second factor making for change was the spread of the educa-
tional philosophy epitomized in the phrase "the child-centered school."
Clearly, the kind of arithmetic content and teaching found commonly
in the primary grades fifteen or twenty years ago was inconsistent
with this conception of education. Little effort was expended to
utilize children's interests; classroom situations in the arithmetic class
were typically "unnatural"; few indeed were the opportunities for
individual creative and exploratory activity; children were told what
to learn and how to "learn" it and then required to "learn" precisely
In a word, many practices in the primary arithmetic class violated
the tenets of the new conception of education; but, unfortunately, this
new conception did not in itself contain a single unambiguous plan
for the needed improvement. To illustrate, it was perfectly consistent
with the new views (or with some of them) to abolish all planned
experiences with number, on the one hand, and, on the other, to
adopt a new and different kind of planned arithmetic and teaching to
start in Grade I. Like the research mentioned in earlier paragraphs,
the new conception of education pointed the need for change but
found equally congenial actual changes which were quite unlike each
A third factor leading to change, if not for agreement in that
change, was psychological in character. Reference here is to learning
theory. Both educational and psychological experimentation on learn-
ing during the last fifteen years or so has raised objections to the
oversimplification of the learning process which had theretofore been
generally accepted. This research called attention to aspects of the
learning situation which had been neglected when the experimenter
artificially isolated the particular "S's" and "R's" in which he was
interested. Nowadays psychol'-cist t,'nd to stress the importance of
the subjective conditions of learmin.:-the previous relevant experi-
ences of the learner, his present goals and motivation, and the like;
they point out that when learning involves relationships and rich
understandings, learning does not take place all at once, but is rather
long continued; they insist that however consistent the learning proc-
Th, Present Status of inmbtr 5
is mn the acqu'siti:on of mianinis: and understandings may be basi-
cally and ultimately with the facts o4 simple conditioning, learning in
such cai>es i- not 'ery helpfully viewed in these terms.
Briefly then, learning theo.:ry now Lmnphasizes increasingly: (1)
the allowance of sufficient time for the completion of learning, (2) the
nece;-it. that the learner have the essential intellectual capacity to
larn,. an.] 13 the presence :-.f a felt purpose or a goal for learning.
Unfortunately. the.e new p:-ycholo,.:ical emphases, important and
welcome as they\ are. furnii-h guidance no less equivocal for the de-
termination ,:. the content and instructional procedures in primary
number than do the factors already considered. Advocates of early
sy.temnatic in-truction insist that by providing the requisite founda-
tionial experience- from the start and b\ deferring mastery perhaps
tO:, Grade III they are making proper allowance for time for learning.
At the -ame time the:ct who viouldi restrict all number experiences in
the primary grades to thostc v. which appFar incidentally and by chance,
hold that they alone are acting in accordance with modern psychol-
g:,_. for they are v.aiting for the child to develop adequate intellectual
power for learning.
/ Coi:fuslion v. ith regard to' primary arithmetic is worse confounded
Sb. the operation:n of a fourth factor, namely, different conceptions of
K tlht1 purpose of arithmetic in the elementary curriculum. These dif-
ferent co:nceptions of course quite largely reflect the influence of
factors, mentioned above, and their differences arise from the ways
in which these factors and the principles deduced therefrom have
been combined. Be that as it may, tliese various conceptions provide
teacher and allministratnr with concrete patterns of thinking which
n- turn influence the kind and amount of arithmetic assigned to the
According. to one view. for example, arithmetic is a tool subject;
it is in the elementary curriculum to equip children to deal effectively
v.ith the quantitatiie problems they can hardly avoid in later life.
Consequently. attention is given chiefly to efficiency (speed and ac-
curacy i. and there is little concern that children shall understand
,.hat the\ are taught. The arithmetic of abstract numbers is intro-
duced from the trart -of Grade I; instruction takes the form of telling
children hat and how, to think; and children "learn" by mastering
the prescrilbed formulas and skills. According to a second conception.
arithmetic is first :of all a mathematical system; the crucial element in
learning is tI1h uiiderstanding of the number system and its operation.
Aritlietic mn Grades I ajid II
Obviously, both in conten' and imiphasi- tihe courics in prinar nunm-
ber which arise from these two <:ntreption- mut Ut -L quite Jiim-inilar.'
THE COMMONER PROGP.'IS, OF PkI'MAR' NUMBER I N.;TRI.CT 10'.
The situation with respect tci prinmarn numibr has Ibcn dlc-.r[iid
as one of confusion, in the sense that no s'ingl,; program -,f inr-tru.-
tion is predominantly favored at present Neverthelek.. it would be
misleading to say that the situation i chaotic, for in the conifusi-.n
one can detect at least fo-ir prrg.ram- of nstructirin v.hic:h are fairl\
(1) The first program i in ore: e.nst: not a pr..,granm at all. ,Unt
rather the negation of program it a,,bislile, all Yvstenma.ntc inirrue-
tion in number in Grade I. or in Grades I and II. o-,r in Griadr- I,
II, and III. This program ma\ L.< de-ignated a, the indilntaiil ip-
proach and may be interpreted a, a most \ g-rius reaction a.gain.t
the formalism of traditional arithimetic tre al-iing. It i, not '.ipp['i:'iid
that children will have no number tcxi-, I cn..:s in the primary grades
on the contrary, it is assumined thit children after enteriing s.i:hol.
will continue to find number a part Iif their natural acilnt'es, a I tlhe
did before entering school ., and it i; further asuiwed that the-e in-
formal and unplanned ccnltacts are uflticient [ti prr.-vi.le a bh is for
later systematic instruction-il indeeId such in-tru.tin i ever of-
fered. Arguments in favor -'f thi- incidJntal appraclih tres- tlie
mental immaturity of children n tcle primary grades. the w.-as.tc in-
herent in efforts to teach arithmetic t.:i children before they are ready
for it, the dangers (largely cmi:'tio:inali whichli nmut result from un-
seemly haste and forcing, and thle niuih .i gr,:ater value iof other type-
of experience than those in\ii ing nunimber. ii the development fi thlt
(2) The second program i- planned frm in ti our.':t, but ,t t
planned chiefly to acquaint children with number a- a iin:rmal part of
their environments. That is to a'ay. the center if intcre-. in the plan-
ning is the social setting or arpplicrtiCi1n .- nmimber. and c:liild ren art
accordingly encouraged to establilli arind operate _grocer, -trres, ,clii'lA
Nor does the confusion stop imr er.'re v. ith a'lierncne ti different cr':ncep-
tions of elementary school arirlhm tic. ,.ni:. t o .f ( '.liich haVe ben menrti.:.ined.
Each conception may be dinrt-rentli interpirited nl iJifferninl., implernenictl.
Thus, one recent writer, while u tl., pirotestiniig hi: co'nm iciion itht all :,rith-
metic must be "meaningful," objects ig.,r...u-l tI., teaching children in Grade
III the rationale of the number *.ieni. -[Ho,, else it i pii.-ssiblre 1t get chil-
dren to understand computational c.l.,crations it.li numhers, this u riner J.:.ci
'F,-.r example: Howard A Lane. "Child Deelol.:rment and ith Three R'.."
ClhiJI.o-d Education, XVI (N.-, l' -). 1.11. lii.
The PF'es'nl Stl,'is ,',f .V\'nii.c,"
bank, p,.ist .,iTfic.,, and then like lecaucse ,.if lie large place given such
experiences in thlis jr,_'.ramn. Pr.giamin ( 2J1 ma,. for the purposes of
tlis isp1,rt. I,: de signat:d a, tihe s ..'ial app',., *. It should be noted
that. a, part ot ti plan. teachers make certain that the socially cen-
tered activitie- '.-ill provi.le plenty .:f .ntact with number, and
%,.n-tact .f a kr,,nii kind At luis pi-'it Program (2) is markedly
dit,:rent fn..im Pr'igrani (1 i. It is h.;.:eve:r like Program (1) in
that it seeks to i: ak,: arithmietic' funciti,'nal, and many of the argu-
iienit. in faltr f Pr:cram i I i and in *oppoitio'ii to traditional prac-
ncee~ may lie clied in cupp:,rt of Pro:rain 2 1.
Si Mllention hals ituit I.,t:n inade .f "-tralditional practice," and
tl-is maiy he r-.jarded as the thiil proiranm :of primary number in-
ructi.i'n..\:cc:'rdir. to: this l'r.,,rani. niumlber is systematically pre-
sentcd from the tart (or tr ir thi: ldieginnirir, i :t"f Grade II). Little
attintiin is given 1 t the. i social applicati rns of number and to the
psychological fai'tors s pr:,iiiient in Programn, (1) and (2). In-
deed. it is assumed lthat c'hihlri:i 'Alien they enter Grade I (or shortly
therevcafterr are ready tir aintact nimilber. After giving some prac-
tic in counting. pcrhals,. t,:ac,:hrs I I:-Ti Ain) the simple addition and
isulItractic-rn fact_. Tihes thy pres.rlt rally and later in written form,
but aa'.\av h ijthe: nurinlr tmbl,._ls and %ith a minimum of con-
c:r.t,. exper'encc Tlie dI-minaicin, purprr.e :,t" the traditional pro-
n, ii. s t1 ,:. get children in the priinary grades to "know" these
number fact. in till: rit':t |;'ib;lle time as tli>. surest foundation
fic.r eCii.n.. ii,'al ut' e in tc i,: co':im uitatiin ,if Gradj ; III.
(4) Unlike the first three programs-the incidental, the social,
and the tra.ditih.al--che fourth program cannot be neatly described
by a simplee name. One reason is that the fourth program gives place
to', \arillus features of th[lie first three and is therefore somewhat eclec-
tic in nature. P,-rhaps a better reason is that this program has no
s'ingl: strickiin characteristic, except that it recognizes the element of
matheimat].il mi:aninig which is, or tends to be, neglected in the other
pro-grahm; N'evcrtcleless, to call this program the "meaning" program
is t.'. lilziht Ither features that are equally essential parts. Advocates
.if tlii fourth pr,.igram are no less concerned than are the exponents
of the first tc'' with making arithmetic socially functional, and no
less; con,-rned than are the exponents of the traditional program
F:.r example- G M Wilson, "New Standards in Arithmetic: A Con-
irolled Experiment in Supervision," Journal of Education Research, XXII
(De: 193'0. 351-36'.:) This is reference 55 in the research bibliography at the
cn'd co thhi mo,',,.;.raph Henceforth citations to this bibliography will be by
8 Arithmetic in Grades I and II
with developing eventually a high degr,'.' of ':.lthicienc int rhe u.e 1i
arithmetic; but they hold that these are lir enoi ugh-that children
must understand what they learn, and so imiu-t have a gra:p ulpon
the mathematics of arithmetic. Briefly. thun, accor-ling t, thie fourth
program full advantage is taken of all incidental occurrence,. *.f
number in "natural" situations; more'rver. o:thIr 'ocical ;ettiilgs Im-
volving number are 'deliberately arranged to Increat;. rcal contact-
with number. But these informal and planned social experiericc: art
supplemented with learning activities dJchbl.rat-:.. d:--i'_ir:id (aI' to
make number and number operations ..n..ible and i I) to- encourage
children as rapidly as they safely may toi adopt tht proctdurc- which
make for arithmetical proficiency.4
THE CRUCIAL Q1iJFTI,)NS
Four programs of primary arithmetic ha\e been bri.tly d.crilbed
Actually, of course, still other pattern ofif content and Initruction
result from various combinations of the four outlined. Thic i_
not the place for a critique of the program-. The purp,-' in men-
tioning them here is mainly to point out that the different progr-au
represent different solutions for the troublec-ome problem which re-
late to arithmetic in the primary curriculum. These troublesome
problems may be reduced to three crucial questions, all of which
must be answered before a satisfactory solution can ultimately be
found. These questions are:
(1) Is the primary grade pupil intellectually capable. of profiting from
systematic instruction in arithmetic? That is, has he the mental powers
requisite to this learning?
(2) If the primary grade pupil can learn much about arithmetic., Jiuld
he be asked to do so at this time; is instruction in number wasteful if
given in the primary grades, or does it produce gains which justify it-
(3) If the primary grade child can and shoudd learn arithmetic from
the start of his school career, what should be the content and form ..' this
The four programs of instruction which have been analyzed ,xii-
ously are based upon different answers to these crucial quc.Ation.
Thus, Program (1) answers question (1) in the negative, while: the
This program is essentially that of the National Council Commiti: (.ri
Arithmetic, sponsored by the National Council of Teachers of Matiemati.::
The point of view has been set forth by the chairman of the Committee. Pr.:--
fessor R. L. Morton, in the Mathematics Teacher for Oct., 1938, ard in ith
Curriculum Journal for Nov., 1938. The forthcoming (1941) yearboo-: .:.f" uii
Committee, the sixteenth in the series of the National Council of Teacher' uf
Mathematics, should be especially helpful in implementing this program.
Thc Piets'nt Status of Number 9
,:thclr three programir aii4-'er it in the affirmative. Likewise Pro-
gramr 1 2 1. .3 1, and i 4 I ansiier the second question in the affirma-
tl't. but an2f r iqueti on i in quite different ways, while Program
i 1 c.nsitcithl sa No: to question (2) and dismisses the problem
i.,f content and meth... bi.\ r,.lling upon incidental number experiences.
But at the. fkV.c:t hIre are four quite different programs of in-
-itruction which airi:c fri:,rm different answers to the crucial questions
and irom dilferent a.\is of implementing those answers. The fact
that there can b,- different arin.vers and programs arises in part from
lack .,f research data and in part from disagreement with respect to
educational philosophy. Question (2) is fundamentally a question
of values and so of philosophy, but an answer to the question, even
if philosophical, is a better answer if it is based upon some kind of
research 'data. Thus, an adequate answer to question (1) should
make easier the answering of question (2). Likewise, data on the
outcomes of various programs of primary number instruction should
be helpful in answering question (3), and indirectly in answering
PURPOSE OF THIS MONOGRAPH
It is the purpose of this monograph to supply data with which
to help answer the crucial questions just posed. The question as to
the "readiness" of the primary pupil for number instruction is partly
answerable from evidence as to the number knowledge and skill with
which he enters school. Several investigations have already been
made in this area. These will be summarized, and to their data will
be added the results of an original investigation not previously re-
ported. The question as to the best system of instruction (it being
assumed that instruction is to be given) is answerable in the light
of evidence of the effects of various programs. A few research re-
ports contain information on different approaches to instruction, and
thii inf-..niation will be summarized and supplemented by new data
ti.-. hIii the results of a particular program of instruction. If it can
he dimi_-nn.trated that children actually do profit from number in-
struction frim the start, then further evidence is provided for an
arfirimtiave answer for question (1) as well.
Chlapttr 11 will canvass previous investigations and report a new
.tud\ :-,f the number knowledge and ability of children when they
cnier Grade 1. Chapter III will present data on a new study, de-
s.,i1ed to ;show.v the results of one particular program of instruction
in Gradrs I and II. Chapter IV will summarize other research on the
10 .-1, ti ii tit ,t 'n G ,,ai 's I Jiid II
effects b,-tih -f pj-tplo.nii.. 'nd f iniiatinl' in-i-truc:ti., in the pr1-
mar, -.:ra,]-.-. Chapter \ ciinta-iii a hinal iiterpretatiii-n -:. rcc-earch
findin.g- and ,.tc fi-rtil the v riter'- -c.:ce.[.ton -i arithmetic for hie
N.:.t the I l.a-t tl-eI l Ife l t cJirt 1 f the. ,in.n-.'raph -bI uld 1.,i thc bib-
liogr'h li if -ixt\ reachh tiUil-r in thi. i.I] f prinimeri aritliiK-etic.
with whi,.h the Ti-.n.Igraph :nd]- TI-hr...ul-.ut the ni:.ri-,cgriaph cita-
tions t... this Irbli.:.grajh. Ll bvy number --.nl\. Facing, [lihe firti
page .:,f the bil'l.,ralph\ i. a cla'il'nIcat.i.ni -I r-:-earch -tuJie- accoTr.d-
ing t,:. I pi..1 le- vith 01 hi: thC del T'1lii' cla"-ificati'-.n -h-ul
be of -cr.icc t, the crcful stu- eint .-. arithm .-tic i.:. ', ihe .S L:'
check tlih v, riter'- re. ie o. ain i trip-rtaiiil.n- b. c:ntiulrine 1Ihe
origl.nal -'iuirc:, .
C HA.PF'TR II
ARITHMETIC IN THE POSSESSION OF
This chapter i~ de'.ot<:d tE:. a critical review of both new and pre-
vi'tU-lY reportE- r :-,:arh *:.n Ile arithmetical skills and knowledge of
children judt entering Grade I.' flie treatment will be topical. That
ii to say. All rn:sarchl which relatc- to the skill, item of knowledge, or
other category in question will be brought together and discussed to-
gether. The list of topics to be considered in this chapter follows:
1. Rote Counting (p. 14)
2. Enumeration (p. 18)
3. Identification (p. 20)
4. Reproduction (p. 24)
5. Crude Quantitative Comparison (p. 28)
6. Exact Quantitative Comparison, or Matching (p. 33)
7. Number Combinations or Facts, and Their Use in Problems (p. 34)
8. Fractions (p. 44)
9. Ordinals (p. 47)
10. Reading and Writing Numbers (p. 48)
11. Recognition of Geometric Forms (p. 49)
12. Time, U. S. Money, and Measures (p. 50)
13. Sex Differences (p. 51)
14. Differences between City and Rural Children (p. 52)
15. Differences in Levels of Intelligence (p. 53)
16. Effect of Kindergarten Instruction (p. 54)
17. Miscellaneous Studies (p. 55)
The chapter is divided into three parts. Topics 1-7 are treated
as a group in Part I, because new data relating to these topics are
to be reported here for the first time. Other research dealing with
thtt-n same1r topics will be summarized in Part I also, but the separate
:.'si-derati':n of this group of seven topics assures some unity to
th,- .r.ngiidl investigation. The rest of the topics listed above are
*:.nsiiired in Part II. Part III consists in an appraisal of the re-
.carch in, this general area, the purpose being to point out various
limnatii.n- ,.n research techniques which have been used, to stress the
:c-,n<|qu-nt necessity for caution in interpreting the results of the re-
Th,. hhlilhed research available before 1932 has been elsewhere sum-
mnar'-.:d ht. T. C.. Foran in "Early Development of Number Ideas," Catholic
EJd...i..,i' i. ,::, ew, XXX (Dec., 1932), 598-609.
12 .4rithn.,'t" inl Giades I and II
search, and to direct notice to a rect: of number knowledge upon
which re.-tarch i- needed.
Not all the available rt-earch data ire canma:.-ed in Part-s I and
II. i 1 Inpubli.lied material il not includ.led. partly becau.-e uch
report arc n,,t acce-::ible to thle average reader. mi-.re becatul- an
exhau-itl. c iinv.'tntorry of thtee reports ,wa: impracticable (2 All
forevln -itdieg. except Briti -h. are onmiitted. Miost :f theie to.:i are
usel.-:, t.: the average reader. .irnct th,-\ cannot be Cibtained, or rtid
if obtained. M:lrce.:vr-anIlI lh at more imiportant-thle ignili-
canc o:t ;(iuich :tudie f',r Amen-rican education i aribibgu.-,i: Di.:-
similar environnienital circum-tance, alnI-,t certainly intrm-Atic fac:t.-r
which make for lar'e difft.:r,.n-r. in number kini, ledge and kill. F-,or
these r.a'-i-,r. een ,sulch v. ell-lkn,:,ri :tudite a- thi-'-e '-f D)ecr.:,lv and
Degand- and. r.i Decrtiidres' ar',- not here ci.n.-ilererd. I 3 Studies
made 'Aith fce ub jctz. a. in the caae of Caldv. in and Stecher4 :or
made under conditions insufficieintl. described ti:O juLtil -, evalnatinl.
as in the ca -,f Hall," are like'. i-e Oi.itted. i4 i It s possible ti::.
that a few v-itiihcant 1 ri ettigat ion,_r: r.rinted in periodicals nIt co.:-crtd
by the Edunati,'n indc.i or or by biblio:,graphice: like tho- e in the Revi.ew
of Ei/ih'ati,',.nl Rt'si'cil ma.' ha1v: escaped ni:tice and 1o are no ,t here
P'.FT I r.pRIGlIAL I N\'E. TiGAlI, 0l'., V. I I[ PlL.\ri:E PESEARCHI"
i Topic-, 1-7: -ee p. 11
Ne, data .:ii the "readiness" :4f 1iir:t-zgrade children for arithminetic
were collected in the fall of 19.I. and the fall :-f 193" from) tv.enty-
four :chi:il: I thirt3-tv.i- cla::' I in f'oiir -tates. The irnstrunLint uIti.1
for thi- t,':-:tin, %%a thet crit iof prete'.tz -.lich are rublilced in con-
nection v. ith the "J.lly NunibLr-" prin.ary material. -,f the Daily-
Life .4riillhiiit' ';
Th':-c pret,'. conlih-t in la, a rrou.ip of -Lub-tet l.azed uponl
picture: and admini:tered, to the cla; a v. .l' hlc, i b i a '.econd -ierles
of sub-test.- intended t-, be gl.'en to. mnallkr. t.le.:tdl grolup- of chil-
[ Dr. 1M I Dce,.:.:,. and I M lle. Julia Deca ni.l. lb,, r jati.:.kn rdlall i. l' I .,-
lutioni ijI: n.:,ti...n ItI quanntil,', o:'ntinLut t d li-i:-niortlu -, '.-h z l'Iin 't," A.r-
chivi .1,- P : '.- ;.,.,,],. X II M .., 1,12,1. 1-121.
S I.M.,: D :-c.:tu.Jrer. L, '.:,l f.'i',': d, I E fr...t -ari s: Drl.-chauz el
N iesile. S A., r.. .' i (1.. 1-2 14.
F;. T. P.al. in an.d L ..rl: Stecher, Ti.,. F c. ,I.. '. ..f li r'r,sc t. 'o1 Chil.
(Ne- .:.r D. .\rr.lrto:,n and C.:... 10 4 r'r'. I 2- 0.
'1. Stanlr. Hall. "_,..-nt.nt. ..I ,.hildIJtn'i Mirndj .:.n Entri ing Sch.:.9l."
PJ J,. 'r i.,(, l .'.- IuI. ,. I I 118 I,1'. 13'I -173
"Gui. T. f.i l. \\'ilhams A E.i.:.. n. 11. n,: Len.:.i: l,:.hrn. pF,a ,. Lit',
.-r,'ll,, .,'cts i E' .:.n : inrt and C '.,, Irj..''l
.Aithnil'tic in the Pkssession of School Beginners 13
*lr.n who pre-io:.usl% haI e had suib-test (a), and (c) a third series of
sub-ter-ts t.., be given inlividually to a still smaller sample of pupils.
In this particular instance. Io,.etver, sub-tests (b) and (c) were
administered in their et-ilret \ t .ll pupils of a class or to as many
puLils :,F a class as circumcaitactcs permitted. In the latter case direc-
ti.'ii.. er t elec t.. the ic rst. fourth, seventh pupil, etc., on the
teach..r's alphabetical roll. or the first, third, fifth pupil, etc.-in other
words. t,: give tili tests s,-, as ti.- insure a representative sample of
the total enlIilneint. In all. 365 rural and 327 city pupils (see Table
I i t::ok sub-test (a) arid all o:r a usable part of the other sub-tests.
The teLtin.; was regularly dune within the first two months of the
term, in Grade I only, and with pupils who had just entered school
for the first time. (Repeaters were thus eliminated.) In no class
had any number instruction been offered.
GRADE I PUPILS TESTED
State Number of Pupils by Schools
Florida................. ................... Sanford:
School 16......... 12
North Carolina.......... Durham County: Durham:
School 1......... 30 School 17 ........ 72*
School 2 ........ 40 School 18 ........ 62*
School 3......... 25 Raleigh:
School 4......... 35 School 19 ........ 28
School 20......... 47
Pennsylvania............ North Central area: Williamsport:
School 5......... 52* School 21......... 25
School 6......... 25
School 7......... 22*
School 8 ........ 9
School 9 ........ 23*
School 10......... 10
School 11......... 33
'*rp_-ri: ....... Vicinity of Petersburg: Petersburg:
School 12 ........ 7 School 22 ........ 14
School 13 ........ 13 School 23 ........ 36
School 14......... 12 School 24......... 31
School 15......... 29
.- ,':' t 365 327
I' ,ch..l. rr,..-Le.1 thus supplied pupils to the testing both in 1938 and in 1939.
r Thi-. r...ri include all pupils whose records, even when incomplete, were used in the
,ih.u ,.'..r, ".1 ri. chapter. The tables reporting data from the individual tests show totals
*.oller thr. the ri.gures above, for the reason that not all pupils in every class were given
[I,,2 ltiil. ,J l h.: lC .
. lr:ll:h ct.' in Gmc',h I 71and II
The testeis rere graduate students in thie writer's c...urs., Inves-
tigation- in Arithmetic. in the case f -choolls 1-4. 12-15. 17 and S1
(for the mon.t part ,. 19 and 20. and 22-24. In the .:'thler ,ch:,.-,l
(5-11, 16. 17 and 1 [itn part]. and 21 i the tz-ter va' Mi_- Hulda
Kilmer.7 a teach:ir in the )us-hore. I-'etnnl\ vania. sch'l'..Il. *.r the
classrcomi teacher h:,i.i ai-.qwtance M[ii; Kilmer hadl .rcurtrd In tie
latter can'on great care "as taken to halia thin: tea.lier,- tunidertand]
and follk," the tet 'dircm:tiot:ins.
Th ret uilt obtained fror. thi testing are gi vn behloe acc.'rdiring
to the o:,rder of the topics as iiuthilined :.)n pagc 1. Thi ordtr .t topics
differs fr.min the ordkr of tle 1ul.-te-_ts theni-ilie:, Lut is hI:r<, adolt,.d
as making for greater c:larit 'ind com:mn\:;ience to the reader. F.:or
each topic there \. il be in a daescrilti.'ni of t tue retin- instrumiient,.
the directrm'n f':or adlmirniternl- the t]l.m-tm:7t. and theil quantitati\'
data secured. 'lie-i data \'ill then be ominpar:d \tinh tie cltr-'re-,pnd-
ing findin.'i -of others \iho: havin: mvestieated tie
way it ill be [Lpossible to. synth'si.'ze the new and tih i.ilder inf:, rmm -
tion willi les[ect to: each .:.f thlie eken arithnietical ideas and skils-
consider,d in thii part ..f the chapt,.r.
/. RoW L,*';tin',
The temt of ability in r ite crouniltinit; .as of: course given indi-
vidual: Thel child \as tohi. "I \ int tio st, h'o." \.ell \.:u can count.
Will yoi c.:.unt for me ?" If tle child as ti id .:.r v a- unc:rtaran A:
the nature .-,f the ta-k. he .*a- tincoruramgeil il"n] the rask was illun-
trated itn some -ilrhi falmi]il a "1 knoV, \otu .ani ..tLtiti. N'. li'tcn
to me: rnu:. t:, i tIhr Cai Ui It amount fr.iin m other: ?'" All children
were stopped u l.n tih,., reachdied 20. Chiklren i ho did no:t tr\ toi
count that far o.r ho imae ina mi ktak :. l.eir.re that point ii, the numbi.r
series uere credited itrk hatin.i cor.unted to: thle laot correctly stated
number. For practically all the children one trial nas eiouigh. \\'Vhen.
however, the examiner had raacin to, h:h.ie that a child had erred
accide:ntall. hie a',s i\e a second trial.
Thime rit half .-4 Table 2 suiimmari..ets the iii.hnit-. It \\ill be i.:.ted
that 47 .s per :,It of thi, rural .:hilrven and 57 4 i.ir cent of tiht cit,'
children counted t'.: 20 and that 52 3 ipr i.ent of the 6.31 chi
the tctal grotii attirnedl thi- inme ioinit. nl,' alout l teinthl
stopped short r.f 10
M i-- Kinlm.r hal i. rc m 'u- .- 1 i h. at..','. .:-mi nti.,-ni ] m..ur it. h tl,
writer, and c..ll.:ct,:-l ;r .la'a ...r I r ,\ M. th.-i+ a\h h t imnp iubhl h.d, i a
file in t1lie Di[ L Uk L' i., .--r,t: I ii-rar,.
Pl-L- .;'-T .N.'Ti_.,i'. ['E-ULT: ,-,r i .- T L.4T- .-F RI-,L. C..,L',.TiN,:. .N i, EN., .I ..N
[f',r, ,' ss Pitf.ls
.4 ;'lu.v i In'1, ra L f'l% Toratl
I .V = 2.';- I f --- 2 r ,' (31 J e
R.-ic (. ifluniiii b., 1'4 ,
2IiJ ... . . 52
I -! . . . 57 1 r,7 2 Wl.
10-14 .. 84 Q4.o .5
I . 111.i 1 11).1 II
Enim ra ir :,r .
10 objects ................ 83.3 88.5 85.7
9 objects ................ 87.5 91.2 89.2
8 objects ................ 90.4 93.2 91.7
7 objects............. ... 91.3 93.9 92.5
6 objects... ......... 93.3 96.0 94.6
5 objects, or fewer....... 100.0 100.0 100.0
The figures obtained in the present study are to be compared
with those of other investigators, as summarized in Table 3. The
first three of these studies have been reported by Buckingham and
MacLatchy (reference 12 in the research bibliography at the end of
the monograph). For two of these studies Buckingham and Mac-
Latchy are themselves directly responsible. One of their studies,
which will hereinafter be referred to as their "major" or "main"
study, involved a total of 1,356 school entrants in one Texas and
sixteen Ohio communities. Their second study, herein designated
the "Cincinnati" study, exactly paralleled their main study but was
restricted to Cincinnati schools, where 1,067 children, about two
thirds of whom had previously attended kindergarten, were inter-
viewed. (In spite of the difference in earlier educational experience
among the Cincinnati children, they will all be referred to here as
school entrants.) The third study reported by Buckingham and
MacLatchy was conducted independently in Cleveland by O'Connor,
whose subjects were 1,242 kindergarten children, only 313 of whom
were however tested on rote counting. O'Connor's study will be
named the "Cleveland" study for the purposes of this report. Ac-
c-.:.ring t:. Table 3, the figures for the Cleveland subjects run high,
pir,:.IahIl because of the influence of some selective factor, or because
..f -,ni:mc -.,ieal instruction not enjoyed by other subjects, or because
rfi I..th -4f th: factors mentioned.
Data for the Woody study (56, 58; see the research bibliography
at thld end ol the monograph) were gathered from 2,895 children in
.4qrii th'ciC in G'(JdS i / Cid Hi
S1U.MMI P\ ...r Fi -t.: .:,. ,.iiH RifE,_r T.I R u.TE COIN'T[',G B" I'.
Inv'.;t uat. Sit.,cs -'er cit.'s il,' t/ C. I Throitugh
li 2,.' .i.# 11.10
Mac Latch% .
M in Stud .......... 1.2"?nl ch.:.-ol .'i0 .I0 2_ 1)
C nriurnaii Study..... I .I.iI; ;cl.:,,:l 8, 10)
Cleveland 5tud,*. 3. I3 I ,r,, r- *') 7r, IS
W ood. t ........... 2..' :. pupil; in 2.. 9S. U.. ,(., '.'4
erouri;. I- nlI -
Present tud, .. . .631 chi(,.:.l 'i 52
*O' C-...,,. r i .cure- r -ur L J. f. r c..r.-l, ;n- r, ri.J L M ,. L., ,.12
t T .e I.,ial r.jri..-r cbi jr.r. iri.I ,eI-. .r.J t6 k.r .irc, r r rn I "-4. I'. Gra.-J IB.
607; in Gr.J]- 1.; I in i.rad, lIB. I. ard -.r,i.J, !A. : Tlh r- 1,. c r *..i.-
reported itb... e I tht i.tI-I,:..lu ri c.:.r.r .p ; i I... ibc rc..rdJ! i..i.Je I.. Ihe-- .-r.: i..
five half-4-rade kgrup< iln th rt- -uine ystcrns., tthe..: Ihial f-'rade '.r,',tlp.b;
being. kindergarten, I, I ht-ginninig bali .:( Grade I i. IA. IiB, and
IIA. In 3acI:h instance cliildr,-ii '.i. rev te-t in the half-grade j.it_ prior
to that in \which '.le matic inctruicti,:,n in aithinietic v.-as bc.Lun. In
so far -a \V\'.,:d\ la publi.-ied data for tli- Ile groul-, l they v. ill hze
included in thie su.i inmary talle., of lhli m,:n,,rraph
Four c.ti Ji ontai',ai data for cruLtm1i. b I's. The two made by
Buckinigham and MacL-at:ch i acrere in slh:n.'i that abuitit onei pupil
in ten has d~vl:--':ped tlii- ability at leas.t tc, IlI a; a hmin;t. \, lih.:.ut
specific. schol:i instruction. bi tlic time hie entertcr Grade I \\.e,-dy's
figure i; much hii,;ler. 3S per cent, or I.etter than :,ne ptupil in three.
The reason for this large di'.crepanc. I; pro:blimatic. but it is not
improbhalle that mtany of \V\-,,d .'s Grade IB _.ubjecit had re:cit:ed
instruction in tii-h ability in conercti,-*n v. ith kindergarten schlolitig.
For countiitg 1\ I's tc. 10, 20. and 50 t[l-e data from;i the ctuidirs
which are comparable .with respect to, subijeci and te.,tini procedures
are in fairly uubaiitantial a4r,mL.iinlnt. It ,eI;l, aL.a i t1 conclude- that
nearly all children upontit entering school -are in coinim and of il h nmiini-
ber series. to 10 and that bet e.i.it half and t,\o thirds know the scrics
to 20. For the: purpoem, of instl action, thlie initicanc>- of tlhi ability
"An A ariv tu..!v, that b '. Yocum 1 591. found rhai 2i0 r.tr cant I.f fiity bo.,
and filt\ %irls just ente.rine Grad.l. I could coini 1t., 21i adala irom Buclingham
and Mac:Latch, I).
.-rithmintic in ilit Pcsscssion of School Beginners 17
in rote counting may be either exaggerated or too greatly minimized.
The chili. who can count to: 20. for example, (a) knows the number
names anI ib) knows them in their correct order. Having this
knowi-ledge. he ic has the basis for understanding the relative size
,of numberr. Thus. in kn-.':in- that 8 comes after 7 and before 9
he has the basis for Iknowing that 8 is more than 7 and less than 9.
Furthermore, when this ability to repeat the number sequence is
coupled with the ability to apply the number names to groups of
objects, he (d) has a means of satisfying by the very immature pro-
cedure of enumeration most of the practical number needs which he
will encounter in and out of school.
At the same time, it is to be noted that rote counting in itself
does not make (c) and (d) possible. On the contrary, in both in-
stances rote counting must be supplemented by other kinds of learn-
ing. As a matter of fact, rote counting guarantees only (a) and (b)
and may actually be quite devoid of quantitative meaning. The truth
of this statement is revealed in the child's inability to put the number
terms he knows to any useful service. He may "count" a group of
six objects and find "eight," or "three," or "sixteen"; whatever
number he assigns the group, he is entirely complacent about its
accuracy. For such a child "five" is merely the word to say after
saying "four" and before saying "six"; for him "five" possesses no
more mathematical meaning than does "e" in the alphabetical series
"a, b, c, d, e, f."
Thus far ability in rote counting by l's only has been discussed.
Some attempts have been made to ascertain how well school entrants
can count by other units, as by 10's. Buckingham and MacLatchy
(12) report that (a) about 50 per cent of their pupils could not
count by 10's at all, even though they were helped by the examiner's
starting them with "Ten, twenty, thirty"; (b) about a fourth (28
per cent) could count by 10's to 40, and (c) another fourth (24 per
cent) could count in this way to 100. The figures for the separate
Cincinnati study are extraordinarily alike, being 52 per cent for (a),
29 per cent for (b), and 22 per cent for (c).
In \\Woo::'dy's investigation (58) about the same proportion of
children could count to 100 by 10's as by 1's. For his five half-grade
groups kindergarten through IIA) the per cents who could reach
100 v.ere .32, 46, 69, 76, and 93. Woody's second group, composed
of children in Grade IB, were most like those in the main Bucking-
ham and MacLatchy study and in their Cincinnati study; yet
\\':.d. reports a percentage figure of 46 for his subjects, and the
Arithmetic in Gratids / and II
latter two investigations, a percentage figure :f aliout 28 .A-s men-
tioned before, the most reasonable explanation for the diffterelce i.
the probability of specialized training ;i man\ :if the children in
Woody's groups, though there is no proof that thi_ hI) 'pcthesi i; valid.
As for other forms of rote counting. \\'rod, i 56 i reports 34 per
cent of his 1,897 Grade IA pupils as able to county Iy 2's t,: 30, 9
per cent as able to count by 3's to 311, and 12 per cent a; able to
count backward from 20 by 2's.
The second half of Table 2 (p. 15 1 summaarize, the results of the
test in enumeration. The differences a- but\.teen rural and cita chil-
dren are slight and probably of no educational significance. Accord-
ing to the last column, nearly seven eighth.i of the children could
enumerate ten objects and about 92 rer cent could cInumie-rate at
least eight. Their ability in enumeration therefore a; but slightly
less than their ability in rote counting, by I's. a finding w Ihich conrirl:
that reported by Buckingham and MacLatchv
In this study the testing procedure "\as as follo-,v. the examiner
scatters ten small objects (pegs, paper clip,. penlie i on, the table
top in front of each child and says, "Here are some pegs. I want you
to tell me how many you have." The child is required to lay his
finger or hand on each object as he tells it off, thus to show that he
actually has established a one-to-one correspondence between the
series of language terms and the objects in the group. This procedure
has been rather generally used in the investigations of others, th,,ough
the objects used and the numbers of objects have differed. Tihus,
Buckingham and MacLatchy used twenty instead of ten object-;
Woody presented a row of twenty-one circles in a printed booklet:
and in the Cleveland study made by O'Connor and reported b3 Buck-
ingham and MacLatchy the children were given two trials in pushing
twenty tacks into a board.
It is difficult to assess the influence of variations such a- those
just described, but internal inconsistencies in bodies of data from the
same study and discrepancies as between 'different studies probail'
mean that these and other variations in testing procedure introduce
factors which, however negligible they appear to adults, cause marked
differences in the responses of children.
Comparative data on enumeration from other investigation are
summarized in Table 4. The Cleveland and the Woody data are out
of line with the data from the other three studies. In the case of
.- l,' linii,'l in the i '. ,.;,',., i oi .cil.i, I I.,l i i'rs 19
T .E:LE 4
S uM i.kY OF Fiti iN ,' IT H ER '-s iF-T Iu- [F I_. '. IEI I-TN
ii'': c lJ.7al.', S ul'ict/.< P.** iC /, ta E toiiiu rate Through
S 10 15 20
Main Stud% .. 1 222 i,.:..-l )7 C43 "J' .0 58
Cle.eland Stud3' ... 1,242 kinder- 2r, ijutb trial).
garteners 22 (more, on one
W oody................. 2,895 pupils, .. .. .. .. 71,79,93,
kindergarten 98, 97t
Present Study.......... 631 school 99 93 89
O'Connor's figures reported by Buckingham and MacLatchy (12).
t Figures are for subjects as follows: kindergarten (94), IB (607), IA (1,877), IIB
(80), and IIA (207). (Woody, 58)
the Cleveland study the discrepancy is explained partly by the fact
that kindergarteners served as subjects, but probably much more by
differences in the procedure for testing enumeration. These children
were required not alone to select the correct number of tacks (20)
but also to stick them into a board. Clearly, lack of success may be
as reasonably attributed to distraction or to difficulties in fixing the
tacks in the board as to difficulties of a purely quantitative character.
The high per cents in the Woody study may be explained by the
hypothesis suggested twice before (p. 18), namely, that his subjects
actually benefited by planned kindergarten experiences before the
supposed start of systematic instruction in Grade I or II.
If conclusions are 'drawn only from the data for school entrants
it appears that nine out of ten children may be expected at the start
of their schooling to be able to enumerate ten objects and that seven
:i thLe ten will be able to enumerate successfully fifteen objects and
;ix of the ten, twenty objects.9 This conclusion is of considerable
SThe 19Rl, Stanford-Binet placed the enumeration of four pennies at age
iour Enumeration to 4 appears in the new (1938) test at age five, where
children mnii successfully enumerate objects in two of three trials, the objects
vanring from trial to trial. In the validation of this item it was found that 40
per cent e:ri: successful at age four, 68 per cent at age five, and 91 per cent
at acLe ;ix. The 1916 item calling for enumeration of thirteen pennies and
JplJced then at age six has been dropped in the revision.
Yviinmi 1 59) in 1901 reported that 30 per cent of his fifty boys and 38 per
20 Arithmetic .a GC'idcs I .Ind I1
practical significance. Unlike the ability\ tI:, i.outni bl, rote t:, ID0. tlte
ability to enumerate ten object- correctly is p hi-ible ,-nl 1.i childrenn
whose number concepts have really. b1gun i.i take -on c-:nientc. That
is to say, such children can use the nuimnbir nanic in functinLal quarn-
titative situations. Ability to enutmeratc t., 10 or .'mec other sich
point does not warrant the infeience iliat children have iell-devel-
oped number concepts, but it d'-cs \\arranr the iwiferentcl that theil
number concepts are in process ..f quantitative development.
3. Ide 'nl;itC ,iU'-
Closely related to ability in enurmeration is the ability to identify
or name the number of object- itn groups of vari -tis 'ize'. In the
present study this ability was tc-stc ] ith cla a -e- a- .h,,l- ,I mnieans
of pictured groups (see Part -\, p. 21) Inctruction- were Vo "put
a mark on the man with four ball:on : on Mary's birthday\ tale itli
seven candles; on the pot with ten Nv'.rs." A -imilar prxedure
was employed by Grant (15), -.h roi 'ade use .:.f Te.t 5 .f llth- Metro-
politan Readiness Tests for this purpose. His suibjcct halal to clect:r
from among four choices the b..x that ciintained three d:ts. six dori.
and eight 'dots. In both of these -tudic-, therefore, something miii.rc
than mere enumeration was in\ole,.I; the children tc-ted were pre-
sented with a variety of number rep.ir-entratii., i and had to -elect thl-
Quite a different procedure v..ai employed L Buikinliham and
MacLatchy (12). In their two inv- .tiatons (in Cincimnati and else-
where) the examiner threw fi t, ix. s-even. ei-Jht, and ten ,lljec'.
in random patterns on the table t-p before the child and asked. "How
many pegs (or what not) ar,- here?'" In their main trndv. three
trials were given and data are rc-port i: for successes on ion trial
only, on two trials, and on three trial'. MAL. they translated suc-
cesses into a "mean index of cifficultc" for the different numbner- in
such a way as to make the resulting index fi' ure.' represent the per
cents of success as compared v ith the total numnier of i-,l.pirtuioties.
In their Cincinnati study evidently but onec trial ,as given. It might
be argued that both of these studied art I'ettcr cla.sicfid under enu-
meration than under identification, hut thev are inclindi.l under the
latter, in part because they are so, cias-ified bh the ail-irs and in
part because of discrepancies between the perceniraute hures for the-e
tasks and those for enumeration I Tallc 4'i.
cent of his fifty girls on entering sclh,.., coubl c:.unt more than tr.cniy i..biecis
(data from Buckingham and MacLat:ch I
Arithmetic in the Possession of School Beginners
Name -----------------------------------..... School--......----------------- Date ...-----..
Prt A P.rt B
Woody (56) presented his subjects with three sets of grouped
objects, the sets being made up of domino figures, of groups of dots,
and of groups of lines. Each set consisted of five groups. The subject
was first instructed to point to the particular domino among the five
dominoes given which contained five dots; then the one which con-
22 Arithmetic in Grades I and II
trained four, etc., etc., until he had used four of the five dominoes in
that set. The subject then proceeded similarly with the set of dot
figures, and then with the set of line figures, in each case identifying
the number in four of the five groups. Three other items in the
Woody test required the subject to divide a line of stars into groups
of three, another line of stars into groups of five, and a third, into
groups of four. The per cents of accuracy for 1,897 IA pupils were
90, 79, and 76, respectively. Obviously, this last task differs con-
siderably from that in the other studies in this section, and for this
reason the data are not included in the summary table or statements.
Table 5 contains the detailed summary of findings from the
present study. Again the results of rural and city children are about
the same, the rural children being slightly superior (about 5 per cent)
on the numbers 4 and 7 and slightly inferior (4.3 per cent) on the
number 10. About 6 per cent more of the city children were suc-
cessful on all three numbers than is true in the case of the rural
children. As to the relative difficulty of the three numbers 4, 7, and
10 for the purposes of identification, the data are anomalous: the per
cents of children who identified 4 and 10 were very much alike,
but the per cents who identified 7 were, both for city and for rural
children, less than the per cents for 10. On the assumption that the
figures for 4 and 10 are correct, and that more than three fourths of
school beginners can identify 10 as well as 4, then the figures for 7
are difficult to account for. On the assumption that the per cents for
4 and 7 are correct and that numbers increase in difficulty of identi-
fication as they increase in size, then the per cent for 10 is too high.
A tentative explanation is as follows: In the third picture the children
were instructed to mark the pot which contains ten flowers; they
knew that 10 is a large number, and the pot showing ten flowers is
rather obviously better supplied with flowers than the other two pic-
PRESENT STUDY: RESULTS ON GROUP TEST OF IDENTIFICATION-
FOUR, SEVEN, AND TEN OBJECTS
Per cents Identifying 0, 1, Per cents Identifying Spe-
Type of Pupil 2, or 3 Numbers Correctly cific Numbers Correctly
0 1 2 3 4 7 10
Rural (365).....6.3 12.1 31.2 50.3 86.7 72.6 76.4
City (327) ......5.5 14.9 22.9 56.5 80.7 69.1 80.7
Total (692)... 5.9 13.4 27.3 53.3 83.8 70.9 78.5
Arithmetic in the Possession of School Beginners 23
tured, which contain but six and eight flowers respectively. That is
to say, the decision in favor of the 10-pot may well have been made
in terms of gross comparison rather than in terms of actual and
exact enumeration and comparison. That children do so discriminate
between groups of objects has been amply demonstrated by Rus-
Comparative data on identification (in the different meanings of
this term as already described) have been previously reported by
Buckingham and MacLatchy (12) (their main and their Cincinnati
studies), by Woody (56), and by Grant (15). The relevant facts
are presented in Table 6. If one takes the median figures reported
for each number (and uses only the Grant per cents for children of
average intelligence), the numbers are found to have the following
approximate per cents of success:
3 4 5 6 7 8 9 10
(90) (90) (80) (66) (71) (68) (75) (70)
The anomaly found in the present study in the case of 7 an'd 10
is to be noted in the Grant study in the case of 6 and 8. If Grant's
SUMMARY OF FINDINGS WITH RESPECT TO IDENTIFICATION
Investigation Subjects Per cents Able to Identify
3 4 5 6 7 8 9 10
Main Study*........ 1,356 school .. .. 72 63 60 58 .. 56
entrants .. .. 82 75 74 72 .. 70
63 52 46 45 .. 42
Cincinnati Study .... 1,123 school .. .. 80 69 63 63 .. 60
Woodyt................ 1,897 IA 80 .. 83 81 .. 79
pupils 99 96 92 95
91 .. 86 .. 75 80
Grantt.................. 563 school 54 .. .. 19 .. 54
entrants 82 .. .. 43 .. 73
96 .. 61 .. 86
Present Study........... 692 school .. 84 .. .. 71 .. .. 79
First row of figures is in terms of "mean index of difficulty"; second row shows
per cents successful in at least one trial; third row shows per cents successful in all three
t Data are presented for Grade IA only. First row is for responses to domino pat-
tern; second row, to groups of dots; third row, to group of lines.
$ First row for 145 pupils with IQs below 90; second row for 252 pupils with IQs
between 90 and 109; third row for 166 pupils with IQs of 110 and higher.
Arithmetic, i Jd,-:: i and II
per cents for 3 and 6 are cor;ccl, liat i:.r 8 -eeni l:. blie ,.::, lwbh:
if the per cents for 3 and 8 arc c. rrect. hat fir (.1 .eemi tIi hIe much
too low. An examination of the pictured gr:oupt, sLlI[.lI: n-.' explana-
tion to favor either hypothesi All gn-.up', ,-f doti. from three t.'-
nine, are distributed over squares .,f the anime ize half-nclh'i. The
number 8 must be identified frim p-i-itures co',mainingi four. ninety. ix.
and eight dots; the number 6. fr'..m piiture- c:,ntramininr three. -ix.
four, and five dots.
Whatever be the true explanation for the twoi Cturi.k- iIncim!isten-
cies noted, the data on identification as a whole seem to warrant the
conclusion that six out of ten children on entering Grade I can iden-
tify or name the numbers to 10 when these numbers are represented
concretely and without regularity of pattern. This figure is to be
compared with that for enumeration, in which nine out of ten chil-
dren were able to enumerate ten objects correctly. The difference
in relative success in enumeration and identification may be ascribed
to the extra task in identification as here tested, namely, to the neces-
sity of selecting the correct number from among several others pres-
ent at the time.
This conclusion and this explanation are by no means certain.
All measures for enumeration were secured from individual chil-
dren. The data on identification, however, were obtained, in the
present study and in the Grant study, by means of group tests. Un-
familiarity with the test situation on the part of school beginners
almost certainly would have made for an excessive number of mis-
takes. On the other hand, another factor in this testing would have
had the opposite effect. In both of the studies named the subject
made his selection from among three or four pictures, and he could
therefore have scored some unearned successes through the operation
of chance. How much one of these two factors-test unsophistication
and chance-compensated for the other it is impossible to say.
Identification is the activity by which one answers the qi-.ticunn.
"How many apples have I?" Reproduction, on the other hand. it
the activity in which one engages to comply with the request. "Gtue
me five apples." In the case of identification a group of objects is
given and their number must be found; in the case of reprI.dIlucti.:in
the number is given or announced and the corresponding _r'up 4-.f
objects must be found. The mental processes required in the r\c,
.- ithmetic in the Possession of School Beginners 25
number feat- are markedly different (though both obviously employ
enumeration), and so both need to be tested.
In the present study reproduction was tested for classes as wholes
by means of three pictures (Part B, p. 21). The directions were:
1. This boy wants to play marbles. Draw five marbles for him.
2. Look at the umbrellas. They have no handles. Put handles on six
3. Do you see the rabbits? They have no tails. Put tails on nine of
A single trial was given with each picture, and the numbers for
which ability to reproduce was tested were 5, 6, and 9. The results
of the testing are summarized in Table 7.
PRESENT STUDIO: RESULTS ON GROUP TEST OF NUMBER
REPRODUCTIOX-5, 6, AND 9
Per cents Reproducing 0, 1, Per cents Reproducing
Type of Pupil 2, or 3 Numbers Correctly Specific Numbers Correctly
0 1 2 3 5 6 9
Rural (365).... 17.8 20.8 27.1 34.2 73.7 55.1 49.0
City (327) ..... 5.2 22.9 25.9 45.9 86.9 66,7 59.0
Total (692).. 11.8 21.8 26.6 39.7 79.9 60.5 53.8
Nearly half (45.9 per cent) of the city children, compared with
about a third (34.2 per cent) of the rural children, were successful
with all three numbers; and only 5.2 per cent of the city children,
compared with 17.8 per cent of the rural children, failed on all num-
bers. So far as the separate numbers 5, 6, and 9 are concerned, diffi-
culty in reproduction is seen to have increased with the size of the
groups to be reproduced.
There are at least four other related investigations. Buckingham
and MacLatchy (12) report data for the numbers 5, 6, 7, 8, and 10.
In their study the examiner began with 5 and worked upward if the
b-hild was successful, and downward if he was unsuccessful. Three
trials were given, and their data are here reproduced for those sub-
jects who were successful in one trial only, in two trials, and in all
three trials. In addition, their figures for "mean index of difficulty"
i see p. 20) are quoted.
26 Arithmetic in, Grades I and II
SUMMARY OF FINDINGS WITH RESPECT TO NUMBER REPRODUCTION
Investigation Subjects Per cents Successful in Repro.ie: m
4 5 6 7 8 I'*' 1
Main Study*........1,355 school .. 74 67 66 63 t,3
Cincinnati...........1,123 school .. 83 73 72 68 t.7
Grantt.................. 513 school 57 .. 39 .. I'
entrants 64 .. .. 58 .. 4. 0
87 .. .. 83 .. .. ..4
Present Study........... 692 school .. 80 61 .. .. 5-1
Figures are for "mean index of difficulty." Per cents of subjects successful ..
trial only were, for the five members respectively, 14, 15, 16, 17, and 16; per cer, :.
cessful on two trials only were: 7, 10, 11, 12, 10; per cents successful on all thr .. r ir.
were: 64, 56, 54, 49, 50.
t First row of figures is for 145 children with IQs below 90; second row, t'..
children with IQs between 90 and 109; third row, for 116 children with IQs ab...- I1I"
Table 8 summarizes the data from the Buckingham rn J
MacLatchy studies. No facts for O'Connor's study with Cleveland
kindergarten children are given here or in the Buckingham ai'l.
MacLatchy report, for the reason that the methods of testing ai,,
of treating the results are not comparable. Buckingham and Mac-
Latchy's statement may be quoted:
Of 1,242 children at Cleveland, 32 per cent were completely suc-.-:s'- ul
in both trials of a test in which they were to reproduce 5, 7, 9, and 11 b:.
putting the required number of marks in designated spaces on a sheet ..I
paper. According to the marking plan for this test, the highest -l.tan-
able score was 10, and the median of all the scores actually made .. .
This is a very creditable performance.
Grant's procedure (15) with Test 5 of the Metropolitan Rj'' -
ness Tests was much like that in the present study. Picture-4 i
objects were given, with directions to put marks on four olije.:t
(houses) in one picture, on seven in the second, and on thirteen tII
the third. His test results were then tabulated according to the i(-,
of the subjects, and the per cents so obtained are included ii, thi-
summary table (Table 8).
The per cents obtained in the present study and in the i-'rani
study are, with one exception (the number 5, in the present st Ii'., i.
lower than the corresponding per cents obtained in the other t'..
Arithmetic in the Possession of School Beginners 27
studies. If it is assumed that the true abilities of the four different
groups of subjects were actually equivalent, then it follows that the
children tested by group tests were thereby put at a disadvantage and
were unable to show their real abilities. This inference is not im-
probable. In the first place, school entrants would suffer from their
unfamiliarity with the techniques of group testing: they lose their
place, misunderstand directions, etc., thus making errors which
should not arise in the case of individual testing. In the second
place, their scores with the larger numbers would be especially in-
fluenced for the worse by the rather crowded appearance of the
objects pictured. Certainly this is true in the case of the test picture
for 10 (rabbits) in the present study (see p. 21). Likewise it seems
to have been true in the case of the pictures for 9 and 13 in the
Grant testing procedure.
For these reasons the two sets of figures from the Buckingham
and MacLatchy studies should be given special weight in assessing
the ability of school entrants to reproduce numbers. But in this case
which of Buckingham and MacLatchy's four sets of measures should
be adopted? These authors point out that no less than three successes
in as many trials can be accepted as guaranteeing real ability, but
they also stress the danger of disregarding or minimizing the signifi-
cance of smaller 'degrees of success. After all, the child who can
produce five objects in two of three trials is well on his way in the
development of a good reproduction-concept of 5, and even the child
who can correctly produce five objects only once in three trials is
not without some degree of ability.
In the listing below the numbers are arranged in the order from
4 to 13, and with each is given the per cent of successful reproduc-
tion in the same way as had already been done for identification.
Each per cent figure represents the median of the measures avail-
able for each number, only the Buckingham and MacLatchy mean
indices of difficulty being used from their study:
4 5 6 7 8 10 13
(64) (80) (67) (66) (66) (63) (40)
If these figures could be accepted at face value, it would be pos-
sible to conclude that about two thirds of school entrants are able
to reproduce all the numbers to 10 (the Grant figure for 4 is almost
certainly much too low)."' Obviously, however, various limitations
10 The 1938 Stanford-Binet places a reproduction test at age six. On de-
mand children must be able to supply three, nine, five, and seven objects (three
successes required for credit). Percentages for the various numbers are not
28 Arithmetic in Grades I and II
with respect to testing procedures and with respect to the determina-
tion of accurate single indices from the data obtained render this
conclusion debatable; it must be regarded as purely tentative, though
it is the best guess, on the basis of the evidence available.
5. Crude Quantitative Comparison
Concrete objects or pictures.-Under this caption are included
terms like largest, shortest, most, and so on, a total of eleven such
terms by means of which one describes crude differences in size or
amount without attempting to fix exactly the degree of the differ-
ence. In the present study single trials were given in the use of six
of these words or phrases. Three of these same terms appear among
the eleven terms of the Metropolitan Readiness Tests, Test 5, which
was used by Grant (15).
In both of these studies the testing employed pictures. The pic-
ture test of the present study is Part C of page 29. The specific
1. ...... Put a mark on the largest one [cat].
2. ...... Put a mark on the smallest one [doll].
3. ...... Put a mark on the kite with the longest tail.
4. ...... Put a mark on the pan with the shortest handle.
5. ...... Put a mark on the nest that has the most eggs in it.
6. ...... Put a mark on the tree that has the most apples on it.
7. ...... Put a mark on the table that has the smallest number of
cups, or the fewest cups, on it.
As shown in the first half of Table 9, less than half the children
tested knew all the terms well enough to score consistent success.
PRESENT STUDY: RESULTS ON GROUP TEST OF CRUDE COMPARISON-
Type of Pupil Per cents Answering 0-7 Items Correctly
0 1 2 3 4 5 6 7
Rural (365) .............. 0.8 0.8 0.5 2.7 5.8 16.2 30.7 42.5
City (327)............... 0.0 0.0 0.6 2.1 4.6 14.1 32.4 46.2
Total (692) ............ 0.4 0.4 0.6 2.5 5.2 15.2 31.5 44.2
Per cents Ansvering Specific Items Correctly
Larg- Small- Long- Short- Most Most Fewest
est est est est
Rural (365)..............80.0 78.6 92.3 86.8 96.2 96.2 65.2
City (327) ...............87.8 79.8 95.7 92.4 97.6 98.8 62.1
Total (692) ............ 83.7 79.2 93.9 89.5 96.8 97.4 63.7
Arithmetic in the Possession of School Beginners
Name................................... School -------------- -- Date ----....--.........
Part C Part D
Three fourths of them, however, identified either six or seven terms,
and nine out of ten identified at least five. Differences between rural
and city children were conspicuous only in the case of largest and
shortest, where the city children were slightly superior (about 6
percentage points). The best known terms in both groups of chil-
30 Arithmetic in Grades I and II
dren were most and longest; the least well known was fewest (or
The comparisons in the present study, as will be noted from the
picture, regularly involved three objects; the comparisons in the
Grant study with equal regularity involved four objects, and his
results are therefore less subject to chance success and error. The
specific directions for the Grant test were:
Mark the other board that is just as long as the first one.
Two tests involve half, and it is impossible to tell from the re-
port which of the two yielded the data reported for the term.
Mark the longest pencil.
Mark the middle hat.
Mark the shortest flower.
Mark the smallest tree.
Mark the tallest boy.
Mark the widest board.
The three terms studied in both the present and the Grant in-
vestigations are longest, shortest, and smallest. Both groups of
children knew the first two of these terms exceedingly well, the per
cents who were successful running at least to 90. The term smallest
was about as well known to Grant's pupils as were the other two.
but in the present study the per cent amounted to 79, perhaps be-
cause of unfavorable factors relating to the test picture (see above).
SUMMARY OF DATA WITH RESPECT TO CRUDE COMPARISON, CONCRETE NUMBERS
Per cents of School Entrants
Tcrnz of Successful with Various Terms
Term of ------------------
Comparison Grant Study Present Study
IQ 90- IQ 90-110 IQ 110+
(N= 145) (N =252) (N = 166) (N =692)
as long as.................. 24 50 68
fewest (smallest number) ..... 64
half.......................55 76 83
largest ....................... 84
longest.................... .92 99 100 94
m iddle..................... 84 96 98
m ost......................... 97, 98*
shortest.................... 74 94 98 90
smallest.................... 66 91 93 79
tallest..................... 71 89 98
widest..................... 72 85 93
Two tests-5 compared with 3 and 2; 6 compared with 2 and 4.
.-Irithmetic in the Possession of School Beginners 31
Thui figures of Table 10 seem to support the conclusion that most
chillr.n :on entering school are in control of the process of crude
co:IiipLari,rn. at least as comparison was involved in the test situations
uice. in these studies. Exceptions should be noted in the case of
iadl il tlii ugh, as stated, Grant's data are ambiguous) and fewest, or
si.?,7. ..;I number. It is a fair guess that the latter term was less
tr:.ubl.bl'.mni because of an as yet undeveloped kind of comparison
th-in lecaLie'nu of difficulty associated with the language for expressing
the result of the comparison.
The ri.-,st extensive data collected with respect to crude com-
pari.-n live been summarized by Woody (56) in a way which
r,-:atl., reduces their value for the present purpose. Woody's inven-
i'-.rv rt't. iven to children from the kindergarten through Grade
11.\. ct.'.ained twenty-two items relevant to crude comparison. The
first eizit were concrete in type-e.g., "Which is more, 3 apples or 5
apples "Which is most, 11 cookies, 7 cookies, or 9 cookies?" and
whichc h i: less, 33 marbles or 29 marbles?" Ten items dealt with
abjtracr imnnbers and involved the questions, "Which is more (e.g.,
3 .:.r "Which is less?" "Which is 'biggest'?" "Which is
"n,,allr .i' some of the numbers being as large as 229. In another
exer-.i-,- children had to cross out all numbers "bigger" than 9 in
a :'t of *i,,,ht numbers, all numbers "bigger" than 11 in another set,
all numiiibcr, "smaller" than 17 in a third set, and all numbers "big-
c-r" tli 25 in a fourth. On these twenty-two tasks combined, the
':,.r cetir ,:.f success was 77, for Grade IA pupils.
Thi- nr.-st careful study yet made of children's procedures in com-
p.ariIg numbers represented by groups of objects has been reported
I. Ru:.cIl (42). Unfortunately for the purposes of this monograph
hi data ar, not tabulated to show quantitatively the ability of school
intrant: ''. make the kinds of comparisons he studied. In his first
x.[Pl In',eint he used thirteen kindergarten, twelve first-grade, and
:,iir e,:,,'nil-grade children, who were asked to identify which of
[t,..- .,,l|. of blocks (such as 4 and 7, 5 and 5, 6 and 8) was "more"
.:.r "In.-i-t Having found this to be a "leading" or a "misleading"
,uit-,t'n'ii. in his second experiment with ten kindergarten, ten first-
SakL. '-ind five second-grade children, he asked for the identification
*if *rn'. : that were "alike," or "the same," or "equal in size."
Thie value of this study lies in Russell's psychological analysis of
.lil'lren'. thought processes when required to compare quantities.
Ruiiell rel-orts that children of age five have good concepts of
,c.t /.,,'ih. and biggest; that not even seven-year-old children know
Arithmetic in Grades I and II
same and equal at all generally; that in making crude comparisons
children of the ages of his subjects rarely use counting to detect dif-
ferences in amount, but rather estimate directly when they can or
break up large groups into identifiable smaller groups which they
then use by some such process as matching. The introduction of
blocks of different sizes or different colors resulted in marked in-
crease in error, a fact which clearly reveals the instability of chil-
dren's thought processes in quantitative comparisons of the kind
Abstract numbers.-The crude comparisons thus far considered
all dealt with groups of actual or pictured concrete objects. It is of
course possible to compare abstract numbers, such as 4 and 6, 3 and
7, etc. The mental 'demands of such comparisons are more arduous
than when actual or pictured objects are present. In the latter case,
for example, the compare has direct evidence of difference in size
or amount in the external groupings. In comparing abstract num-
bers, on the other hand, the compare must himself supply whatever
content he uses.
So far as could be ascertained by diligent bibliographical search,
this ability has been studied only twice-and then not very satis-
factorily-in the investigation here reported for the first time, and
in Woody's study. The procedure in the present case was to ask
which of two given numbers is more and which of two other num-
bers is less. The exact questions asked of the subjects, individually
in each case, were:
1. Which is more, 2 or 4?
2. Which is more, 7 or 3 ?
3. Which is more, 5 or 8?
4. Which is less, 3 or 5?
5. Which is less, 6 or 4 ?
6. Which is less, 10 or 8 ?
To be given credit for a "pass" the child had to answer correctly
twice in the three trials for more and, similarly, twice in the three
trials for less. This method of scoring is open to criticism, first of
all, because it by no means eliminates the factor of chance. It can be
criticized, in the second place, because really neither success nor
failure tells very much. Even when chance is ruled out, probably
many children were given credit for knowing more or less or both
merely because they employed correctly their knowledge of the se-
rial order of the numbers. Moreover, it is not improbable that chil-
dren scored successes with particular pairs of numbers whose
Arithmetic in the Possession of School Beginners 33
relations as to size they knew, whereas testing with other and less
familiar pairs of numbers might have revealed ignorance or inability.
For these reasons not much weight can be attached to the results
which are summarized in Table 11.
PRESENT STUDY: RESULTS ON INDIVIDUAL TESTS OF CRUDE COMPARISON-
Type of Pupil Per cents Comparing Correctly
Rural (337) .................... 81.6 48.7
City (296) ..................... 84.8 40.9
Total (633) .................. 83.1 45.0
The difference in per cents of success for more and less, and in
favor of more, probably is sufficiently large, in spite of the limita-
tions mentioned above, to be regarded as a fact. Whether this means
greater ignorance of the word less than of the word more, or less
ability to compare abstract numbers in descending than in ascending
order, or both, it is impossible to say from the data at hand.1
6. Exact Quantitative Comparison, or Matching
The requirement in exact quantitative comparison, as this ability
was measured in the present study, is essentially that of matching a
given number of objects. For this purpose Part D of the group test
(see p. 29) was used. The directions are as follows:
1. Put as many candles on this tree [pointing to the tree with no
candles] as there are on this one [pointing to the first tree]. [The num-
ber to be matched is 4.]
2. Here are two boxes. One has eggs in it; the other one is empty.
Put as many eggs in this box [pointing to the empty box] as there are
in this one [pointing to the full one]. [The number to be matched is 5.]
3. Draw marbles in this ring, so that this boy [pointing to the second
one] will have as many as this one [pointing to the first boy]. [The
number to be matched is 7.]
To score a correct response the child must first enumerate the
objects presented in the picture and then reproduce this number by
drawing. In each instance the drawing requirement is simple-
almost any kind of mark satisfies the demand-so that errors are
probably quantitative in character, the result of incorrect enumera-
Woody (56) asked his subjects, "Which is more, 3 or 5 (5 or 8) (13 or
17) ?" and "Which is less, 15 or 22?" but the response data are not reported
in a way which permits comparison with the figures from the present study.
34 Arithmetic in Grades I and II
tion, of failure to hold in mind the numerical total, of inability to
reproduce that total, or of some combination of elements. Correct
matching, in other words, is possible only after rather complicated
The results obtained in the present study (the only one yet re-
ported in this area)12 are to be found in Table 12. Three fourths,
slightly more or less, were able to match 4 and 5, and about half,
to match 7. The matching of 5 was easier than the matching of
either of the other numbers, even 4. Whether this fact is to be
explained as evidence of a richer concept of 5 than of 4, or as the
result of differences in the test situations which unintentionally made
the group of five eggs easier than the group of four candles to ap-
prehend or reproduce or both, it is impossible to say.
PRESENT STUDY: RESULTS ON GROUP TEST OF EXACT COMPARISON-
CONCRETE NUMBERS 4, 5, 7
Type of Pupil Per cents Answering Per cents Matching Specific
0-3 Items Correctly Numbers Correctly
0 1 2 3 4 5 7
Rural (365) .... 13.9 13.9 32.3 39.7 69.0 79.5 49.3
City (327) ..... 12.8 11.6 30.3 45.3 72.5 80.1 55.4
Total (692)..13.4 12.9 31.4 42.3 70.7 79.8 52.2
The city children were better able to match the three numbers in
the test than were the rural children; 45.3 per cent of the former
scored successes on all three numbers, as compared with 39.7 per
cent of the rural children. Moreover, this advantage is consistently
in the favor of the city children in the case of each of the numl.,er-.
though the difference in the case of the number 5 is negligible.
7. Number Combinations or Facts, and Their Use in Problems
The ability of school entrants to deal with number combination
or facts (the terms will be used interchangeably) has been studiiel
in various ways, and the investigations unfortunately have for thel
most part dealt with different combinations. Comparisons of rc:ultcs
are therefore of limited value. For the purposes of the present -.mn.-
mary the studies are grouped and the findings reported under three
12 Russell (42) had his subjects in his second experiment tell whether
groups of objects were the "same" in size or "equal." Possibly success meIiin
the ability to match; but in any case his data as reported cannot be coririp d
with those obtained in the present study.
Arithmetic in the Possession of School Beginners 35
heads: (a) the number combinations when represented as a whole
or in part by means of concrete objects; (b) the number combina-
tions when presented in verbal problems, and (c) the number com-
binations when presented abstractly.
a. The number combinations when presented concretely.-Both in
the main Buckingham and MacLatchy study (12) and in their Cin-
cinnati study ten addition combinations were presented individually
to children, first, in an "invisible" test and, then, for children
who failed in the "invisible" test, in a "visible" test. The procedure
was as follows: In the case of the combination 2 2, for example,
a child was first shown two small objects which were then covered;
next, he was shown two more small objects which were also im-
mediately covered; then he was asked, "How many oranges are two
oranges and two more oranges ?" If the child scored a success in his
answer, he was given the next combination in the same way. If,
however, a child could not give the answer, he was given a "visible"
test on the same combination. This time groups of small objects
representing both terms of the combination were exposed and left
exposed until the child arrived at an answer. Thus in the "visible"
test both numbers involved were present together for the child's use.
The Cleveland study (12) included only five combinations which
are identical with those in the preceding studies, and data are re-
ported for these combinations alone. The first trial in this study was
similar to the "invisible" test already described.
Grant (15) made use of pictures instead of real objects. The
subjects were instructed to look at a row of ten apples, after which
the examiner said, "I had one apple and, mother gave me two more;
think how many I would have, and mark the number of apples I
would have then." The requirements of the subjects here were more
like thone of the Buckingham and MacLatchy "visible" test than of
their "invisible" test in that all the objects were continuously present
L'fore- them. On the other hand, the Grant test was probably harder
in that i a) the groups of objects for the two terms of the combina-
tirn 'eren- not separated and identifiable as such and (b) the total
had t-i. be isolated from a larger total which was also continuously
present On the other hand, the Grant subjects did not have to
it-e.rd their answers in any numerical form, either written or oral,
:,inl tibm fact probably compensated in part for the greater difficulty
ciiu:ed b (a) and (b) just mentioned.
Table 13 contains the per cents of correct responses in the four
-tmluie: described above. For the Buckingham and MacLatchy studies
Arithmetic in Grades I and II
the first column of figures refers to the "invisible" test and the sec-
ond column to the total of successes on both the "invisible" and the
"visible" test. The assumption here is that those who passed the
"invisible" test would certainly have passed the "visible" test also,
had they taken the latter test as well.
SUMMARY OF FINDINGS WITH RESPECT TO NUMBER COMBINATIONS WHEN
Per cents of Subjects Successful
Buckingham and MacLatchy
Main Study Cincinnati Study Cleveland Study* Granti
| (1356 school (1,123 school (313 kinder- (563 school
entrants) entrants) gardeners) entrants)
0 Invisible Visible Invisible Visible Invisible
2 + 2..... 66 89 70 90 66 ........
8 + 1..... 45 76 46 76 ..........
6 + 1..... 51 77 55 81 ........
1 + 7..... 53 80 54 82 53 ........
3 + 1..... 64 89 71 90 ........
2 + 4..... 40 72 39 82 40 ........
2 + 8 ..... 37 73 36 74 37 ........
2 + 6..... 50 78 37 76 50 ........
3 + 7..... 33 73 27 72 ..........
4 + 6..... 32 72 27 73 ..
1 + 2 ..... .. .. .. .. 32, 56, 67
3 + 4 . .. .. .. .. .. 12,36,51
6 + 6 . .. .. .. .. .. 11,30,50
5 2 ..... .. .. .. .. 17,39, 54
3 1 ..... .. .. .. .. .. 19,44,69
O'Connor's figures reported by Buckingham and MacLatchy (12).
t The three figures for each combination stand for the records made by the three in-
telligence groups: 145 pupils with IQs less than 90, 252 with IQs between 90 and 109,
and 166 with IQs of 110 and higher.
The first three studies in the table are roughly comparable as to
procedure, and the results obtained in them are in extraordinarily
close agreement. A fair summary is that a third or more of the
children (except in the Cincinnati study, for 3 -- 7 and 4 -- 4,
which were "hard" combinations) were able to get answers for the
combinations in the "invisible" test, and that this proportion of chil-
dren increased to about three fourths when the combinations were
"visibly" presented. The per cents for the addition combinations in
the Grant study are lower if they are compared with the figures for
the "visible" tests in the other three studies, but are about the same
if they are compared with the results of the "invisible" test. The
same may be said with respect to the two subtraction combinations
in the Grant study.
None of the investigators in commenting upon his results inter-
prets these facts to mean that such-and-such per cents of children
.-l4ithinctic mt the Possession of School Beginners 37
"'kne.v" the-c, fact' : they rtay well within the conditions under which
their data v, er,: ,Itainted when they say that such-and-such per cents
rere alle o "',':t the anns.u-rs" or were "able to handle" the combi-
nati:,irn- a- the', \ere [pr-sc-ented. Others in citing these figures have
inil.ie.I that tli.i research "shows that children on entering Grade I
already kni.'', many ,-_f their addition (or addition and subtraction)
:':nbinrti:in'." Statementtic of this kind are wholly unwarranted. It
is one ithln for a child to "get a correct answer" for a combination
:.r fi.r a problem ontrainig a combination when he is allowed to
'csee r tc. manipulate r,.pr,.-entative objects and quite another thing
to:I .1,. s, '. Ieii he ,:an aut.unmatically and meaningfully supply a correct
ab-tract arn-wer fir a c.'-mbination abstractly stated.
The minml, CO e.ill. lionss when presented in verbal problems.
-In IX\ .tudi':-- niiuber combinations, predominantly in addition and
f-:.r thile m.i-t part ,liiferin g considerably from study to study, have
been pre-'ented t., ch,- I entrants as parts of verbal problems.
File ii-n.'c'tti':at,.n 0'.n the use of number combinations in verbal
pri-iblemni- nil be later '..'-nsidered together. The sixth study, by
\V'.Jod I 56 i. i- treated fir-st and separately, chiefly because the find-
in-: cannot be br'-,ker-Ti RJ'in for comparison with those in the other
Fv:i Thi \V::od\ in%.vent:or, given to children from the kindergarten
through Crade II i but always to the class immediately preceding
that in v.hich- .stematI: instruction was offered), contains eighteen
Ierbal pr.i:.b-im. lhese range from simple one-step problems, five
in a.hlditi,'n and firr. in subtraction (like "How many pennies are 2
pniies and I pennyy" -ind "If you have 2 pennies and give 1 penny
a.a,. I,.-i'A mania, pennies- do you have left?")-from problems as
ea-\ a- thes-: l.' t\.' .'n,-'-'rep problems in column addition, to four
'..ne-:tep pr.:.blem- in hizh,.r-decade addition, to two two-step prob-
l,-m in In addition. The per cent of success for the 1,666 Grade IA
pupril ic 39. buit 'vi,-u-i'ly this measure tells little about success or
tailiire *:.n particular combinations as presented in verbal problems.
In the present -tuJly t'-.- addition combinations (2 +- 2, 2 3)
ard tuo. -'ubtracti'-'n c:,inbinations (3 2, 4 3) were presented
indi idutall to:, 6.33 3s:h',I entrants. The actual problems used
1. On ,iri.'s biirthla:. her daddy gave her 2 dollars and her aunt
ca.e her 2 d.-.llar- m.-.re. H.:..v many dollars did she have then?
2 Mar\ ial 3 .1 :-.lls. She put 2 of them to bed. How many stayed
?. i. :.,l ca;n t., Frlay with Tom one day. Then 3 more boys
canni H-.:'.. nm..n- b.: i in ill came to play with Tom?
38 Arithii,',c io Grades I /mJd II
4. Tom put 4 pencils on the t.ble. Three r:.lled. off. Hov. imny were
left on the table?
The results obtained from, tlhi tc-ting are .uinrnarized in Table
14. Only a few more than 10 per cIent -,f the children wcre able t':
give correct answers for all four prohlenis. and shlghily les. than a
third were able to answer- a, niinny as three The city children Aere
somewhat superior to the rural chil.lren. b,-,th in thi,- function as a
whole and in each of th,: -.eparatc problcizm. TI o of the prubleicm.
were correctly solved by about half the children. one .:-f thete being
in addition (2 + 2) and thi .:Iher in suilitracti:,ln i3 21 The
other two problems (2 + 3 and 4 3- 1 ere b.,.ei by a few m.-,re
than a third of the children.
TA BLE 14
PRESENT STUDY: RESULTS c' N"L', BLE CI '.[fiE n .,',.~ I. FP. PA ,i-,t,
Per cents .': ll.) i '-1 Fer .''i,; riin.; Spc 'ifpL
Type of Verbal Problt,,., Cl ',r..r il' '.., '"i." .,i.;.i: .'.."L r'cti
0 1 2 .3 4 2 3 2 4
+_ --2 +3 --3
Rural (337) .... 22.3 24.6 2410 I'. '1 2 51 6 47.; 351) ,35.50
City (296) ...... 15.9 20.6 2'" 4 21 1" 122 2 4 s5 4 3J', 4) 2
Total (633).. .19.3 22.7 -, 5 21:'" 10 F. 2 .'* 5 37.3 37.4
As has already been explained. the teist ued] in the l're:ent tu'ld
was originally planned for til special .urpq.ose f ,:las.f.i, Mg children
very crudely at the beginning ,of a particular c::ur,.e -:f instruction.
The sampling of number co:'imibiatins is thetef',re much t.-., liritc'd
to serve as a means of imn nter'ryIn. the ability cit chilllren in general
with respect to the number ccomi]natir.ni The Duckin-ghani and
MacLatchy and the Cincinnati -tudiee, on ithe other hand, %\ere in-
tended primarily for this wider purpose.
Buckingham and MacLatchy (12) made use of ten number
combinations, all in addition, in as many verbal problems." lie
problems deal with familiar situations and with easily imagined ob-
jects (pencils, apples, marbles, paper dolls, books, oranges, prnnie-.
jacks, and the like). The ten problems are read or stated to children
one at a time, and responses are scored, as in the present stud',. a:
successes or failures. The data from these two studies are repro-
duced in Table 15, along with the corresponding data for t:ie -f,.ur
identical number combinations used in the Cleveland study.
MacLatchy (32) has summarized these and other data for a hyp:oth-ii,:al
group of thirty-five first graders.
.1,itli ui' lc tII lihe P'ossession of School Beginners 39
S%..IM.I. M \ OAi N -ili V. I'. rH RESPECT TO NUMBER COMBINATIONS WHEN
FiF;ri;TE. IN VERBAL PROBLEMS
B , ,. ..,,, .. ..;J .ll cLatchy
......r. r 1.',.. i. ',, ', i.' l .,eland G rant P resent
L. ..rn ; 11i. s -'., i udy* (563 school entrants) Study
S.. i. 1 1.' i 13) IQ 90- IQ 90-109 IQ 110+ (633 school
1, 1 a I,.'- 1 nder- (N= 145) (N= 252) (N= 166) entrants)
,..lru l ., ',ir ..' i .,ir eners)
S + 1...... "2 72 48
7 + I... c4 (.3 30
1 + '- -t 5,..
4 + 4 .... 31 3 37
I + r...... 4*' 51
5 + 2...... 44 43
S + 2. 44 4 :,. 36
4 + -s. 22 22
5 + ... 32 34 ..
3 +5 27 -. .
I, r.... .. 17 26 35
. 0.4 18 31
2. .. .. .. .. .. 54
+ ...... .. .. .. 37
2 .... .. .. .. .. 52
4 3 . . .. .. .. .. 37
"'C....r,...r'. I'i,-,rt: r-.,rr.l t, P.iJckingham and MacLatchy (12).
r The tr-a ,-,r, ... i all, rr,I,.-.l I"- 2, though Grant has reported data for 10- 1.
It :: ': ir., .J rl,;r I, I a: ,,p..-; phical error.
Data i' r tv.o ,f Garait'- number combinations (15), 10 2 and
3 : 2. ;re inclulr-d in TaWTlt 15. Yet, these data are not exactly
ca:.tipaarabl- t, h'.-. : frii,m tihe other studies. Grant's subjects, once
th,..v 10hal leaide,: ulp,,'n an answer, were required to identify that
anr ier ir.ni anL :,i hi'e numbers printed in the booklet. Thus the
ans.:r I.n.r tlhi:- niuiltipilh aticii problem (6, for 3 x 2) had to be se-
lected fri.:'1 arnia 9. 7 10i. r.. and 15 ; and the answer 8, for 10 2,
In. t,. Ie-,:- ilotcl fr..-i am;...ng 8, 9, 2, 6, and 4. Errors might there-
.-rc have- r,--Lult>-d. n.I:a I'r.:.m inaccurate solutions, but from inability
r., rt:, ..aiie t vle 5. rittri -\ nlhols for the answers.
A: ha- iben in-.tc l lIe',.r. the per cent figures for the two Buck-
iiliain and MacLatdv. tulies are in remarkable agreement. The
ihurre for thlt Cits-Iaial -rihjects are the same for 4 -- 4, slightly
la,:,r f,.,r 8 + 2. -:ndl con-i .rably lower for 5 1 and 7 -4- 1. One
xplaniiati,:in fa-r flht l:i.,er ui.uccesses is suggested by Buckingham and
Mac. Latclv 3piari':ntl\ their Cleveland problems dealt with less fa-
imilia.- ituati.-,, ,i-a v ithi l%<- asily imagined objects. Thus, the prob-
leni i-,r 7 +I a a.
Bili.ie is :e..i tc.ir- ...I Tomin is one vear older. How old is Tomr?
40 .*1i "lijti'c itn Gjles ,' J ld Ii
None .of tile four cr,iribinatio:n- emnipi-'',ed in tile pre-ent ,tuCJd
were als e-mploy-e.d in an\ other investigation, L.ut tie hEfitlr,: : f.r
the two a'dditi',in combiiiat.iorn ( 54 per cent and 37 per cent > are not
seriously ''lit Of li:ne .'1 tll thi.-e in tIe ihr' three columnist of the table.
The :act- iin t',e table .:L a v. hole m11, le -iiiiiarized a i f:lli'-'
for the addition combination- in.voilkii'n 1 itr c.. 5 + 1 1 + 9I the
median per c-ent of u: -...e- i- lightly mrui:r thin 50: for tilth ,l.i-
tion combiniatin- :, in, liin, 2, th,-- iiediai per cent i1 letterr thaI
40; for lih-- rt-mnaining, additi.:.n c':ibinati:onl. ti e nicllin1 [-r -enti i
about 30. Tli-:t- m,-edianr fir addition criiiinaiiti'in.- in '.riLal prob-
lems are alimn:'t identii:ca.l '.ith the me.liani ffir -itnilar :'rliii', of
addition co:ili-li.rson; u i'hren tlhty vere [.r--enell in "'i:ibl, ti-bie t
(Table 14 ). liut tle\ are -,f c,'ursc much I':,, er tihani the nediran per
cents for the samit ionijiilt L: i hen [rentite'J : ",. isl. 'b test4.
(In the aitta:r ca-e. thil iimedian t.i r cnit, are .. .75, an-]: 72. re[e:-c-
tively, fo'r tihe three _''-roup, i:f c,:il'in ti(oi I T"I cr cihlL e iii.,
summary :,b'i. %., tiht niiirber clnliiiati,-', iii .ubtractilrin ani]
multiplication are t:oo' f-, L:. lur1isli i]ati:i of any imp.,irtance.
Succe-.- in ,Jealiir- '.'-ith number ci-mlinations in verbal pro:bli-mis
would seem-n t: Le IlILICil ilinre lifficU.lt tiaan .LiCcess i\ len thi riiiiiher
combina:ioii:n are presentel c-'incretel., I.%\ inean 'of grouii[p' In s, (l -
ing verbal problems of thLe kind co: :-re- d L-. Table 15 ;ile child I.,
without external Cul-- a1' ,'- ti l,: Cont -i t ,:-i the niimhler; imn-r-i'ce'l anJ]
must therefore r up[il. their c:inti:nt himsnielf Tl. Iii- e ma'1,. I a I bv'
using abtrict ic:ncqpr- tiel nuiniler-s a units I thiu, "i'.,," ft',,r
example, a- a 'rotip i., i I-y de'.eloeping cl-ar imcntal imnaes of
groups of .-,i..ject I inllutliIlin t,.,:. or mi-,re -uhgroup- fo:r .acih nuim-
ber) whi.:-i he then lconLbme; iIrectly. o:r i -, I l.y ubistitiiivin' ill
thought aogre-..at':-. :f _~i .le oiibject- to' talii.n, ti t\- .: nuii i l.'er' ani,.]
then by c:'unini], ti-r.-ether tle-e -ulbttirute mental -,ltiject-
In the- "i i:ible" tenti n:o such a'stract concript;. imhac o gr,:uip-.
or image o:f i separate olject-. ,rvie re, -Uired actual ,b.ijctc. \,ere
present ,:' senie. an']. a.cc':r.ligl,'. the cliidrern ..*ere in cenerai iiuch
more succet-ful in thi- I in,.l f tert. The fact that 'nme cliilren \were
equally cr'iio pct-'ntll in th!e "ini ible" 1t,.t. rd in sl' in; .e-rhal pro:l-
lems (it l eiin; a.-urni.d:l thatli a i nminber comblinilatic-rs ,'.,re ,of aibut
equal difficui' i nI a I\ meani that thIl:\ actually\ eilli,-l'. Iei ai:,iiut tlhe
same mential ipr,::e.es ii tile tL1' -.itlationl In the in' is-ble" te t-
the children 'sa. thie if '.-t gr'.oup. remembler,..l that i 'aii' a cleir ima',:
of it), sa%, ihI- -cc l co i ':iup, reinmem i ele that i had an-'tier clear
image), in-il thin coml in.'d tle- iiija-e-- .1recti' ,:,r Iby c1 'ntin, '. In
.-Iritl,,iH ic in the Possession of School Beginners 41
the ciase of [Ile ,rrtial problems they could not of course start with
external oiup-,. lit thev could have developed clear images of the
r,'rouli:,p an.] dealt with these images by direct combination or by
countinct.i. It 'A ill be noted that practically all the problems used in
thLe .:Lrio- ,tudies ,':ure carefully prepared to enable children to
conjure ui precls'ly- thi- kind of subjective substitute for actual
UniJortiiiatel. in none of the investigations summarized in Table
15 were data collected on children's thought processes in dealing
. llh the nuinmer combinations. Woody (56, 58), however, reports
-L fi:. fact- in thick connection and stresses the importance of evaluat-
min'' performance, not i,.irely in terms of correctness of response, but
al.o in term ..it tih: niamrity of the process used by the child. Data
of this kind are badly. needed in order to combat misinterpretation of
the result, of the inue-ti-ations which have been summarized above.
This point nas made in the section immediately preceding, but it can
I.,ear re-tatement. It i- a serious error to infer that because a
child ca-in furni-h the correct answer for the combination 3 -+ 4 in
pioiblen-i. lie kno- that combination. If the child has substituted
-ei.arate ,imiital o:bject- and counted them ("one, two, three, ..
siien." or "three. four, five, six, seven"), he has not, in strict liter-
qdinecs, reacted tl the combination as such at all. What he has done
i; mc-reli to: re'eil liat he has some way of dealing with the quanti-
tairie -ir uitin in -in effective, if immature, manner. Only further
olervati on and (ue.iti.Ioning can reveal the precise nature of that
te-aclhrs a1r. 0to iunJei-land how children develop number concepts
and ho,;, they' c:,ome t.: an understanding and meaningful mastery of
[lie IIntl 'ih-el -om binti ticon- (5).
\'hlit ha: Pi.t beun ,aid has reference to the child who approaches
number in vwhat man\ b,: called a "natural" way, through the exten-
:-ion and refinemrent ,:f procedures which are at first crude, uneco-
nomical. "ind inmmattire. and who later surrenders these procedures
frr othl'er,- .l'1chi are imre refined, economical, and mature. But not
all children ta-L thi' route to number knowledge. They are some-
time' mis'directed 'b i ell-meaning efforts on the part of older chil-
dren or adults. In tuch cases rather peculiar consequences follow.
In the pre-emnt itudl _c' eral children were found who were better
Jile t. supj.l\ an-t:.'er: for abstract combinations than for combina-
tion, in erbal .r.:obleim-. When the combinations were presented in
ahtract ,formi. the', .e the answers glibly and correctly; but when
42 Arithmetic in Grades I and II
combinations of supposedly equal difficulty were presented in verbal
problems, they failed.
These children appear to have been coached to memorize the
combinations as abstract facts (a practice which is by no means un-
common in classroom instruction, but which was hardly anticipated
in the case of school beginners). When the combinations are pre-
sented in abstract form, they are able immediately to produce answers;
but when the combinations occur, not as simple uncomplicated num-
ber statements, but as parts of verbal descriptions as in problems,
they cannot identify the combinations as such; and, having no other
means of dealing with the numbers, they cannot supply the answers.
c. The number combinations when presented in abstract form.14-
Eight number combinations, half in addition and half in subtraction,
were presented to the subjects in the present investigation. In addi-
tion, the question took the form, "How many are . and . .?"
In subtraction this question was, "How many are . take away
S. .?" No objects were used; the combinations did not occur in
verbal problems; they were merely stated abstractly as questions.
According to Table 16 nearly a third (29.5 per cent) gave the cor-
rect answers for half of the facts. The per cent for the city children
was 32.9, and for rural children, 26.8. As a group, the addition
combinations were easier than the subtraction combinations; the
mean per cents of children succeeding on the two groups of facts
are 42.8 and 27.3, respectively.
As would be expected, the abstract combinations were in general
too difficult for these school beginners to negotiate. Only two com-
binations, 1 2 and 2 + 2, were correctly dealt with by as many
as half of the children. Moreover, it would be stretching probability
to infer from these results that even those relatively few children
who answered three or more, actually "knew" the combinations in
any real sense of the word. Attention has already been called to the
fact that many of them evidently recalled memorized (if meaning-
less) answers which they had picked up somewhere. Others of them
undoubtedly counted some kind of images to secure the answers.
Still others were lucky in guessing.
Woody (56) collected data both on the abstract addition and on the ab-
stract subtraction facts. In both cases, however, the facts were included in a
test which involved more advanced forms of the two processes. It is impos-
sible from his figures to secure measures of "knowledge" of the subtraction
combinations, but his later report (58) carries facts on the addition combina-
tions. These are summarized in the section above.
.- ithi nlii iii the P,'ssession of School Beginners 43
F'PEsNr SrIUN *- kE-':LT4 OrN THE NUMBER COMBINATIONS IN ABSTRACT FORM
.', Pe'r u,': .-ic'. 'r.., No. Combi- Per cents Answering
C -,*.r i... T. t P,-ir. nations Correctly, According
f n ..,"; in Test to Type of Pupil
fly' ''' ,J ..r .r (C, T i A Rural City Total
,. '.rct'ly = 3 ,: N = :, ,N = ,03) (N = 337) (N = 296) (N = 633)
. .. 145 125 13 1 + 2 ..... 49.9 53.1 51.3
I 1. 1..4 14* Ir,7 2 + 2..... 52.5 55.5 53.9
2 ..... 21 7 "' 2v 1) 1 + 3..... 34.1 38.9 36.3
3 ... 1. 7 I'.'1 1'0.3 3 + 2 ..... 26.4 33.4 29.7
4 s' ..s 7.? 2 1..... 37.4 42.2 39.7
5 5 r. 2 ... 3 2 ... 29.1 35.8 32.2
S ... 4 2 5. 4 7 4 3 ..... 19.6 25.7 22.4
3. 5.1 4.3 5 2..... 19.0 27.9 22.7
4. .. ..j ............. .. . ... ...
N,, dala V.-rre- rcrIIried ,Lth respect to the incorrect answers
iienI. Liut certain inter-sting facts were observed. There were wide
dilfferennc'- alm,.ng thI clildr.:n in the extent to which their answers
erred. S'c..:e :I them gltii .'.el blindly: 3 + 2 were "sixteen," or
"tele." :.r "It\.nti\" 4 3 were "ten," or "fifteen," or "nine."
th0er ,t er able l.:. I,:
nI i,..r 1li erincs i- .1 + 2 were "four," or "six"; 4 3 were
"i.o.'" r "lithr." Thei lar,-Ir children seemed to have mapped out
ith numi l.-,t % st A i.\
The i.,rm-iii.r <.htidr: II i.n1 the ,r-.her hand, had little idea of the com-
.arati'.e izi >.," inulil.t.r oI:ir i.f their relative places in the number
-Ise-r. .At thi. po:iit ,3alual.Ik research might well be undertaken, to
deternniie tlie ciiionite-, of the practice of "locating answers in
their al.,p|'i. 3iale area" andj t!ie relation between different degrees
.f Iti; ahitlit .and stubi-iirIent success in learning the number com-
,ir3atit.ns. It i-s ii nt unliL-el that experiences in "approximating"
an- .tr niiiihJit t.ill I-e .carterl very early, as a preliminary to the
cl -er .ud :,of thi: initl.r ,:.c-mbiinations as such.
Data .:o l i. :trcn _,.litir .. ..>mbinations are reported by Woody
158 i for i lie fivi halft'-.raih I_ i oups to whom his inventory test was
tliiiinitslerl S.iriie intrerq i, largest in his Grade IB subjects who
w:r,: it C nt:it in,' I -I li, tlie figures for this group are briefly sum-
iiari?''I I I:'l.l. in i t .i-i .f .,:r cents of correct responses, and the
rt i:,r.l- I,..r th,i:-t r.i.14 chlilruci are compared with those for Woody's
I 897 Glali I.\ cil:dil.iin -if ile sixteen combinations in the inven-
ir..r. the l":t ii o'. \a-, 2 + 1 (52 per cent in Grade IB, 78 per
,ent in Gral I .\ i: nxt i.-rler of increasing difficulty was 2 2
i 42 l.er c.enti in Ciale 1[. 72 per cent in Grade IA) ; next, 1 + 7
44 Arithmetic in UJ./s.' I .-Id if
(36 per cent and 65 per cent i th-in. i1 + 2 and 7 + I 1 35 per Crilt
in Grade IB, 56 per cent and e:5 per Ccent, resp'ctieli;. in itird l IA .
6 + 2 (27 per cent and 57 per cent ; :' + I aindl 5 5 (25 Tpir
cent in Grade IB, and 85 per cciit and 54 pei ccnt. ri-cpucticl\. inl
Grade IA) ; 6 -+- 3 (24 per ciint anid 53 p._r cent I. The co:mnbirati:,nii
5 +- 4, 6 + 4, and 3 + 9 'v.,r: r,:rrectl\ resp':'nded t .: b bhrt-.eei
10 and 18 per cent in Gra6c: iB; thI co'miinlbnati,:'ii 5 + 7. 5 + 8.
7 + 6, and 9 + 8, by between 5 per ccint and ')9 ic c-rit. TI,,
combination contained both in tIh[ pire-nt tudl' andI in tli. \\,:mud).
investigation was 2 2, wlhichi 'Aa- caI_',rr>Cctl; aii.,'crcd by 5-4 p:r
cent of school entrants in the fi.'nirmr and by -42 per cent in thc latter
Conclusions with respect i:,' it- t | ,,I.,',- ...iil. ,i",'i.<.- It is p:s-
sible to read both too much and t...... littlit inilic'I itii'' tie re-
search findings on the ability, -f c ii....'l citratl-d t.-, deal ,'-ith ti h
number combinations. One p'r_.r.,'ii cm[rlasii.:es tiec pr:p':reti
children who secured answer- lu tic v3ri'ill' i'.om.inaii'iti>in-, pa3, r:
attention to the methods used L,\ children I,..' '. tl"-ir an. v,<:t.., atd
recommends immediate instrucri,:ii *: 1'hr ab.sti:ct 1ci.Z'mbinatiun- as
such. The other points to t'K f'rat ti it the cI.mlination., included in
the testing are all among the simnph:r ,_.nicS, .trc.sses the pr.r:ortionI
of children who can do nothing. .with combi atlii:n. nd asr.ues
for postponement of all instructi.-in ,-n, th :, combiniiiati:ns as suchli
Neither extreme view is calculated t1:' prmite _i-i:.unr d iintrutc-
tional procedures. It is trui. that rather isurpiir.in;'lQi la.ic numbers
of these school entrants managed 'mhid,'.'- t,'. -,et arin.\ers .. L ih
combinations presented; it is true that thlie, co.mbinati.ii nsrc :riin l:ii
the easiest; and it is true that pr:.L. abl. lfe.. if dan, :, I tn. suiibjects
found answers by mature inethid- i f lthinkirni' Tirhese stittmint-
lead, so it seems to the writer, i,_- thi- .:-.incsii-i:'n that t: b>t pr[ritabi,m
the experiences (if any) which First-grade children hIi.uild ha': \with
the combinations must be \.cil clhj:',en ail \V.icm. (lir,..t'd. Th..
adjectives in this sentence ari- :rttl notuiii
PART li. -i FIic .- .-17
.'. F actii'/ i
Three studies are summiari]2:-d in thi- ,cCti,_i. ianam l\. thll:-ce mad'-
by Woody, by Grant, and by P.:,lkinghiii, nc.
Woody's study (56, 58) iiH ol\-ci fiar mi:,'e ub.jitt t1li i did
either of the other two, a tont! ,f 3.002 children, dividedd] amiio"i.i,' If-
.-Irith,'ic mn the Possession of School Beginners 45
-:iup,, as sho.in below,, hut none of them as yet exposed to system-
.tlic rinthineric instruction The first series of three items called for
the identlicati,-n i-,f pictures showing apples cut into halves, into
f.,urtli-, :rnd in. thirds His data for these three items (56) are
i2'en bil.lw i per cr1it of successful responses, the results for
L.,:' and zirlk]- I.:ii combined:
hI,,,..r.L,] ,', Grade IB Grade IA Grade IIB (N=238)
F,.rm..o '., = -- =oI (N =594) (N= 1896) (N = 80) Grade IIA
S. ,4 67 77 82 88
S. . 45 52 67 69 74
13. ... . -I4 45 65 61 76
.rated brinetr. tiw,' thirds -f4 school entrants (Grade IB) could deal
-ati-fact..xril. with 1'., about one half with Y4, and nearly one half
i 1 1..
The sec.-,nd s.ri:.s :f three items took the form, "Into how many
hale t'ithirds. l':.urth can you cut an apple?" The results for
the:. items. reported as are those for the first three items above, are:
hale. .. .... -I 44 48 44 45
thirds ...... 26 29 35 43 46
furtl. 2. 22 34 29 47
..'ain. the co:nicept :,f Ihales was better understood than that of either
.-.f the other fractions. and the concept of fourths better than that of
thir-.s Thi p'.r cent. foir thirds and fourths are considerably below
th:os fL:r alves except in Grade IIA.
The- third serie- of \:oody's items called for the comparison of
the fractli:,n- 1_. an. Jd I3, as to relative size. Eighty per cent or
more :4 hii- Iibject_. r,-grdless of half-grade, answered correctly
that a .t-,le i larger than a half; between 40 and 50 per cent up to
Grride II l:nrew that j half is larger than a fourth; the other com-
pari-on ,. 1i :nd ';,:. 14 :,nd %, and 1/2, and %) were success-
illY 'tcalc \,. ith l .-nl. About one third prior to Grade IIB. Woody
'. iels calutii:n: acaiin.t inferring from these objective data alone that
lhildr,'n in lthe pi-rnuar. ;r-ides actually possess rich concepts of these
unit fra,:tion: er.en in ite limited ways in which they were tested,
and he rerp'rts remnarikl made by his subjects which indicate that
-ucce-_ ihn ni-,t a f,-: instances was more or less accidental or the
Inilt of Iut v' rY Spri:-it :al knowledge.
Itmni 27 i-. Test 5 .4- the Metropolitan Readiness Test used by
.ir.iiit 1 15 1 rei- ti :ll.. s:
-46 .-AtiiMuetict il (i' rads I and II
Loo:k at the r.:. ofi tour crcle,. Maik halt it all the- circlei-julst half
,:tf the circle-.
Item 28 reads-
Look at the ro,, oi circles th lines in them Fmind the circle tht is
m.rk t' ii :..i/: ,. that i s cut in /half rind remark it
(Grant ha- reported data for "half, ut v. hther his data refer tou.
the results on item 27. :n it lii 28. or on the two i.ombined. it i-
impo:-_sible t.:, s-a. \\haiteer the basis. his figure s-hrw that "'half"
was s.iccesfull, negotiated by 55 per cent of hi 14-5 "-dull" pupt;1
(10 Ihel,"- .(. I: ) 7Tr per cent ii Ins 252 "average" pupil- i l,
90-109' and I.v 6,3 per cent .If his "bright" pupils 1 itt 1 10 and
abtve'i. Item 27 atpplie- the fraction !' to a grtutp of obli-:ts: item
2&. to:I -iinglhe r'iijec:t- (- tiii,-,u l thte e f",:% dJata. particul rl ial ici:
their reference: i, ;miiigui'u-. tell little ab:iut the abilitNy of school
entrants to diLal % ith frac:tions.
The most tili.r:-.uglomrin investigation in thiN area is that uiade
li P'ilkinlghiirne i 40A: \ t. gond as ais thi, itudl. it i ituljert I,:-
lihim nations liich affect tlie general %aliditl\ f the re'ult- [ii thi-:
ri place. *oinl part lie-r 22W -ubicts- icre schi.Il entiranti. Part
%%ere kindergarten pupil:. and part err et-nrollred in GradJc II and
Ill Nre_\rthl.t,-. thu data for the group as a S\h,:l M arc heri treeate:d
tigeriher. smine v. batt:\vr the childri.n could do \\ith the est,. the-
liad:l Iarned t i' r. 'Ithout sistemati.; minru.:tion. In the -ec-ind
-lac.e. the test has not bc,.n printed. Otl, a -saiple r-f the fi-ct-t%-ko
itemni it a\ailaLlc. and lack of inf formatic-n at this point mal-e int-r-
pr,-tatul n difficult. In thte third place the -utljectt :., ere undlrultedly
if -i.i, riotr alili ha ing been dr\unn from the Unir-it\ Lal.ra-
tior School if the Uiiver-it\ L.f Chicago In thec fourth place.
in tle brief article heree the findings ar. reported, ihe data are
treated in a\ %a i.liich maiakes inipo-Vible the kind .f anal yi- 'Ie-
-iral:l (for theI rep-let t.iurpoC c
The frt\ -th-o item, .if ti,: test. I[ich '.a- cien iniJi.iduallvy t-
each child. are cla-sifed under thli he:id-,. unit fraction: i such an:
and 14 i. ,oihi:r properr fractiir. 1 ;. '- .-.. '4 1. im p i oper fract1in-.
idiitfihcatioin of fraction. and erqui\alnt fract:'iins. Tli. r:spo,:iise
i:r .acli item ,.a-s e.itler nmilltrical ur a perfirllnance v.liich could lbe
,coreld oliectl ,l Examples ar'. "Here are tw o, pencil- \Will yrni
give noe_ one half of tlhcin ?": "I aIn, *,ain t,-, dra\, a line [dras a
line about -i' incht-u longg. Ira', anro-thr line one half a lo,"':
* Here i- a: dra\-.i,; of four marble:- Coli-r thrce-fuurth i:f thiet "
.-,',tIinut/ic in the Possession of School Beginners 47
Filty- ihree per c~ni of ihe unit fraction items were correctly
an ,erdl [,v the p'.rouii as a %%hole. The next largest per cent was
fhr othlier proper fraci,'n-IS-l per cent. The per cents for improper
fraction and identilcatiion of fractions were both 8, and that for
eiluiialent fraction-, 0. The kindergarten children averaged 3.7 cor-
recr iitemn. all dealmc .i th unit fractions; the first-grade children
aira2,:dl 5.9 correct item-, all :f them again on unit fractions. The
a' cr-Ae score on Iunit raciion- for second-graders was 9.4, and for
ihird-graders. I1.' \\ ith other proper fractions, the averages were,
iesplieccuely. 1 7 and 3.8. Thl Grade II children had negligible suc-
ce-s v. ithl te other thrre clae's- of items. In the case of Grade III
children, the aTvrare i.,orL- on the three last classes were 1.2, 1.6,
jnd 00. \\hen tilth sub.ijct,- .',.re grouped according to mental age
S5-7. 7-9. 9-11. aid 11-13 scores comparable to those for grade
er:.lup[ I erer Tllade
I-'olkinigh:.rne conclude- that in the first grade unit fractions may
ie -afely prei-,ntedl. Firrs in c..nnection with single objects, next in
cnnnction ,.'ith 'roupr, atnd finally for comparing, first, two objects,
andl. ithen. tu group. of olbjuct. In Grade II fraction concepts may
and a beginning 'nay Ihe mad: in the understanding of improper frac-
rn:,n whether r or not rhe-c recommendations, made on the basis of
tetnmr' ratlhlr than i:f te' '.mig and on the basis of testing superior
children. are -.alid for sch: 'o.ls in general is a question for future
re-earch. Data -ifll be prrLsntcd in Chapter III to show that at least
-,omtn of thc ,e cInic.p-Is ar, \.ell within the powers of first- and
c-econd --; radc childri i n.
Grant 1 15 i hai rpunrte] the only sizeable study of school en-
trants' knowlvJdge -..f ordinals. For this purpose he used items 20
and 21 of TL-t 5 of tie Nletropolitan Readiness Test.
2'1 See t he farmer fretdinic the chickens. Mark the third chicken from
2'1 I-e the p, ruinin, -.. thle fence. Mark the sixth pig from the
Thi- re- ult- of th 1e 1 ing may e summarized as follows:
0, i./,,./ 'D.l!" frpils "Average" pupils "Bright" pupils
!,/ !,is than IQ 90-109 IQ 110+
0 1 = !45) (N = 252) (N = 166)
/thhid ..... . I 39 71
. l .. .. 21 44 67
Arithmetic in Grades I and I1
10. Reading and Writing Numbers
Reading numbers.-The study by Grant (15) samples the num-
ber series to 100, testing only on the numbers 4, 9, 16, and 75. The
study by Wheeler and Wheeler (54) gives data on all the numbers
to 100. The item for 4 in the Grant investigation is typical: "Look
at the row of numbers where the hand is. Mark the 4." To be suc-
cessful the child must select the 4 from among the numerals 6, 3, 4,
8, and 9. Grant's findings, in per cents of successful responses, are
"Dull" pupils "Average" pupils "Bright" pupils
(N = 145) (N = 252) (N = 166)
4 26 60 83
9 34 48 63
16 19 40 67
75 21 43 62
The data from Wheeler and Wheeler are too numerous to pre-
sent in full. They were obtained individually from 157 first-grade
children within a month of the start of the first term. The numbers
from 1 to 9 were known by an average of 64 per cent of the children
(4, by 68 per cent; 9, by 33 per cent) ; the numbers 10 to 19, by 15
per cent (16, by 8 per cent) ; those for 20 to 29, by 13 per cent; and
so on. The per cents for the numbers for the decades above the 20's
range from 6 (for the 50's) and 8 (for the 30's) to 13 (for the
70's and 80's). The median score obtained was 9.21 (one point being
allowed for each correct response) ; the range was 0-100, with Qi
at 0.99 and Q3 at 11.8.
Woody's inventory test (56) contained thirty items on the read-
ing of numbers. Six exercises were based upon the numbers in a
calendar and involved the numbers 3, 8, 6, 11, 23, and 28. In another
exercise the subject was asked to identify six of seven numbers
printed in a row; these numbers, in the testing presented at random,
were: 37, 19, 72, 41, 131, and 113. The next exercise consisted in
twelve numbers, ranging from 13 to 10,103, which had to be read
exactly, with all such words as "thousands" and "hundreds" cor-
rectly placed. Finally the subject had to locate correct pages in a
book, these pages being numbered 13, 21, 70, 57, 89, and 123. The
thirty items were presented to 1,897 pupils in Grade IA, but before
they had been subjected to systematic instruction. The results are
not analyzed for the separate numbers, only the per cents of success
by city (there were eleven cities) being given, together with that for
the total, which was 63 per cent.
.-rithimclic in the Pos.',.-sion of School Beginners 49
So far, onl\ the ability tc- recognize or identify the numbers as
such has Ien con-zidered. .A child had only to say "thirty-six" upon
bein .ir mir "r',3" to score a success. Knowledge of the meaning of
the number played ,:, part. The Grant study contains 'data on the
ability oi sclh..Il enttrant; to, "interpret" the numerals 4 and 7. The
item 'for 7 is much like that f.:,r 4. which is: "Now look at the next
niumb, r [it i- 4). Mak, a mrnani dots after the number as it tells
,\ou ti miake." Ii ,:lr tic.e conditions Grant's pupils were less suc-
ces.ful .ith 4 than they had been in "reading" 4: the per cent of
"duill" pupil fell from' 26 to 10: of "average" pupils, from 60 to 40;
and ,-,f "'bri'-lit" pupil- fioim 83 to 70. Unfortunately, there are no
-iuch comllparable "'readjincg" '.lat for 7; as "interpreted" by the three
zrouilps of children, the per ccraitz .ere 9, 33, and 61. Nevertheless,
the Lff,-ct of calling f,-,r "interpretation" in the case of the numeral 4
m;ia be enr:iu2li If a caution against the acceptance of ability to
.read" numerals a- e.tti alnit ., a grasp upon their meaning.
l'riti.ug uninm irs.-Gravt i 15 i had his subjects try to make the
ntunritrals for 4, 7. 2. 5, '. iiii The per cents of success for his
a"verae'" pupils- were- ,0 17. 17. 27. 22, 9, and 21 (mean, 21). Ar-
ranfed in :,rdier. the numrral. 1. ert, from easiest to hardest: 4, 2, 5,
. 7. and 9. The mean pcr cent of the "dull" pupils for the six
numerals .. 5. andi for the "bright" 49.8. For the last named
gro.:ip. the i-.rder .-f dilihKuilv, as: 5, 4, 2, 6, 7, and 9. Grant's figures
re\cal a ratrlhr surprising degree of ability to write the numbers
iprir tn school iriitrniction. particularly in the case of the "bright"
11. Rcc.p-n*iic-i., .f Geometric Forms
Grant'- studiv 15 1 reports data on the ability of his 563 subjects
to recognize figur-,s in the iliape -,if a square, a circle, and a triangle.
In each case the correct form had to be selected from a group of
four diaramn,. the correct ,inl and three others. The per cents of
siuccezs folloit, :
T7 ,'.,, ,[ li" pupils "Average" pupils "Bright" pupils
i- i *., ..:, ,, IQ 90-109 IQ 110+
'. = !J (N = 252) (N = 166)
si/tn, ......... 79 92
c -c,-" ... . 76 85
i, i ,. ,: . . . 1 19 34
Thri-e iotirthls or inore o, Grant's school entrants already knew
the tern-i /,',u, and ciro/lc well enough to use them intelligently.
The ca-e '..ith rc-s[p-,it t, .. /i,.l,- is quite different. Clearly, these
50 Arithmetic in (Grdes I and II
children 'did not know the ia.rm., .cn ti the ext-int reJquired by the
test. How much of a task it t would be to teach its mIcanii.n. toU ftirst-
grade children is not known. Their ignorance, accirdlinc to Grant's
data, may be the result (ai of lack o.f experience % ih the term and
the corresponding form, i of real *]ilt-icult\ in acquiring the
concept, or (c) of both (a > and i bi.
12. Time, U. S. .lloait'. oad .llcasiir's
Included among the 204 temns in Wood\i 's inventory <56) are
eight dealing with time, thirt-ren 'with U. S. cm.inIs. in.d eight wilt
linear and liquid measures. Tabll IT s;urnnnrizes the result; of test-
ing 1,897 IA pupils who liaIl had no s stenmiatic inr-.truicti'inr in
arithmetic, the items in the table having to d.1Ju ith the qut-cstiuns
on time and on U. S. coins. .\bout half -if those children cuild tell
time on the hour, and about fiur out of ten cCould locate the position
of the long hand on the clock at the c: en ilhour. Timeri at the: hall- or
quarter-hour was pretty well bevorid their powers. Practically nint-
out of ten could identify each of thie ie cin;s to the-- halh dollar, andl
PER CENTS OF 1,897 GRADE IA 'liri fi L trH.:,.,r I -rr.'.iti.'. ,*.,. N 'if'tr!i.
WHO HAD VARIOUS L'**: .,r-pr' .f TIME .\':- Ll S MI''El
I .,F \A '..r \ I
Time Items Pr .1
Tells time on clock 1
(9 o'clock) ............... .. 1
Tells time on clock 2
(1 o'clock) .............. ... 53
Tells time on clock 3
(12 o'clock) .............. .
Tells time on clock 4
(3:30 o'clock) ............... 15
Places long hand to
show 7 o'clock........... ... -4
Places long hand to
show 5 o'clock............... 42
Places long hand to
show 10 o'clock.......... -43
Places long hand to
show 11:45 o'clock........ 12
Points to penny.............. '"
Points to dime............... "
Points to quarter............ s.
Points to nickel.............. ..')
Points to half-dollar......... .'is,
.1/.....,- ,'; .,:s:
\\ brch lll, m:, 'lOr .
a p.min .:.r a ik'l .
., rprin:. .o.r din'ri :
a lime -r a rc:' .
a3 li-rr, i,r A qL.IruarE .
a haI l .I:llar .-r iquarti:r
ai l. intan a i. amr
am..untl j A
l. lr'i. *Ii. rti r .. . 31
qjarit:r .... ,-'
rilckil- .1illn c .......
,lu r te r- . -,halt-,lh ll:,r
,] inr, . -. h al '-,l..ll.r ...
.-jri,,iJ',i!" in ith P'l'lSc'ston of School Beginners 51
about thic ;amin: ratio. kn,:w the gross comparative worth of pairs of
tlr,:-e coint, but third precise relationships between the coins, as be-
tween nickel' and quarter, %ere not so well known. As would be
ex['ect.:d, the nuimbi,:r of peiinie; in the nickel and the dime were
Le-4t knlio' n
Unfortunately. W\\ood doe not report the corresponding data
for childr,'n in the other lalf--,rades tested, except in the case of
tellin. timn:. 158 In gc'n,:ral. pupils in his IIB and IIA groups
.lho.wed aLbihty beyond that .:o his IA pupils, for whom the facts are
tatited above. O-Inly the iact fao-r his 594 IB pupils are given here.
\bout 30 per cent :f the'e children correctly identified the clock
,li'ho uii nine o'cloi:k I data ior the sexes are combined), about 49
l-cr cent identifi.-d that shi:. ing one o'clock, 40 per cent that showing
te. dlve o'clock. 7 per ce:n that ;h.wing half-past three, 24 per cent
that sti-: .v.ing ei';l'it ':,'clock. 20 per cent that showing five o'clock, 22
.per rent that -l\,-, mi te-n o'clock. and 8 per cent that showing a
'liiarter to:' t el\e o'cl,'ck
W\\ood\'; finding. for linear nieasure and for liquid measure are
c,'-.n-olidated int. per centii- ,r his Grade IA pupils as a whole and
b', citi:.-. Fort.\ ,.er c-rit of the responses to linear measures were
correct i the- ranee 1.v cities, v.a, 20 to 60 per cent), and 33 per cent
of th,: rc-i :,ner, 10 liquid nme3iurc, (range, 33 to 67 per cent). The
it,_im- :'rn liii'ar nrauiire, caliAl for the drawing of lines of the fol-
l, ing l':nigth: j fo..,t., ix inchiis. three inches, two feet, and a foot
and a hall'. Th,: item on lilinii measure were: "Which holds more,
a pint little. '-r a quart bottle''" "How many pints will a quart
b.:ttle lihld'" "How man. quart- in a gallon?" Obviously, success
on iht-,:- item-. -nmall a- it wv.a-. indicates little of the meaning and
-iit.'nntcance uhich tili'-e meaur,:- held for the children tested.
/L S,'.i L fft't'JI,'.;
Buickinr.hain and MacLatch'. I 12) found a small but rather con-
i-tent -up.ri rit'. :.f ir- ':''.er b''ys in the various functions which
th;t tu.lied. In rot': co:U.iitnL. b1', I's the median limit attained by
l.,.:, na, 24.5. anld lv .irl-. 2'-,. while 27.3 per cent of the girls
ac1ie\'.d, the urimt -:'f 50 as c-.impared with 19.1 per cent of the boys.
Similarl. in rote c,-,ntitrt.n l,\ 10' the boys fell behind the girls, but
I, a 'eer\ narrow martin. In 'nurmmerating twenty objects 58.2 per
Lent of the ",:'v- and .11 7.0 per cent of the girls were successful; and
S5 3 per crent o'f ithe b,:.., and '.-I 1 per cent of the girls enumerated
at kl'-. t ten ,:,l j,:-.:t_.
52 ).-l;iiinicte in Grades I and. 11
Moil gyir thir bol1 '.-i-re abl- to r._l-'J-dlucl t ac: ..f the niinbe.rs
5. 6, 7, S, and 10, thit margin of dirtier, :ce biinlg al.,iut 5 p-erc,-taigg
poiItc I tlhutl. 81.0 and S5.8 per :ent for 5 1. Thin tuip.,riorlt\ f1:,
cirl 'i a n maintain'
:f the sime hfe nuimlblt) r
Thil median [It.rci.-ita: -.''rt-r iad.- [ bi. and girl irr-, th,
-7inml in :'I;: tin to rLi:di pronblcmn, ,i ith iplN[: alddlitio:n ,::,l.ina-
ti[':ln ain] in Irnddiwin an'-mers for lt:.n other addiitio:,n fact: when "vi-
iblyV" pr'eenteu' \\ hein thei "viible" ind tihe "in-\i-ible" p.reeintationii.
\ier,: c::imbhieird. thie cirl exc-elledi tlh r h .:, by fr-m I to nearly v
pe'rcr'ental.re pointrit :n all ten ..f thile latter -et of iadd'iti'-n cOicbiiia-
ti'-n- BIuickin-lia ham and MacLatchy conclude tlhat. mlioe\ver co0ii-t-
ent the a'J iantag IIcd b tihe 'irkl, it i nmail anrd cin[:'ltet-ly irth-' t
-i ih:ticarin.e from'il the -ta.'li:-.'iit ':f clia, r,:,:n ini-trrtctio.n
\\ood"J'- hrinm- 156, 5S8 i ith re'pe-t to -ex dJifere,:nt. ;inr.
unlikt- tho-_. titlt l'r<'-r:ntle, for in hi- inte-tigation the a,''ana1ea2
l.%i .ith tilt. L,'.: \\I h.i"the sci r :r- f r the l h':ile test .err:- tabu-
lite']. th, 1.-'.y- :iiirpasld th pir in all hailf-.'rad,'e groiips eXYC r
G rade II .\. Non'. : of t' he ,ir -irr:i'r. -r,:.. t:.r, v.ere tatisticallF
reliable. .About the inle rrlatio.-ii'li.c Uere f1:Iun] i. whe thii: [aliil.-
ti -. were rectrio-t:,i to s'.-:ar--. i-.l r.ga-irdle- of 'grade. lThe Teupe-
rio-rity of the b,'-yc wa- monirt: iiar-:,:-d in tl'- part- oif thle t'_-t d:l:etted
to ltimi -. rniion-y,y. -tt: H iwhler [,-r centc :I." _o, than ,..f girl- f'a-:,c
,;c h of the cigl, t itcmi- O:n tin;,_: in th- iJ ntiiication :,I ,.:,in
it-cin I the hoy- '.'Iunlrd, or ,urpa-:', tile t'irl- in each cae : rand thr
-alle \.Va- true on tile tt.l(:I int-sn relating to the co mpi arlr.riie v.tltei
In *ze'n,:ral. thte problem of -'e., ditxerenice: i of Ie- irntrr:.t nc ni..
than it ias -ome itv.',-, ,e]cae.- arind rinore -,,o. The diff-rt:ice-: found
between the uexre are mircih le,- in ari-'untlr til-in thi,: 'lirtte'r-- b>.-
t.-een1 inJi\idiid-als within n i\teIn x. That i- to -a., uu-all', nearly;
50 rper cent ':if :m- :.e. will be found to equal tlh'- nit-dian for thlk
ot!ier tx. \\'ltrre d]irft'ertnc-I- i nre f'outind, they are -li.-l:t aid i:an
rra,-'iiably he attrirtitId to diulcn.ii..i- in cultural itimilation, rather
than t,. in rini- Ji, l rff,:r. 'c, o:f a .,,l:,o ,: .,:rt
1J. Ci (\ ;'. 3. 3'.ii Rir-l C/iildr'cn
In the criminal r,-:ear.:h rpo'rte-.l in Parr I of this chipttr th,-
.ity cdl,!lrn -:rpas- i] tht riral children rather cs-:nliitintli in
the ariIthiiitic:ail fni.ii,, t:.'d. The akarima.-e lay,' cl,-arl.\ with th,
,-tyir\ cil]Jri in role o'inrmtii- L I' l:len-,_ratic-lI of en :iLbjct-.
.-lritln, tic ini thle Possession of School Beginners 53
rcprodu'cti.in i.f ih',e. six. anmd nine objects, crude comparisons with
cotncr.-te or pictuiredJ i.bjc'tr, exact comparison of pictured objects,
and the nunibcr contIbin-Ati,-'iis whether presented in verbal problems
or as the ab-tract facts The advantage for the city children was not
c.. Clear in lthe idenitifictiiin of four, seven, and ten objects and in
crude .i mparso;ns insm,. .ing abstract numbers.
NMacI.atchly (28 i has reported an analysis of the data on ten addi-
tiin cnmbin'ititons obtainedd from 1,226 school entrants in ten Ohio
cities. 1.123 sch-iol entrants. in Cincinnati, and 130 school entrants
in a itinumber of one-r,-.m schools. In every combination except one,
the cit, children :xcell:dl b, margins of from 6 to 17 percentage
p.inmt Thus, the per cents f.:.r the three groups just named on the
combination 7 + I *.'ere 64. 65, and 58; on the combination 1 + 6,
the per cents .-ere -49, 41. and 32. The one exception to this general
condition was in thc ca-e i.-f the combination 5 + 1, which was
"ni -. ln'" bi 91 r r cent i..f the rural children as compared with 72
per cent for the .., city groups. No reason either for this exception
or for the high deIgree of -uccess in the case of the rural children is
offered b\ the investlat,-tr.
The c:ml:piriaratie arithmetical abilities of city and rural children
liaie ben tr:ated in cther r,: ferences than the two cited, but for the
mloist part ,ulh rtfe2ren'c:e. and all of the research relate to arithmetic
at higher le:l: than that lihere under consideration. Future studies
may .,ir ma\ not co:inirni tilht superiority of city children as mentioned
in the frireci;iti: para.raplhs. but it is fairly certain (a) that the dif-
ferences fiiund \vill riit be large, so large as to demand markedly
ditfertent pr granmu -f instruction for the two types of children, and
b I that tlihe rc:a.ins ft'r thi differences will lie in unlike experiences
ila ing their orir:in in unlike: Linvironments.
15 L'iiffcireccs in Levels of Intelligence
Ac v. cill bi: ex-pected. such research findings as are available
reveal a po-.iti\ie rlatiii-n bet.'een intelligence on the one hand and
,ari,,.Li arithinretical abilitie o:n the other hand. Evidence with re-
gard tvi this r,:.lation is iurnirished in two studies in particular.
MlacLatchi\ 301 clazitied 291 six-year-old school entrants on
the liai- .:f tlieir latng, i.n thile Pintner-Cunningham Primary Men-
tal Tet., di idling th-im into,: tie lowest fourth, the middle half, and the
hihehi s-t fourth. Tl-h: rii2hre-_t group on the average could count to
27 and "'km.." tilI: ,Is of 6 8 of ten fairly easy addition combina-
ti...n. Tihe ',orri:sionding hlures for the other two intelligence groups
54 Ailn, tc .111 Gr0ah I a1nd 1f
were: counting to 21 arid 4.S umn, :. or the a\eira;e roup,. and count-
ing to 12 and 1.5 suri-_ for the -ln.est ;rou'[j The niedian superiorr
child could reproduce *'ii-;r .-_f Fe. 4rx. ,ever. eight, an.d ten ob-
jects correctly three .:nut of three trial-,. and tlie acraV'., child. unly
once in three trials. AbilitN to repr-lduCe number- was limited to
groups of five for the l:ia '.-t ;r(up,. and then ionl. once in th],;,
Grant's data (15) lhax been reported in full in \arro )r prec:.dnmg
sections. His subjects tcre re-arded as ":lll" if their l('s ,-r,- 89
or less, as "average" ii lect.v:.-ii '.0 and I10. and a- "brilit" if 110 or
higher. In every one of hIn comiparison the per cents .f accuracy:
increase from the "dull" tiroaiue the averagee" t.. tlhe "brihit"
group. The comparison c:,ier: identilicatin -.Af gri ,ups of three.
six, and eight objects: rep-ioducition of g: roups of f:,ur. se,-rn, and
thirteen; crude compare on: i -uch C:o-nc:ept as "ac long as.," "largest.
etc.); number combination-, pre'-rented ci-ncretel,' i ( 1 + 2. 3 + 4.
6 + 6, 5 2, and 3 1I and in xvrbal problems 10 2 an,]
3 x 2) ; the fraction a. alppilii.,a tu a rouip and t-. a single i.biec:t .
ordinals (third and s$.ri,' : reading, "mterrpretati,.n," and] % writing of
numbers, and knowledge of tiihree -e:mnietric formii-.
Wheeler and Wheel':r 154 I repa,_rt a :..efficient -.f crrelation of
.51 between M. A. and ability of unintructed hr -t-grader. ti, read
the numbers to 100.
16. Effect of Kird,f rriarIlta Instruction
It will be recalled that Ruckinehani anld NlacL-at:lih 1 12, 291
include in their report data from'i abut -one th, oucand schm:a-I entrants
in Cincinnati. Almost exactly tv. tlird> o:f thicse entered Grade I
after some time in the kindergarten. .-\ compari-un of the te-t re-tilrt
of these kindergarten children x itli tle re-.ultr a;f nofnkiindergartenir.
should provide some c idence un tilhe ffcct of the prech,:l.ool train-
ing. Admittedly, notbin. i L.v,,.f.n of" th,. nature oft this trarrin, in
general, to say nothin.i of the extent to,- which artlthmetioal skills
In rote counting by l's. 1.1 4 per ,ent ,:,f the K-zrtroup I kinder-
garten children) and 4..' per cent of ithe NK-gri.nup iarnkindJcrrar-
teners) reached 100. "lie wiedlians for thlie v.. .;io inrp v.ere 2'': 3 and
19.1. In rote counting, bIv l's t11 K-grro:up field C,.'.mpial.rle adlala-
tages. Of the K-group, 70.3 per cent enumnerated ta.,:nt\ Object;
correctly, compared v. itl- 47 7 fei cent ,f tl,,- NK-.r,:itip. Ten ob-
.-ntlhn'lic in the Possession of School Beginners 55
Ject, \;ere correctly enumerated by 93.7 per cent of the K-children
and bi 82.3 per cent of the NK-children. The K-children both re-
prdun.c-. and identified or named each of the numbers 5, 6, 7, 8, and
10 n;-ore successfully than did the NK-children.
The K-children gave correct answers to 5.4 combinations and the
NIK-chtldren to 3.5 when ten addition combinations were presented
in 'erhal problems. Their margin of superiority ranged from 5
percentage points for 3 + 5 (29.3 and 24.3) to 20.7 for 7 + 1
722 2 nd 51.5). When ten other addition combinations were pre-
s1intrcd "visibly" and "invisibly" the K-group excelled on every com-
bination and by about equal margins.
These figures bespeak far greater ability on the part of the
lK-children. However, no simple conclusion can be drawn from
tli~ee heures with respect to the effects of kindergarten instruction.
Buiickin gham and MacLatchy report that the average mental age of
the K-children was six years six months, compared with a mental
ac.e of five years ten months for the NK-children. Stated in terms
If IQs, the difference in average brightness was the difference be-
t.,-.een IQ 104 and IQ 93. Whatever may be the contribution of
each factor, the K-children had advantages not merely in their kin-
dergarten training but also in mental maturity (MA) and in bright-
17. Miscellaneous Studies
Preschool studies.-A number of research investigations have
dealt with the development of arithmetical abilities of children prior
ti: entering school. These can be but briefly referred to here, since
thlir value is greater for the psychologist than for the practical
teacher ind administrator.
In the developmental biographies of individual children more or
less attention is commonly given to the appearance and use of various
irithmlcical concepts and skills. Two such studies which deal pri-
nmarilv with aspects of quantitative development are those by Court15
I,nl by Drummond.10
ID)'uglass (14) administered three tests to children aged four and
a Ial to six. The first test consisted in recognizing the number of
S.:r.hie Ravitch Altshiller Court, "Numbers. Time, and Space in the First
Fi,' \Y,:irs of a Child's Life," Pedagogical Seminary, XXVII (March, 1920),
71 -S'. Also by the same author: "Self-Taught Arithmetic from the Age of
Fin-. t... the Age of Eight," Pedagogical Seminary, XXX (March, 1923), 51-68.
M igaret Drummond, Five Years or Thereabouts (London: Edward Ar-
rn..il1 ,I.] Co., 1921), pp. 119-135.
5'. .-pitiii'ic" in Grades I and II
do]tc in pattern-i of ditfielet e ize- : the -cond, of sclelcting certain
patterns a- represcnritin the number- announced by the experimenter
the third, of estimating' the number of marbles expo.etd 1-r,.rmientarily
by the experimenter. The conclusion drawn was that the experi-
mental children had completelyy accurate concepts of I and 2. very
-erviceahble and accurate concepts of 3. and a very vcriccable con-
cept of 4. and of 5. 6. S. 9. and 10 rather vague concept:. though
c.rviccable to a slight dc-gr.e."
McL.'iiiulin 1 33 i ia\c indi'.idual test- to 125 children enrolled
in nur-ery 4chool- and I- indcrgartcns. Their aces ranged fri-im thirt'-
six tV -cventv-tv.o nm-ontI. One seriess of tcts measured ability in
rtCe counting: an'-ther. ability to recognize mall aorgate, of oLb-
jects: a third. albilit,. to comil.,ine nuim bers tinder varying_ conditions.
Thie data. reported onlv in part in the reference cited, arm. tabulated
lr three age gr.:iup. Children a'red thirt\-i:ix to fo.rt.-eight riionthc
could count by rote to 4.5 on thie average, c. iuld enumerate an aver-
age of 4.4 object-. and could counItt back,.ards niot at all. The cet.rrc-
s.pondin:.-, limit- for children aged fort,-eiht to s]ity month- were
17, ), 14 5. and 1.6. re-pectively: for the oIlder children. 33.4. 2S.2.
and 5.5. re-pectivel Practically all :f the hv;e-vear-ol.k could reEu-
larly rcogri:e ,gr: up[,c of t\'.o and three obiect.. and mietimes
groups of four. The combiniiii of inti, her '...a- :..' i hard for three-
e\-ar-old,.. oas soim.. ihat easier for fo-mr-'.ear-old-., .hi-' ui-ed counit-
ing pred.iminantil', anr.] v a i much casi er for i'.e-yar-old \.h. utied
countinm, to supplement groupin.-!. The ref cr'r-nc co'.ntains a fairly\
extensive jdisc sioni of the r,.latio,:n b t.t'.e,.n -uccess ,:on the nimib'. r
tacs;s inpom-ced An'i the mi iital prce-.e o:f the children in arri,.i,
at avic r erc.
.\bilitv to compare cr.-jiipm- of ob.icts numb:rin ten or fcocr
v. :i studied int>,nsi4.:ly bLv Ruisell 142 ). Hi- ucibjects in pon> e:peri-
ment ".ire thirteen I.:jicrygartener., [.' .1. First-_.raid,.rs. and four
seciond-graders. In this m.xperiment the children v ere in-tructed to
tell I.hich of t',-, expo:,sed groiip o ,f ojects a-, "more." \'ariatimon-
,,.ere introduced bv. u-iin bl cks of different size-s and of dit fere.nt
colors i Th-e chan.iizc se-ned t,:, c. 'niuse- the children, a fact v.hich
i- perhaps best interpr-.ted (ithoiigh it i- nI -i o interpreted lb tihe
,-xperiiijmenter ai meaning_ that to the: lubject- thle -ituation had
really cea-ed to he a 1iuaniititatic -one in the mOrdinary -seie. 'Birnet7
Allre. Bii- e "[.. Pe'rccpr.i;ion .k,- L .,,;gu, r-4 tt !zt. N...m-L.re ." F .:'i :
I'i' l. .,,:/,,, ,.-" X X X Il ul.,. 'i ':.- 1.
Arithmetlic 1n the Possession of School Beginners 57
nearly fifty years Lb'fort hadi noted the same phenomenon.) Russell's
second experinient in ,old t.'.iicty-five subjects (ten kindergarten,
ten I.rade I, and five Grade 11 i. This time the children were re-
"equal"' in szc. the word i"more" in the first experiment having
proved to furnish fals- le:i. Russell concluded (a) that "many-
ness" is tih fir't iquantitatke concept, (b) that cardinal and ordinal
number con.cp:ts de.elo,'p toetLhr, (c) that counting is not a re-
liable measure .:4 this dLcleopmenI'nt, (d) that children under five
\ear- of age understand "'most." "both," and "biggest," but not
"eqt:l." i e that 'eien-\.car-old-; know "more" but not "same" and
"equal" at all fully. ind I if tlat counting as a means of differen-
tiating grro:,up. d\el...pz late Tlih last conclusion is understandable,
-ince cou:,ting is hardly as direct and serviceable a procedure even
%%ith adults a. the matching of small groups which can be really
apprehended at a glance. The author's views with regard to the late
development of counting seem therefore to be in error, since his
experimental situation-'-s \ere such as to place counting at a serious
R,'aeicss tcstian' -The Metropolitan Readiness Tests were ad-
ministered h% Hildreth I 181 to two groups totaling about one hun-
dred children in the lirst month of school. A year and a half later,
hlicn one -.f the-.e groups w\as int the second grade, they were given
a special arithmetic test designed for the program of instruction to
hllich they had been zubJected. From Hildreth's description this
,rogram appears to hai. been o:,ne based upon "activities" in which
considerable: u.se wa made of number situations which occurred
natural' in the classroom' Comparison of the readiness and the
aclidveinent te't corer. for twenty-six children remaining in this
gr.:>up yielded a correlation ,:olicient of .50. The other of the two
group, then inii.Cirinc, thirtm-three, was given its arithmetic test
two and a half \ears afte-r takl-iii. the readiness test. The coefficient
of correlation in thi ca.e as 5; 9 Neither coefficient suggests much
useftulness for this particular readiness test as a means of predicting
achietementit. hio,.cier %aluiiile it may be for inventorying purposes.
Hildreth points *out reasons \.h. the predictions of the readiness test
%\ere no higher. but does not mention the fact that to be most effec-
tive the readiness t,:t nimu-t .e closely keyed to the program of in-
structinn which i to: folla,'w. a' closely as must the achievement test
by Which the eftects of that p-rogram are later assessed.
58 .-,'uliiiii c,' in: 'J a..h. .s I and If
PART Ii AM I F.*i.AL OF f E: \- F1'. iDNiN.
tric l Si lUaIll V o'f FiJdi,'..:s
The research i..n tle arihlie.tcai kin,,v. !ed.4e and -.ilk .if children
just entering sch::il in ,ipre-tive b.:il i in it extent and ii' the fac:t:
which it has revealed Partc I and II -.f thil cliapter uiinnmarL.e
data from twelve -eparaLtc in. etltiiai.n relating' t:. tv.el\e different
arithmetical topi':>. A u ill be p,.:rited oi.t in a later -ectijon. -nie o01
the studies are (,pen ti. ,l1e:tic-i .' .t *Aill ri.urn ..r an'.ticr ncvc\r-
theless, the fact rcain- that children .. hen the\ come It,-. i ch.-..-.l k:nF...'-
a great deal about iinmdel it i wi..rlh .hilc tI. cla-i-:v the recearcli
findings in three catc.jric. ,lav.;y,' rcmenimber.ini that certain limi-
tations attach to thii- rtc-arch
1. The follow in'z -l:,ll .,i. c_.. ,cer'- eei' t:, e qu;r,- ;..1 ell I.e. lo[ip
by the time most cliddilren -t.rt -ch,:,l
Rote counting I. I'.- throu-.. h i'i at lea't
Enumeration: thr.:,i'h 21.1 ,it lea-[
Identification: tlir.ii.l I.I at le'-ar i the Imlit t ,..I.ed. in re-.etr':li and
probably tli:u. u-'h 2'l
Crude comnpar-:.:l a .,ith l:c:t tih coicept- l.tiiet." "inidJ-
,lie." 1:',.-1." ":h,,rte-_t." "' i niller "t.. lle' ,"
I i ... ih tl..:Itract iitiber- 'm,:,rc." th thl e ntlnl-
ber: thrr..u'h b10
Exact compari-,:i .:.-r miar'lCiinz Icact tliroin.-h 5 :,r 7 i l-e limit of
Number combiri *ii i. dt ,:bect- t-o c-in- of Il1
tions: ill erL ajl [prolh:iin v. n1th e-,zily i Iiin."iied ob.iecit
.ln, etuati,:.l-. ;,,JJil', I nd _". l, In,| pr. .l, dl,l y
,o,-t f V1'. t: 1- ith nl ii- t.:, ..r "7
Fractions: unit Fractioni thir.on li lIial~e. aiid tIiurrbi .i- a, pli'Il t)
single objct-, an] perhli. h lh' e- a-: ue.] ih *small r.:,u.. ,n
Ordinals: thrcw..b o i
Geometric fig-ire: 6rcle" jii -iuari"
Telling time: at ile .:,ouir
U. S. coins: rec,:.: ,ori of -ill coi m tl-he half Jollar. r1..I ,, F.
understandir-' -.f reliat e ilue .f [.c ii.:' ar.J] other nii'ller coin-
2. The follow' skill- aiilnd :oncepr.- ire i-..t -o fully I .. n to -cl:l, :dl
entrants, but are fLi'rly v. l -tartedi amo:.-' a rei-.:il.ly l.ir'.e per cent ri
children of this ace:
Rote counting I '- t1 1111
liv i I' t, 1111.1
L.. -.'s. t, 2d' or 30
.-ilinhctic in the Possession of School Beginners 59
Crude co:ri.ir. :,n: a i with objects: "as long as," "fewest" (or "the
b i with abstract numbers: "less," with the ab-
stract numbers to 10
Number c.iiibina3- :, i in verbal problems: probably all the facts with
ii. .n : sums to 9 or 10
b i with abstract numbers: few research data
available, but apparently less than 50 per cent
able to deal successfully even with the easiest
facts (e.g., those involving the addition or
subtraction of 1)
ReaJdint number : only a few know the numerals to 10
3 The folo'.' in- slill and concepts are possessed by less than a third
*-iI i.',el -rienr:.iiint., ani then in limited degrees of richness or proficiency:
Rk-tc c*.-tM inz. : l:,' 3'_, to 30
Crudle comiipari on objects: "same" or "equal"
Frariti' a I proper fractions other than unit fractions, ap-
plied to single objects and small groups
1. i improper fractions
c p relative size of fractions
Rei..nrr an.l writing g numbers: virtually no ability to read beyond 10;
\rtu:.dll i nonIi I.'. rite, even to 10
Ge.,,metric iguri-. "triangle"
Telling time- at the half- and quarter-hour
U S mii,,ne' : relIati: value of coins other than pennies
Liqiid and linea-r measures: relative size of units
Rc:-':arch has contriluted little or nothing with respect to several
traditional t.oic: in ithe primary course of study. For example, crude
c:':mpar.I ins are s.,nmetimes made by the use of such terms as
",:'iiiiet." "ol..let." "heaviest," "lightest," "darkest," and the like,
*:On nii'ne 1if .lhich are there research data. Likewise, research is vir-
rcallv -illent in the matter of the subtraction combinations. Too few
:if tllh:c have 1l.,cen included in inventory tests to reveal first-graders'
familiaritN uitl thint. At other points, for example in the case of
I ractirim. rei-earchl secms to indicate possible success in teaching
ide-cs n,.' *.ener-illk '[ withheld to the later grades. In such cases more
re-eard;- i. niided. to determine whether the inferences drawn with
reg-ard to, rhi pr-.:.l,able effects of instruction are sound or unsound.
Limitations of Research
Iin the -clitnce immediately preceding the lists of concepts and
skill :|aT,<, -. ca.tiu,:,n was stated to the effect that these research
ending!. mu-t ni:t be accepted too quickly. The investigations which
.-ritliintic H 'i Gra&ds I and II
have been summarized are riitmetime li hmted in ,alue becau-e of
errors in technique or of n' utficient cc'.eragt.
In the f.rst [pace, fe\ arithmetical .skill- and conce-pt' ha'.e been
at all exhau.tielh v tudied. Reference here i toi what may be called
the horizontal dimincnion. Mlo-t skill; and co'ricepts have bt-cian merely
sampled, which is ti, -_. 2 that they have been studi'-d only in part, in
but some of the -ituations in which they function. An example i-
the number cominlination-, ,nly a few of \lhich I'.e been included
in research ineri iorkes. The fec, concepts and s.ills that h12e Ieeni
studied more comn.letel:,, for examrnpl., coiuntinlg irid enumeratin,
are among the ,io'lTet and leait cn.mplicate i in th eld of arithmetic.
In the second place. a rather large part of the data ha. buen col-
lected by means of group test. Tht ttechnitq:ue of group t-twing vI
somewhat Ulki.crtain l|articulrlv\ "itlih pupils a; immature as chdl-
dren just bc;innin. Grade 1. EBery fir-t-grade teacher knoi\\ the
difficulties of conttroilling attention ..f all pupil, throughout the test-
ing period; he lkiov.s als the extraneous errors mtroduced by in-
ability to understand and to foll:,. direction. and to tnter an 'r.
in the right place. Group testing i a xe for rniltipli.-choice types)
has, however, '-on. distinct adlantag. : the rt-ult- obtain d almost
certainly under rate the actual cxtirent of kno\riid'-]c. In [art, but
only in part. tlhi- advantage compen-ater for the incomplete saimpnll:r
of concepts and skill as these are rer'resenited in tlh tt.':..
In the tfird place, in ery tfe invest I;atiin4 haxe data been ob-
tained on children' pIrocedures in dealing v. itl' tiS c number task;
tested. That is, experimenters report the an.: \ers children give, but
not how the antl w,'r, have been arrived at. 'Of coMurs. in the caL e COf
rote counting arnd enumieratiiin children necessarily re.eal their ro-
cedures, bu: nt :'' itn .Kuchl quantitati'te f.atw an compilaring number>.
solving combination.'"? e-timA.itinsi le ngth:,. and the Iille Yet. tEie
procedures u;ed by children are eu.ential to a true :apprai..al -'f their
developmental StatuLs. Correct arin .'.ers c-an be obtained by immature
procedures; incorrect ansi ers may merely mean iiicompkte coin-
mand of mature procedures. On this a-ccount future resear,:li rma
well include thI te:hniiiiuc of careful ob:errataioni and of the inter\iie
18 The co-r..j-'.xit. f nI ,n I .l.' iii... i.:.1 Ir mc-mn r ful na-t, ry .4 the
number combl.:i-ri..ns Io b.n anal ed tI.. Br.. .-.r-II \\ill i A.m A. Pr'.,'-rll.
"Teaching M caning:.' F.,'ir ri.d iii in .-1,,a 'i';, '.- i Eullitin :.1" hir A -s.:iati.:.n
for Childhood Elduca n.r. I'." [P. I -1. Th,.: holee bulletin .. ill L.: 1...un.:l
of value to teach rs int,:r.-t..l in .:lv :l..' ,.ii qunr 'im atli .' il'alinei -.
.Ariiitmehc in the Pisse'sion of School Beginners 61
or conicretice a- mcans o:f s'upplemnenting the information procurable
thriiugh group testing.19
Cautnons in Applications
The research findings .summarized in this chapter have been ob-
raincd friiii pupil- _upp,:,,-.dly t~ pical of all children who enter
GriadL I \tar after \ear. This a.iumption may not be valid in every
ca.e: a.tpical clas..,c mnay unint:-ntio:,nally have been selected for test-
ing. But if .o. the error i- not probably as serious as that arising from
.a L second assumiiption. namely. that the first-grade pupils of a single
clay., or .:.f a community arc ncces..sarily comparable with the research
:ub[j::lt Otn thel c,_-ntrarn. a local entering class or the classes of a
whole system liay be quitn unlike the experimental subjects. It is
therefore un\, it,4. apart fr':m surp.':.rting evidence, to regard research
finding-' a- directly applicable t, any and all school situations
Research findingi. tell the teacher or administrator little about the
class as a "hole. but they tell very much less about the number
abilti:t. .:,f particular childir-n. If educational psychology has estab-
li-hed any2 fact, it has emstabli-licd the fact of individual differences.
Instructl.:.n in Grade I land fo:,r that matter in any other grade)
thouid ble l.aerd up':in accurate knowledge of the abilities of each
npupil in the clias. On this account careful inventories should be
miadle :f the arithm.tical concept- and skills of every pupil before
t.achingii. i. undrtakie n. The inventory need not-indeed, should not-
c,:ver all coniccrits arnd -kills and be completed all at once. Rather,
the ihnentory rl. would be carried on progressively, by stages. Thus,
inf,:irniatio:n should firt be obtained concerning ability in the first
arithmetical -kill: ,:ir concept to be taught; the children deficient
thi.r:in can then be formed into a group for instruction. Meanwhile
i,,r aiter,.ardsi tie tiiventory can be extended to cover the next
t.:[pic- In this %%a\ children can eventually by small-group instruc-
rtio n be Lroought t.-i the point \hli-re the class can be taught as a whole.
\'" ihr i pr.I.ably thil rr- th.- r.-rugh and penetrating study yet made of
ili. r.imn:.. **.i rrim.nr, r e J.: rhildr>:n for instruction in arithmetic is unfor-
Sinr t.elt n.:.t a'ailA'lc, f..,r ihi: .umm:ir.. It is the doctoral research of Miss
L':.ris Carrpr an! nma. .:ortl., 1,,: :tlined from the Duke University Library
un.k',r the til,. ".1 Stu.I, ...t S':-m.: A,'.:cts of Children's Number Knowledge
Pri...r t.-. n rlrucrii.:.n i- Carpl r I,:res:lf interviewed 270 school entrants on a
C-niiderr.'hhl r,'n.e .-. f numlb:1 [a4:k.. and her report contains not only the factual
dla lta e c.:.1lrC, but l1:.:. 3 .:a .rhihng iniuiry into the procedures employed by
childlr. in arriminr i at their a .ert Evidence at this point was oltain':lJ
c.:ure i* r-..i tl, .._i, er:': action' nd l iJ .,ing which accompanied the :" .Ie
62 Arithmetic in Grades I and I!
Significance of Research Findinos
It is theoretically possible to take three different pi:.-I ,:tn with
respect to the capacity of first-grade children to pr.,'Ft fr.,m itn-truc-
tion in arithmetic. According to the first positirin. fir.,i-yra.le liil-
dren are too immature for arithmetic; they simply arc tniabil t:. learn
arithmetic. Considerable doubt is thrown on this r'.,si: b, t1'c
facts which have been assembled in this chapter .\.lmitt-dl\. tthec
facts do not completely undermine the position: i...nl direct pil"..f
that children when taught arithmetic in Grade I actual l klari arab-
metic could do this. Nevertheless, the facts cc n:cernin, ,ch..::l en-
trants' knowledge of arithmetic warrant the rather c:.rnfid.ent ilteleni:c
that systematic instruction in this field should yil'l g,''l hcrturnis.
In the absence of more direct evidence of an experimental character.
it is not unreasonable to believe that children v.h:. already kni.,.v a
considerable amount in a given area have thereby demio.rstrat:l their
ability to learn more.20
The second position is to recognize the fact that Fir-t-,gradcr.,
already possess a considerable fund of arithmetical kn.li'-. kd, ,i and
hence are probably able to extend their learnir.,. I.ut t.' il.trtl-t it
"maturation" and to incidental experiences the acqui:sit,:in Of further
knowledge. The argument is that since children ha.t already don'.
so well "on their own," they should be allowed t.:. .r.ntiiine their
learning on the same basis. Systematic instruction is n1-.t nee,-de aind
may be very harmful.
Nothing in the findings of research is in conflict v iti this \t....
As a matter of fact, these research findings arL hardly r
the issue; they tell what children know and can d-:. ii tlicv c,:. c
to school, but they tell nothing about the way childr,.n have ::iir
by their knowledge and ability. It is possible ti7. infer. a- the sup-
porters of this view seem to do, that children. lefc t,. the icl\czs.
somehow grow into number knowledge. If so, thiire is t.',r\ r'a- .ii
to believe that the passing of another year or tiw.. will i,- ..',: Pr-
tunity for more of this same growth.
But number knowledge is hardly the result ...f ani ', such gr-.'. th
or inner maturing: it is the result of directed experienci.e, fitnequnti[
taking the form of direct teaching. Woody (56, 58 hai s rnt i. [l
certain facts which have not received wide enou';h attrnti'.n '. u.s-
tionnaires were sent to the parents of the first-grade children iii ite
20 Dickey in a recent article has rightly challenged tl. it:rnlnc. r,., :I:,:rt
this inference uncritically. John W. Dickey, "Readines; in .A\rl.rnti.:." n .'-
mentary School Journal, XL (April, 1940), 592-598.
.-Ir4ittci:l" 1, th, P'.s:.
een elmtirenitai', chilk ...i f .\nn Arbor, Michigan. Replies came
fr-iiin a i:1l of 164 hliInil Eighty-three per cent of the parents
rpll, 1n -ittd that inli thel honie instruction was given in rote count-
5in t 54 pir cenit i.f tit parentrt. taught counting to 100, for example);
78 per c.r.it of ith1- parnti, caught their children to count objects;
87 per cent. t:. recogni.zjl coiris: 68 per cent, to know the value of
imnev: abot iv: tlird]. ti,: read and write numbers and to solve
simple v-vrbal pr'obltiem. : abolit one half, to understand the size of
,nmllr,. t,-. 10 p,,rfrr', r-.pi ad.lditions, and to tell time; about one
ihird. t'-. deal with -inidple Itnations involving subtraction.
It \i.iii.l be I-:. -i-l- t:o. i:ntinue to absolve teachers of the re-
[:',on:ibilit3 f:.r mliumbr tachin- and to leave to parents the task of
enoicjuragiii, further d,.vel.oppniit of number ability. But if Woody's
it.ctit mea anythi.liii. ti-' rerai: that some one must assume this duty.
Mere att.aiiimtlint ,if mirit anld niore birthdays cannot be expected to
hrtine th- iint.JtJd cica:e- iIn Inumber knowledge, apart from directed
e rlr.c r/nce it .;iid nit of course be inconsistent with the second
i,:.sitt:.in t.:. agrce that this re:-,:nsibility is the school's, but to deny
that it -should I,..: met inl thlt primary grades and to insist that it may
more ea:il. iand pr._Fitabl bIe tmet in Grade III or even later.
Ti : psiti:ns w't'i re.'ardJ to: the ability of first-grade children to
learn arithmetic Iiae h,,en considered. The third position is that
ithe di 'v :f c:
i prprl tl',e funclti-_ii ,:f primary teachers. Long before this
1ioi:it the n.adcr ma] havt_ dt >.cted evidence that this position is the
,-iie entertained b, thl: a riter. Briefly, the position is this: Research
liis s',: :n that ch::OIl t rrantr already know much about number;
the inference i. that ihe can learn more; society requires that chil-
dren niu-t kr'..v arithlmictic: nothing is gained, and much may be
li.tt. if thi. ch-oI .' ila t. later grades the discharging of its
,1i) i 43ti I/f
.\t later p,.iint. inl the m:onn,-,_.raph, particularly in Chapter V, this
p:ti..t.ii ill I. d'ci cl.i-ed and supported at some length. At this
p,.it it 1niiiUt uft:ce to .av that one does not need to advocate a
return to the unintclli-ilit-. al,'tract drill now happily on the way
out : there are other li1ndi of: learning than memorization, and other
Lin,- ,:,f Ikarvin activ.it .- than repetitive practice. Provided that
x-periernce are adju tiii.' t: their interests and capacities, first-grade
Jildren *:an an J ill :.\i-end ilheir number knowledge happily, intel-
li'gcn.tl. aud utitfuill,. Fx:lcniice in support of this belief will be
iprcientel in itle i/ext chapter.
RESULTS O)F A PARTICULAR PROGRAM OF
SYSTEMATIC ARITIIMETIC INSTRUCTION
IN GRADES I AND II
The inference dranii fro.in the research findings re netted in
Chapter II is that c-hildren in Grades I and II are ready fur sy\tem-
atic instruction in arithmetic. The data re'ieu ed di. nut in themn-
selves prove that :hi. irnstructjion if given v.ill produce mera'urable
evidence of sound and xaltabc le warning. f,.r the data relate only to
the arithmetical equipment v.ith liilh children -tart school. Pro.:.[
that this inference i valid imust co:me from a different kind ..f re-
search. It must be shown that in-structicon actually d,:.e yNield the
kind of effects which are desired.
It is the purpose of this .chapter to report the result obtained
from a particular pro'._ram for primary grade arithmetic. a program
which, after two years :,'f experimentation. %\as finally employ 'ed
under circumstances 10 which permitted e'aluati:,in by means :f te-st.'
It is not the purpose .,f thick chapter toj try to pro.ve that this program
is the only possible program, no.r that it i~ tile be-t program available
at present. As a matter 4,_f fact, there are probably many' different
programs now in u.-, and s:.rmie of thm may b.e LettLr than that here
tested. The intentio.in in thi ..hapter is. rather, t,:. dl:w v.hat ci.; be
accomplished when rcas.:'na-ble: outicomnj are -el up, half-grade by
half-grade, and when appropriate pupil actiiitie- are dico:xered and
arranged with a viei t.:, the realization of thiee -,oiito'rnme.
In the writer's ,pini,..n nu all'gy i needed for reporting com-
pletely the results .f any pr,:,ogra3n ol primary arithmetic As the
reader well knows, the literature co:ntaini tmany -tatementcls :t thie
effect that children in Grade I <,:.r Grade II i can i or camniot i learn
this or that, these ctatemeriits generally being based upon notlhii:z
more substantial than .oplinioii _*.r limit-d expe
experiments thus far made. available -;ipplly et-scntial information
neither with regard to ouitonite- asumied nor '.ithl regard t,. inritruc-
1This program is built .around Teachr-' Mlailals and pupil,' v.-.rl-b:,,:,ks
and readers published u,'-dkr li, giencril title .-.I ".:ull Number'." as part :.'
the Daily-Life Arithmeii.- Seric- i:.r Graidc I thr.:.uch VII inclhiii.c Thi;
series, the work of GOu T. F.u-vell. L iiin,.rc J..hi. in'd ilit v. writer i publi lbed
by Ginn and Co.
Results of a Particular Program of Systematic Instruction 65
tional method emplo,:ed. Lacking this knowledge, the reader is
hardly conipectcnt to, pas- critical judgment on the significance of an
inle'tiigation. and certainly lie must be cautious in accepting the evi-
dence either as prtoinrg or as refuting the case for systematic arith-
metic in Grades I and II. In the long run, programs should be
evaluated. not a;s .holes, but rather for their success or failure in
furthering certain end- Programs which at most points are very
bad indeed may [ie excellent at other points, so good in fact that these
features should bt ad,-pted and widely used. There is every reason
to behete thai~t from experience with 'different programs will even-
tually emerge a program far superior to any we now have. This hope
for the ultimate -mergence of a superior program, and not the desire
to e-tabli- special merits in the particular program here under in-
v.c-tination. is respo.:nsible for this chapter.
T i 17 i ., N OF INSTRUCTION
Th-- plan of instruction followed in the present investigation is
fully described in the Teacher,' Manuals which can be consulted by
interested readers. In this pla ce, therefore, only the most important
aspects of the program are outlined.
(O.tficoi'..-Tlit chart hliich begins on the next page contains a
concise -urnmmary of the outcomes by half-grades. The reader who
it acquainted v. ith the traditional course of study for the first two
grades will Find it this chart many familiar outcomes-or at least he
will tend to identify' those *inen as familiar. The outcomes with
rerp[ect t,-, counting and enumeration, for example, are not unusual
for the primary Nrades Moreover, it has been customary to assign
some part of the number cminbi nations to the first two grades. When
thin has been done. however, the course of study usually has required
notlnii short of "'master; In this program, however, the term used
P1 intelligentet control over." and the difference in terminology is a
matter of con-iderable moment. "Intelligent control over" implies
that children mLu-.t lie able to deal with the number combinations in
smlie eftectri. sen-ible manner: it does not imply that children will
from the start. or even very soon thereafter, have the ability auto-
matically t, recall correct an;s\ers as does the adult. Such mastery
is %iewed as tle product of a long period of growth, the result of
dei eloipment throu'.lh a -erie- of stages in thinking.
The pre-cnt lit of outcome s differs perhaps more noticeably in
the pre-ence oif -such term, a- "understanding," "appreciation," "so-
cial valued:." "dispoittn to iise," and the like, all of which imply
66 Arithmetic in Grades I and II
that the arithmetic learned must possess meaning and apparent use-
fulness for the learner; the child must see sense in what he learns
and he must have experience in using what he learns.
Methods of teaching.-In general, teachers who follow this system
of instruction rely little upon telling their pupils and much upon
showing them, or, better, upon having their pupils under guidance
make discoveries and then verify those discoveries for themselves.
The authority made use of is not their authority as teachers, but
rather the authority of truth revealed by their pupils' own practical
concrete experiences in number situations. Understandings and gen-
eralizations evolve from pupil activity; they are not given out by
teachers as neat formulations arrived at ahead of time.
Perhaps the clearest and briefest way to summarize the general
principles of method which are recommended for this program of
instruction is to quote directly from the Teachers' Manual:
1. We must insure orderly development in quantitative thinking.
2. The child must see sense in what he learns.
3. The child's activities and the purposes of arithmetic must harmonize.
4. Meanings must precede symbols; understandings must precede drill.
5. The way children think of numbers is as important as is the result
of their thinking.
6. We must teach at the rate at which the child learns.
7. We must present arithmetic as an object of "natural" interest.
8. Instructional materials should be organized spirallyy."
9. Children must know both what they are to learn and how well they
are learning it.
OUTCOMES BY HALF-GRADES
Grade IB Grade IA .Grade IIB Grade IIA
1. Counting and enumeration:
by l's................. to20 to 100 X* X*
by 10's................. .... to 100 X X
by 2's................. .... .... to20 X
by 5's................. .... .... to 100 X
by 3's ................. .... .... .... to 30
2. Understanding of place value
of numbers in the series to 10 100 X X
3. Understanding and use of
the ordinals to.......... sixth or eighth tenth thirty-
4. Reading and writing numer-
als to................... 10 100 X X
5. Understanding of significance
of 10 as the basic unit in
the larger numbers to ... .... 20 100 X
6. Recognition of regular
groups of objects to..... 6 or 7 9 or 10 X X
Results ,f F'.rti,,cular Program of Systematic Instruction 67
intelligent c.:.ir.,l ....,r the
I inL ber c .m l.in ri ...r .n-
ihr...hph . 6 9 12 18
1St. lclient c.1.ni-I . .,:r the
U1-.: .r. ati.:n .i .. ........ X X
i9 ..R:afin -:, a d .* rating c.:.inbi-
lia i.i ii. crticalli and ,itr-
izrniall thr...ugh........ 6 9 12 18
1'.1. LniiJ r-.tanling and i.e :,i
ih.: ir ,c,-e ;' i *. ddiii.:.r .
and _.i..itract,.:.n X X X X
II Lndtr ia i id .:>if rclan..rn-
hi r. lct ..ccn .l.1 .' in aind
-ubtraCuli-n and inc ci
the addi[tirn combintiin_
rand ll,' re-latd *liitra.:-
ti,_on c.:,IVmbinatio)n4.... .. ..... X X X
-ujt[rai.i.ti n, cjrr.'rin
...r l._rr *. *img. urn aind
rrinuiii nd i.... .. .... ...... 19 99
11 .' iim r. and -.._riz...n. 3] adi-
t...n Ii m I..ire ihtni i,.-,
riumbnter.. ilh -,nd ..uil,- 3 digits, 3 digits, 4 digits,
out ) .... ... .. .... sums to 9 sums to 12 sums to 18
14 U. .:,f fracrti.n. a. al.plie' J
t.i sing i tlic .is and] I ,
gr.,li.- .:,l .. -ic ic s Icre-'
,d i, i:i.:,n . ........... .... 1,/4
13. L;u .,:.l nuiii ,b r in ,:nn.c- U. S. money X
ti..ir v.ih. ... .... Telling time X
PI kcauing andJ ratingg R:i.man
numral ..... ..... .... ... XII X*
17 Finctitnal r.. dir ip :.ca l,-
larv .,i nunmitr u,:,rdl and
:.'.t.-mb l, .... ... .... X X X X
KI~. A. .pr:cci:iio.)n of ':':,ial alucI
.:." aritlh ti ic. li4-..-inrn
tI' u-, 3 r ri n T'ei. and
<;v.tcrrnte ,) t. I'arn mn.:-r
ariih,.: ............. X X X X
cs.*-,,: ,ih the i .t .kil ...r o..i.cept is extended, enriched, and/or practiced.
,aat/ ria1 oif istruntic,.-If natural and planned number occur-
ren:ce- in the clas-room may for convenience be classed under the
headin "inaterial: ,of in-truction," teachers in this investigation had
the f',:ll:,win;Z suircc, iipon A\hich to draw: (1) Teachers' Manuals,
I 2) 1-iipil-' :orkb:,ok ,. (13 pupils' number readers, (4) unplanned
number ,ituatiions -.lich appeared spontaneously, and (5) planned
or prearranged iuiiil_,r StiuItions.
Auliihi'clic iu Goades I aiid II
(1) As has already been explained. e\ery teacher had tl- manual
for her grade. The'e manuals are unuisuially c-.mplete. ,','itainin_
both theoretical discuI.si,',nr t.. mak' clear the undrl\ ing phil:s,-'phy
and practical sugge:ti',_.n '.ith reward to teaching procedure: an.]
materials. In the ca.e of the la:t three half-grade.? Grade IA
through Grade IIA) tlli page- ,f the pupils' :.r'.b:'ik, ar,: rpr--
duced in the manual. ,-, that te-ichers are able t,- relate the manual
suggestions for teaching : dir'ctl, I. thil page t,:1 which they c,',rrep:on,:l.
(2) Pupils in Grade IA had a ',,rkih.-,k ,'.f e\entH-t',,- page;.
about twelve of which c'nained review. au.. testing niatcrial andI the
rest, developmental material Thie '.wMrkb.k f,-.r Gralde. IID and
IIA comprised eighty paees each. .:., ich a-ut t'.'el' c are dIe.'. ted
to reviews and tests. In Graide I I thle pupil. Ih:ad n.' '. irkb.'..ks..' and.
accordingly, their teacher, made much grc.ater u.e ,-f i'4 and t'5
in the list above.
The typical page in cach w':,rkl"'i::k tart ,ithl an interesting pic-
ture, especially designed t, re, cal thle number idea. and relati'.n-hip:
set for that unit of inst.ructicni. Chil.lren are expete'.l under the
teacher's guidance, 1,:' arrive ,at ;eineralizat.i -,o :,r relationship oun a
concrete basis and to exp'-,c-es their i.hoi-',eries in tlhe lian:ruag,': of
abstract number. Space are left in tilthe: '-rkbl_-_".k p.agc- for entering
the results of the vani.u,, nuntmber .\xpericnc i. The intention is int
to hurry the immediate mnmiirization i.'f thll fact, dis -'; ered, 1it.
rather, to show number a- a dish:rtthnd methlicd .:f translating and
recording both the quantctative pr..cCs:- uti:'d and the r,:-_ult: -.f thil
process. As will be Lx.'plained bel-i'..' under 4 and i 5 i. \'.rkb"--k
lessons regularly foll' v experiences :4 a less artfici-il character.
(3) Some, but by n,'' means all. f thie pupils. in thi mn' c igan-ti'in
had number readers,. ihich in their c'.ntent parnillel tlie dte\-,:li-,pmlnr.
of number ideas an'] relaticiinships ,jutlined. in tice manuals. Ther-_
storybooks are planned t,-. slhu:, chii.ldrn that number is a natural ,r
normal element in readindm matter. he I t,-,ri,- c'intain numll..ers and
quantitative situations in iun-' ttru'Liv ',a\ -s th cli j' o n,:-t interfere
with the story proper. .\fttr readini2 a story. clIldren comine t. nuni-
ber questions, they turn back t, thlie apr,',priate c-'ntext. ,cl-ct tl-,i
relevant quantitative material. and v.]:rk :uLit the required rcelti'n.l;ip .
(4) and (5) Each nw% phase '-.f devel- *p.nental in tructi,.,n -tart.
with an actual numLer cxr.ercincr a.ipart from ,:,iiryb',.'k ...r .:.rkb:l:.
SA workbook for thi' half-grade has sii-ce 1.e'r .,uhhlhC.l un.kr ith tilt
Jolly Numbers Primer.
Jolly Number Tals. t...,. (_ nc awr.. ./.:/Iv \,Vi !.,-, Ti, ,.' Ta"o i.:.r
Grade IA and the whole ol Glral. II, r.:. ,i.'A:l..
ResIclts of a Particular Program of Systematic Instruction 69
I Indido. i, Grade IB in this study these extra-book number experi-
ences provided the sole basis for instruction.) Instruction is begun
in this way in order to impress' children with the need for the new
idea or skill. Sometimes a fortuitous or chance occurrence serves the
purpose; at other times a situation must be prearranged.
Chance happenings and prearranged situations involving number
are utilized not alone to initiate developmental instruction, but also
to provide opportunities for children to use the arithmetic they have
learned. As a consequence, arithmetic is not confined to the arithme-
tic period, but is part of the whole school day. It is expected that
by encountering number at all sorts of times and in all kinds of ways
Children will become more sensitive to the values of arithmetic and
will develop habits of use that will function in many practical ways.
(Outcome 18, p. 67).
Experimental subjects.-Usable returns were received for 223
pupils in Grades IB and IA and for 280 pupils in Grades IIB and
IIA. By "usable" is meant (1) that test records were available for
each child for the two terms in his grade and (2) that the test blanks
showed awareness of the purpose of the test and intention to follow
directions. In spite of the second requirement some exceedingly
poor test papers were retained, as will subsequently appear. "Re-
peaters" were not included in the study.
The 223 pupils in the first grade came from ten classrooms lo-
cated in four states-Massachusetts, North Carolina, Ohio, and
Pennsylvania. The 280 second-grade pupils came from eleven class-
rooms in the same four states.
Tests.-The tests were of two kinds: (1) group tests and (2)
individual tests. The first kind was administered to all pupils in
each half-grade; the latter, to a sample of the pupils, selected so as
to be representative of the whole class.
(1) Group tests. The group tests were specially printed blanks
which reproduced the tests provided in the manuals-pages 74 and
75 (Grade IB) and pages 142 and 143 (Grade IA) of the Teachers'
Manual for the Beginners' Course; pages 100 and 101 (Grade IIB)
and pages 153 and 154 (Grade IIA) of the Teachers' Manual for
folly Numbers, Book Two. Samples of these tests appear as needed
on the following pages, altered as required to show the problems
used and the directions for administering the tests. In each test the
verbal problems were read or stated by the teachers and did not
appear on the test blanks. In the test for Grade IB all directions
70 Arithmetic in Grades I and II
were given orally to avoid reading difficulties, but Part II of thi:
tests for the other half-grades regularly involved considerable readinrc.
(2) Individual tests. The contents of the individual tests and tl'e
procedure in administering them are described at a later point (be-
ginning with p. 90).
RESULTS FROM THE GROUP TESTS
Gross results.-Table 18 contains a tabulation of the scores made
by the experimental subjects on the group tests for the four half-
grades. In the group test for Grade IB one point was allowed i'r
each correct response, so that the highest possible number of points
was 55; 65 pupils made scores of 53, 54, or 55 (actually 26 had
perfect scores) ; 48 had scores of 50, 51, or 52, and so on. The
median score was 50, which represents 92.7 per cent of the possible
score. One fourth made scores below 42 (a percentage maximum
of 76), and one fourth made scores of 53 or better (a percentage
minimum of 96).
SCORES ON TERM GROUP TESTS, GRADE IB THROUGH GRADE IIA
31, and below....
First term Second term
.. .. 40
.. .. 30
.. .. 25
.. .. 10
.. .. 11
.. 65 13
.. 48 7
.. 24 5
.. 19 4
.. 17 6
.. 17 0
.. 12 1
.. 6 3
.. 15* 7
.. 50 66
.. 42 59
.. 53 71
First term Second term
The 15 scores distribute as follows: 29-31 (4); 26-28 (5); 23-25 (0); 20-22 (1); 20
and below (5).
t Medians obtained from the raw scores without grouping. The percentage equivalents
of the medians are: Grade IB, 92.7 (50 out of possible 55 points); Grade IA, 90.4; Grade
IIB. 89.2; Grade IIA, 93.8.
I Quartiles obtained from raw scores without grouping.
R ;.'W,ts of a Particular Program of Systematic Instruction 71
Th,. figures for the other half-grades were about equally good,
.1 I, mei-dian per cents of 90.4, 89.2, and 93.8, respectively, for Grades
IA. II11. and IIA. There were nineteen perfect papers among those
f.r the Grade IA pupils, seven perfect papers for the Grade IIB
pupil-. ,-id thirty-four perfect papers for the Grade IIA pupils.
Thi.:rc .. ere, however, a number of very poor papers. Eleven (5 per
centci I the Grade IB pupils made percentage scores of less than
'). ain. there were eleven such pupils (5 per cent) in Grade IA,
fifteen such pupils (5 per cent) in Grade IIB, and five such pupils
(2 per cent) in Grade IIA. Still, if it be granted that the tests sample
achievement satisfactorily and that the method of scoring was ade-
quate, there seems to be sound evidence that the experimental sub-
GRADE IB TERM TEST, PART I
6-1= 5-4= 4+1=
2+3= 5-2= 1+5=
4-3= 6-4= 4+2=
B. --------------------------------------------___ --------- I
3 6 3 6 5 4
+2 -5 -1 -3 -3 -_
3 1 3 5 2 6
-2 +2 +3 +1 -_ -_Z
(I had 5 pen-
nies. I spent
1 penny for a
piece of candy.
How many pen-
nies did I
E. (Ann had a
party. If 4
came, how mary
girls in all
were at the
(I put 2
and 4 green
books on the
were on the
(.Card for 3, 2, and 5 shown;
children to write one story
(Same as G, but card for 1, 3,
I.- (Same as C, but card for 1, 2
(Same as C, but card for 2, 2, and 3.)
S (I saw 4
birds on the
wire. Then 2
How many bir s
were left on
72 Arithmetic in Grades I and II
GRADE IB TERM TEST, PART II
(1. Draw 7 balls in a row.
2. Put I on the second ball.
3. Put C on the fifth ball.)
B (. 1. Make 10 crosses in a row.
2. Draw a rint around the fourth cross.)
3. Put a box around the sixth crose.)
(C-F. Write the number to show how many flags (etc.) there are.)
E.I I F. 1 . I i
(G. \Write the numerals from 1 19 in order.)
G. I I
(H. Write the figures for the number words.)
11. one ----------------- three ----------------- two ---------------
four ----------------- five ----------------- six ----
jects in all the half-grades learned much of what they had been
taught. Three fourths of the IB-pupils made scores equivalent to
77 per cent or more of the possible score. In Grade IA three fourths
made a percentage score of 81 or better; in Grade IIB the corre-
sponding percentage score was 80, and in Grade IIA, 86.
Results in Grade IB.-The extent to which the Grade IB test
covered the outcomes for that half-grade is shown in the upper half of
Table 19. The ability to enumerate to 10 (Outcome 1) was tested in
items A, B, and C-F of Part II; knowledge of the place values of
the numbers to 10 (Outcome 2) was tested in item G of Part II;
ability to read and write the numerals to 10 (Outcome 3) was tested
in items A-J of Part I and in items C-H of Part II, and so on. All
the outcomes are represented by test items, though obviously in each
case by a small sample of the many possible items. No outcome was
tested, or could be tested by the usual group techniques, in its en-
tirety. Outcome 6, "Intelligent control of the combinations," is, for
Q 'Z'''' ^''. T1 I ''0 '.
R, salts of a Particular Program of Systematic Instruction 73
example, deliberately vague: it means little more than the ability to
get the correct answers for combinations by any method whatever.
Scores. in this part of the test tell nothing about the degree to which
the combinations were habituated (they were not supposed to be
habituated until the second half-grade or even later). Outcome 10
cann-.t hb tested adequately by any paper-and-pencil procedure. Sen-
.itivenIe-- to the quantitative in life is to be observed only as one
a.oitd or meets satisfactorily the number situations in which one
Fids ,_,neself. Accordingly, measurement, as here made, through suc-
cess :in the usual kind of verbal problems is very indirect and prob-
ably none too valid.
CLASSIFICATION OF GRADE IB TEST ITEMS BY OUTCOMES, AND
SCORES BY TEST PARTS
Test Items, by Parts
Part I Part II
1. Enumeration to 10 ........... .... A, B, C-F
2. Place value to 10 ............ .... G
3. Numerals to 10............. A-J C-H
4. Ordinals to sixth or seventh.. .... A, B
5. Recognition of regular groups G-J ....
6. Intelligent control of combina-
tions through 6............ A, B, G-J ....
7. Reading and writing combina-
tions horizontally and verti-
cally ...................... A, B H
8. Understanding of addition and
subtraction ................ C-F ....
9. Reading vocabulary ......... A, B H
10. Appreciation of social values,
etc. ........................ C-F
Possible ....................... 29 26
M edian ....................... 25 26
Q i ........................... 17 24
Qs ........................... 25 26
Percentage equivalents of:
M edian ....................... 86.3 100.0
Q i ........................... 58.6 92.3
Q 3 ........................... 96.6 100.0
The lower half of Table 19 reveals that the children tested were
much more successful with Part II than with Part I. On Part II
three fourths or more of the pupils made percentage scores of 92 or
better. On Part I, on the other hand, the corresponding percentage
score was but 58.6.
Arithmetic in Grades I and II
The results on the various items of the test are analyzed in Tabl'
21 for a sample of one hundred Grade IB pupils, so selected that the.\
are truly representative of all the pupils tested. The degree to which
this representativeness was attained is shown in Table 20 below. It
will be seen that 10 per cent both of the total group and of the samnli:
group secured scores of 55 or better, 20 per cent secured scores -'t
at least 54, and so on. The medians and quartile points are identical
(or nearly so) for both groups.4
COMPARISON OF TOTAL GRADE IB GROUP AND THE SELECTED
SAMPLE OF 100 PUPILS
Point Total Group Sample Gro',r
90........................... 55 55
80........................... 54 54
70........................... 52 52+
60........................... 51 51
50 (M edian) ................. 50 50
40........................... 47 47
30 ......................... 44 44
20........................... 40 40
10........................... 35 36
Q 3 .......................... 53 53
Q .......................... 42 42+
To return to Table 21, the per cents of success on Part II of thI.
Grade IB test speak for themselves. Clearly the children had acquir':d
high degrees of skill and knowledge in dealing with arithmetical tasl.-
set for them in this part of the test.
The story for Part I is 'different. All the items in Part I relate
to the number combinations with sums and minuends to 6. In A arc
nine abstract combinations in horizontal form; in B, twelve abstraLt
combinations in vertical form; in C-F, four combinations presented
orally in verbal problems; in G-J, still other combinations to be rec-
ognized from pictured groups.
The situation in Grade IB may be fairly well summarized by
saying that the children demonstrated substantial growth toward all
outcomes, except possibly those connected with the number combina-
tions as such (and possibly those associated with the development of
habits of use, an outcome none too well tested).
In the treatment of the results for Grades IA, IIB, and IIA the same prac-
tice is followed, of analyzing the data for a selected sample of one hundred
cases instead of for the half-grade group as a whole. In each instance the
sample was as closely matched with the total group as above in the case of
Results of a Particular Program'of Systematic Instruction 75
NUMBER OF ERRORS AND PER CENTS OF SUCCESS ON ITEMS
OF GRADE IB TEST; SAMPLE OF 100 CASES
Outcomes Number of Errors Per cents of
Test Item Tested and Omissions Success
A ............ 6,7,9 214 76.2
B .............. 6, 7, 9 331 72.4
C-F ............ 8,10 59 85.3
G-J ............. 5, 6 114 71.5
A-B ............ 1,4 33* 91.8
C-F ............ 1,3 5 98.8
G .............. 2, 3 11 99.9
H .............. 3,7,9 43t 92.9
Out of 100 possible errors on each of the four ordinals tested, 5 errors were made on
second, 7 on fourth, 11 on fifth, and 10 on sixth.
t Out of 100 possible errors on each of the six number words tested, 6 were made on
one, 5 on two, 4 on three, 9 on four, 12 on fire, and 7 on six.
The significance attached to the figures for the number combina-
tions varies with what one expects from children in this half-grade.
If one calls for automatic mastery of all the combinations taught,
then the low per cents signify a wholly unsatisfactory state of affairs:
the children had not "learned" these simple number facts.
On the other hand, if one is willing to wait for mastery and if
one views the experiences these children had with the number com-
binations as being exploratory in character, then one is not at all
disturbed by the apparent lack of mastery in Table 21. As a matter
of fact, on the assumption that the subjects in these experimental
classes were typical of those studied in investigations of the number
knowledge of children on entering school (Chapter II), these chil-
dren show unmistakable evidence of growth. A median of about 35
per cent of school entrants "knew" the abstract combinations pre-
sented to them (Table 16) ; the corresponding figure from the Grade
IB test is 72 per cent. A median, of about 42 per cent "knew" com-
binations in verbal problems when they entered school (Table 15),
though some of these combinations were harder than any here tested;
the corresponding per cent for these Grade IB children is 87 (only
four combinations, however). The amount of growth revealed by
these comparisons is enough to satisfy the student of arithmetic who
judges growth in terms other than those of mastery. Such a person
is not disturbed by the rather superficial mastery shown by these
children since he is confident that the understandings they acquire
76 Arithmetic in Grades I and II
through meaningful experience will eventually make for a more usable
kind of knowledge.
Results in Grade IA.-The Grade IA test is analyzed in terms of
outcomes in Table 22 as was done for the Grade IB test in Table 19.
It will be noted that all outcomes except two (namely, 1 and 2) are
represented in the Grade IA test, though with varying degrees of
completeness. Moreover, as in the case of the Grade IB test, the
measurement of Outcome 12 is very imperfectly and only indirectly
GRADE IA TERM TEST, PART I
A. Write thle answers:
1. 9 2 2 8 G 7 9
S + +7 -5 -2 -3 -7
; 4 4 4
+2 +L +3 +2
*C. Write just the answer :
*Proil'ms ior Ex.C d. There we re jiit 7 len;ve o onone branch of a
a. Pat has : marbles in hi, right hand and 6 ini tree. Along caime the wirl : [dl bilew off all but 2.
his left hand. Jlow many imlarllel, ihan he il lhoth lion many leaves iwri liliini oil the liranch?
hands? e. Jane ha:i's 5 s.'hoiil iidre. and 2 party dresses.
b. Anin counted tlie egg, hlie find ill the larn. How liaiiy d'ss
There were 5. She put therm with tie 4 egg in the f. Eight of .Iabel's drawings were hanging on
icebox. Then there were how many egg in the thlie wall of her bedroom. She took down the one
icebox? that got dirty. Now there are 1 drawings on the
c. If you take 8 sandwiches to the picnic and wall. How miany of the drawings did Mabel take
only 6 are eaten, how many sandwiches will be left? down ?
Results of a Particular Program of Systenatic Instruction 77
GRADE IA TERM TEST, PART II
A. Draw a ring around the right answers:
a. I ow do you find how many in all?
b. WVhich picture shows a half?
c. How many ones in 16?
1 4 6
d. Which number comes just before 18?
17 19 7
e. Which means to add?
= + -
f. Which number is less than 75?
79 SO 63 92
g. Which means one half?
2 four 2
ii... u.... uadd?
i H. ,... tens has 15?
5 1 2
j. Which are parts of S?
2 and 3 5 and I G and 2
k. flow many parts has the ihiole story ablut
3, 6, and 9?
2 4 5
1. Which meiani 7?
tille i-ht .re en
m. How du you find hlw nminy are gone?
n. Which shows 2'?
o. Which is the answer for 7 2
9 5 3
p. Which number is more than 50?
57 29 36 4S
q. How many parts has the whole story about
4, 4, and 8?
5 2 4
B. Write the numbers that are left out;
a. 36 37 .- 39 .
b. 58 .--61- -..----.
c. 10 20 30 ----- ------60
d. 40 --...... C------ SO ----...
proidid f,:,r. The omission of items for Outcomes 1 and 2, most of
the :killi in which can be tested only by individual interviews, is
lhardIh .erious. In the first place, practically all children may be as-
sumed ir. be able to enumerate and to count to 100 by 1's at the end
,if Grade IA if they receive any instruction at all on these skills. It
, ill he recalled that about one tenth of school entrants already possess
thi- bilit\ (Table 3). In the second place, the ability to count to
I(C" L-v 10'I is also rather easily acquired; about one fourth have the
78 Arithmetic in Grades I and II
ability when they enter school (Table 3 again). In the third pla..:.
the high degree of success of these same children at the end of Grade
IB in dealing with the lower ordinals (about 90 per cent knew fifit
and sixth) warrants the belief that, a half-grade later, they would bt
equally successful with the new ordinals seventh and eighth.
The median score on Part I of this test (the lower section .,f
Table 22) amounted to 93.5 per cent of the possibility, and three
fourths secured scores of about 85 per cent or better. Part II N:a-
harder for these children, the median dropping to 88.9 per cent and
Q, to 74.1 per cent. The reason for the lower scores on Part II
cannot be certainly ascribed to any one cause: (1) the form of thul
CLASSIFICATION OF GRADE IA TEST ITEMS BY OUTCOMES,
AND SCORES BY TEST PARTS
Outcomes, by Test Items:
1. Enumeration to 100 by 1's;
ordinals to eighth..................
2. Counting to 100 by l's; by 10's.....
3. Reading and writing numbers to 100
4. Place of numbers to 100...............
5. Significance of 10 in teens
num bers ........................
6. Intelligent control of combinations
with sums and minuends to 9....... A
7. Understanding of addition and
subtraction as processes............
8. Understanding of the relation
between addition and subtraction....
9. Column addition, sums to 9, three
10. Understanding of Y as applied to
objects and even groups............
11. Reading vocabulary (all of Part
II, but especially) ..................
12. Appreciation of values of arith-
metic, etc. ........................
A: d, f, p; B : a-.!
A:d, f,p; B:a-J
A: j, k, q
C A: a,m,o
B (5 examples) A: h
Possible .......................... 46 27
M edian ........................... 43 24
Q I ............................... 39 20
Q 3 ............................... 45 26
Percentage equivalents of:
M edian ........................... 93.5 88.9
Q 1 ............................... 84.8 74.1
Qa ............................. 97.8 96.3
A" sits f' a Particular Program of Systematic Instruction 79
it::t 1u,, Ihi.. bci.n relatively unfamiliar to the children; (2) the
rea.hirnv r,:eq.uirnmt-i mnay have baffled some; (3) the ideas contained
in tile test itt-ii1 nia not actually have been possessed by the chil-
drt.n. '.iIlv tli. third .of these three sources of error is the one which
,.r- actu:ilv [,r.._j,..-.d for testing; the purpose, in other words, was
t ... i,..t.ria i the lItcr'. to which children had acquired the concepts
:indl undreraiiaIi, represented in the items of Part II (uncompli-
ca'teld :o, Jel:icieincie iii reading skill and in the technique of taking
,st-t inrlirii,,ne of the first two sources of error is to depress
urilil tli: -: ...rte .:.l-htined. There is at least one counteracting fac-
t.Li. niCilI. thi. chli..ce selection of answers: by pure guessing cor-
rc,,:t an:v:.e r- xculld [I, marked for one out of each three or four items.
It is r-It. h... .,etr, that the favorable effect of this last influence was
mI'.r Liai :.rf.,t lIv the unfavorable effects of the first two factors, so
th'it thi ., -.-e_ Inre reported are almost certainly too low.
.\>c..rdig t.:. Table 23 for a sample of one hundred Grade IA
iu[.pil.- 91.5 pi cint of the thirty-five abstract combinations were
o.:rre-cthl 'in-vi :r-., :;r well as 81.8 per cent of five examples in col-
mint a i,. itn...n. d 4.0 per cent of the six verbal problems. (One
Iniidred liii..l.I tn1n ach attempted 35 abstract combinations; 297 an-
z-_.tr-: were ., Vrr.g iand 3,203 were correct, to give a figure of 91.5 as
til, ft-r :ilit -A' .iCu CC--.)
'lith .ri..ui- '- r:uipl of items in Part II varied, as to per cents of
succ:.:.-, V'froin, I 5 t,. 93.0. The lowest items, separately considered.
Nu i-itp .'.r Lin.:R.-: A i PER CENTS OF SUCCESS ON ITEMS OF THE GRADE IA
TEST; SAMPLE OF 100 CASES*
T7, ; 1it,, Outcomes Number of Errors Per cents of
Tested and Omissions Success
.A . . ... 6 297 91.5
b .. .. ... .. ...... 9 91 81.8
.. ..... ...... 6, 7, 12 96 84.0
f p. B. a-d ...... 3,4 197 84.8
A. c ... . ...... 5 57 71.5
A ). r... .. ... ......... 6,8 66 78.0
.A m., ...... 7 49 83.7
A\ b ...... 9 19 81.0
.' b. n . . . 10 21 89.5
A ,.. I ...... 11 21 93.0
80 Arithmetic in Grades I and II
are A: c, f, h, i, k, q and B: d. A: c and A: i (81 and 62 per cent
respectively) relate to the meaning of ones and tens in the numbers
11 to 19. Sixteen of the 19 errors made on A: c came from the selec-
tion of 4 instead of 6 as the number of ones in 16; 34 out of 38 errors
made on A: i represented the selection of 5 instead of 1 as the number
of tens in 15. Item A: f required the selection of the number (79, 80.
63, or 92) which is "less than 75." The errors were about evenly
distributed among the wrong alternatives, a fact which seems to in-
dicate ignorance of the meaning of "less." Item A: h ("How do you
add? up . down") probably should not have been included in the
scoring. The "correct" answer according to the manual direction,
for teaching is "down," but whole classes tended to mark "up," thus
suggesting that in those particular groups the manual instructions
had not been adopted. In these classes "up" was of course the correct
answer. Items A: k and A: q deal with the number of parts in the
"whole story" about 3, 6, and 9 and about 4, 4, and 8, respectively.
Seventy-nine per cent answered A: k correctly, but only 57 per cent.
A: q, which contains a "catch." Apparently this idea of the "whole
story" was none too well understood by these children.
The general conclusions to be drawn with respect to the Grade IA
test results must be somewhat guarded, in view of the uncertainty
surrounding the results on Part II of the test. So far as the abstract
combinations are concerned, these children indicated very satisfac-
tory progress in learning. In column addition, which was introduced
in this half-grade, the results were none too good, but the newness of
the process may account for the comparatively low degree of success.
It is difficult to account for the failure of these children to do better
with the verbal problems, in view of their supposed familiarity with
this kind of arithmetic. The system of instruction outlined for tht
experimental schools calls for a great many experiences with de-
scribed quantitative situations, and if these were actually supplied ti:-
the children, they should have been able to solve correctly more of
the test problems.
The results on Part II of the test have already been considered
in some detail. If it is a fair interpretation to regard the per cent-
of success reported as really too low, one could infer that these chil-
dren were acquiring verbal statements of mathematical principles.
generalizations, and relationships of large value in their understand-
ing of arithmetic.
Results in Grade IIB.-According to Table 24 all Grade IIB ou -
comes except one, namely, Outcome 11 (Roman numerals) are rep-
Results of a Particdular Program of Systematic Instruction 81
resented by at least two items in the group test. The coverage of
O(-lutcinle 13 is, as in the case of the other tests already considered,
very inadequate indeed, and the types of behavior associated with
the other outcomes can of course be considered only as sampled
rather than as exhaustively tested. This limited sampling is inevi-
table, ;:. far as the present study is concerned, for the test was de-
\itsed for the measurement of achievement over a half-grade of
instruction and not for the purpose of diagnostic analysis.
GRADE IIB TERM TEST, PART I
*A. Your teacher will tell you what to do.
a.......--------- b..---....... c.
B. Look at the sign. Then add or subtract.
1. 11 3 6
-7 +9 +5 -
12 10 2 5 II
9 -4 +8 +0 -4
10 9 3 12 7 11 9
-7 +3 +7 -5 +4 -6 -9
0 11 7 11 10 12 5
+7 -s +5 -3 -8 -3 +6
12 8 4 12 10 4 11
-1 +3 3 +6 -7 -6 +8 -9
C. Look at the sign.
Then add or subtract.
15 + 10
17 13 19 10
-5 +4 -7 +8
17 IS 19 10 19 12
- 6 -S -3 +7 -0 +6
5. 3 + 0+ 7=
Proems for ,E. A
a II .rr, .....r,, .j his toy soldiers. He had 1 1
. ... i,.,. . c ins and 3 toy soldiers with
.,r, II. n r, more of his toy soldiers had
r.r I , I J I., aJis?
h ft. P ill r ii, Iropped aboxof bottles. When
.. i.-l i i 'ji only 5 of the 12 bottles were
-...d H.... .. L..ttles had been broken?
ci [... l.. t i...'ces of peppermint candy and
4 pieces of chocolate candy in her bag. How many
pieces of candy has she in all ?
d. A farmer was mending his fence. He had to
put in 12 new posts. After lie had put in 9 posts,
how many more did he have to put in?
e. Mrs. Spider spun a fine web. The first day
she caught 2 flies and the next day 8 flies. How
many flies did she catch in bIoth days?
Arithmetic in Grades I and II
GRADE IIB TERM TEST, PART II
A. Draw a ring around the answer:
1. Which means to add?
t + +
2. Which means one fourth ?
1+3 4 1 4-0
3. To find "how many more" you
4. Which shows 4 ?
5. Which is less than 57?
67 90 58 49
6. Which means cents?
7. Which number is I I I I I I?
17 12 26 15
8. How many tens has SO?
3; 0 8 7
9. Which number is largest?
70 62 S9 45
10. Which number is twelve?
2 36 12 17
11. To find "how many gone" you
B. Write the missing number or numbers:
1. After 15, 20, 25, come - - and ..- - -..
2. The hole story about 7, 5, and 12 I,..
3. 5 is one part of 11 ; the other part is
4. At ten o'clock the short hand is on - - -..
The long hand is on ---.
5. If you take away all of a number, -..
6. After S, 10, 12 come ---- and --. ..
7. The ring is around one --- of the d..
8. 17 has -.----- ten and ------ones.
9. The whole story about 5, 5, and 10 has
S- - -parts.
10. The -picture for 30 is .
11. A dime is the same as ------ cents.
The scores in the lower section of the table indicate the degree to
which achievement, broadly considered, was satisfactory among the
experimental subjects. One fourth of the children secured scores
equivalent to more than 98 per cent of the possibility on Part I, one
half, scores equivalent to 93 per cent or better, and three fourths,
scores equivalent to 82.5 per cent or better. The corresponding per-
centage figures for Part II are lower-92.3, 84.6, and 69.2, respec-
Results of a Particular Program of Systematic Instruction 83
CLASSIFICATION OF GRADE IIB TEST ITEMS BY OUTCOMES,
AND SCORES BY TEST PARTS
Outcomes, by Test Items:
1. Understanding of numbers to 100,
ordinals to tenth...............
2. Counting by 2's to 20 and by 5's
3. Intelligent control of combinations
4. Understanding of the relationship
between addition and subtraction
and the corresponding number
5. Intelligent control of the 0-combi-
6. Understanding of the processes of
addition and subtraction........
7. Column and horizontal addition,
three digits, sums to 12........
8. 'Higher-decade addition, to 19....
9. 3 as applied to single objects and
10. U. S. money and telling time.....
11. Reading and writing Roman nu-
merals to XII .................
12. Reading vocabulary (all of Part
II, but especially) .............
13. Appreciation of values of arithme-
A: 5,7-9; B-8
A: 3, 11; B: 3, 5
A: 2, 4; B: 7
A: 6; B: 4,11
A: 1,2,6, 10; B: 10
Possible ........................... 57 26
M edian ............................ 53 22
Q i ................................ 47 18
Q s ................................ 56 24
Percentage equivalents of:
M edian ............................ 93.0 84.6
Q 1 ................................ 82.5 69.2
Q 3 ...... ........................ 98.3 92.3
tihvlv Part I is devoted to abstract arithmetic, chiefly computation;
PF'art II. 1,. mathematical relationships, generalizations, and the like.
A sample of one hundred Grade IIB pupils (Table 25) solved
s2.2 per cet of the verbal problems (5 problems per child, a total
:.f 50XI pr._,blc,.ms, of which 89 were missed). On the abstract facts
S'.tlh 'ums and minuends from 10 to 12 (twenty-eight facts in the
test i. tlit- [i,:er cent of success was 87.7, and on the 0-facts (only four
in the te.;S i t was 96.3. In higher-decade addition and subtraction
.',ilh sums and minuends through 19 (fourteen examples), the per
c,ntm of accuracy was 82.8. In horizontal and column addition, three
digit. uim~n to 12 (six examples), the per cent of accuracy was 92.5.
84 Arithmetic in Grades I and II
Again, as in the case of the Grade IA test, it is difficult to account
for the comparatively low per cent in problem solving, except on the
ground that these children did not have all the expected experience
in dealing with verbally described quantitative situations. The per
cent of success on the new addition and subtraction combinations is
entirely satisfactory in an instructional program which does not hurry
mastery. The figure for the 0-combinations (though based unfor-
tunately on only four combinations) is especially interesting in view
of the still common belief that such combinations are especially hard.
According to the scheme of instruction here under consideration all
0-facts are presented by means of four generalizations and are taught
as groups (0 + n, n + 0, n 0, n n). Under these conditions
the combinations are readily understood, and the learning is accord-
ingly made easier. As for higher-decade addition and subtraction,
the process was new to these children, and 82.8 per cent seems to
represent entirely satisfactory learning at this stage. The last item
in Part I consists of examples in horizontal and column addition,
and the children were very successful with the six examples given
NUMBER OF ERRORS AND PER CENTS OF SUCCESS ON ITEMS
OF THE GRADE IIB TEST; SAMPLE OF 100 CASES
Outcomes Number of Errors Per cents
Test Items Tested and Omissions of Success
A ..................... 6,12 89 82.2
B ...................28 (facts to 12) 345 87.7
4 (0-facts) 15 96.3
C ..................... 8 241 82.8
D ..................... 7 45 92.5
A :5, 7-9;B:8 ......... 1 129 78.5
B : 1, 6 ................. 2 76 81.0
B :2,3,9 ............... 3,4 82 72.0
A :3, 11;B:3,5 ........ 6 84 79.0
A :2,4;B:7 ........... 9 74 72.0
A :6; B:4, 11 ..........10 37 90.8
A : 1,2,6,10; B:10 .....12 51 89.8
On Part II, success varied from 72.0 per cent for the third group
of items (the "whole story" idea) to 90.8 per cent on the sixth group
(U. S. money and telling time). As already mentioned, Part II
seemed to be harder for these children than was Part I, but the reason
for this greater difficulty (if indeed there was greater difficulty) is
uncertain. Like Part II of the Grade IA test, Part II of this test
Results of a Partiiular Program of Systematic Instruction 85
called i 1) for reading and (2) for knowledge of the technique of
m:irking answers. Either or both of these factors, extraneous to the
real purpose of the test, may have introduced an undue number of
The following items were missed by 10 per cent or fewer of the
children: A: 1, 2, 4, 6, 10; B: 5. The following items were missed
by 2) rper cent or more : A: 3; B: 2, 3, 6, 7, 8, 9, 10. Forty per cent
ctaited that to find "how many more" one should add (item A: 3), an
error that may well mean the early but undesirable adoption of
"mor." as a specific cue for addition. B: 2, 3, and 9, missed by 26
per cu-Lnt, 22 per cent, and 34 per cent, respectively, all involve the
"v.'-,.le story" idea, a fact which suggests the need for special atten-
ticn t..' the relationship between combinations. B: 6 (25 per cent
mrle errors) involves counting by 2's, but the framing of the test
itern may not have suggested this fact. B: 7 calls for the identifica-
tion *:'f group of three dots as one fourth of twelve dots. Only one
third *:.f the children (36 per cent) succeeded on this item as com-
paicd ".-ith 93 per cent in the case of A: 4, where one fourth of a
-sigle Object was identified. The difference in per cents confirms the
behalf that the application of fractions to groups of objects is much
n:ore difficult than to single objects, though in this test the low per
cent may reflect ambiguity in the test item itself. B: 8, missed by
3r. per cent, requires two entries, and one of these was frequently
,,mitted B: 10 tests ability to use a special symbol for 10 in the
co'n-truction of a "picture" for 30. Forty-two per cent could not draw
the correct "picture"; this probably should not be surprising in view
O:it c'.imparatively slight experience in this kind of activity.
The form of analysis adopted in the foregoing paragraphs and in
the corresponding sentences for the other tests has the disadvantage
.'f str.essing deficiencies rather than progress in learning. The reader
may" therefore be predisposed to object to any statement which im-
prlies complacency about the situation as revealed by the Grade IIB
te-t results. It is true that the showing at some few places (espe-
call, those mentioned in the preceding paragraph) is none too good,
but there are many evidences that these children were advancing
t.rc, 'ird the outcomes for their half-grade. After all, outcomes must
be ie(xed as directions for development to take, and not goals which
either are or are not attained, the quicker they are attained, the bet-
ter In- the instruction to follow in Grade IIA and in later grades
thIre are many opportunities to enrich and deepen the learning which
Vi b,1gui in Grade IIB and to carry that learning to the desired limit.
Arithmetic in. Grades I aoi'd li
The crucial point is to make sure that earl learning; i- .-f the right
kinrd-meaningful, intelligent, based upon under-..inlirg-and the
experimental subjects in this investigation '.cr -pproacah.ii.. arith-
metic in this way.
Results in Grade IIA.-The use of ordinals (Outcome 1) is not
represented among the items of the Grade IIA test (Table 26) ; and
Outcomes 3, 4, 9, 10, and 12 are but slightly represented. However,
GRADE IIA TERM TEST, PART I
*A. Write just the answers:
B. Do what the sign tells you to do :
1. 15 5 7
- I +9 +f f
3. ----. ...--
[I 7 IS 14 15
-7 +S -9 -I; -7
14 4 17 9 15 14 ;
- s +9 -8 +7 -9 -5 +9
23 1to S 15 9 14 1:1
-7 -9 +9 -S + -9 -S
C. Do what the sign tells you to do :
1. 94 5 47
-4 +34 -3
-3 +6 H
E. Count by 3's:
1.9 12 ---- ---- 21
F. Write the missing numbers:
1.37 3S ---- ---- 41
37 26 49 23
-0 + : -I +5
4 3S 32 99
S+4 -6 +7 -2
2.IS 1 ---- ---- 27 ....
-..--- 71 ----.... 73
Proems for -r. A 4. E[lla n rote 14 words on tihe board. Tom w rot.
1. If you have 15 peanuts in a bag and eat 9 of 6 words. How many fewer words did Tom rite?
them, how ninny will be left? 5. Dorothy has 8 hair ribbons. Her sister JuIhi
2. A book costs So and a tablet GO. HIow much han 9. HIow many hair ribbons have the two girls
ido the book and tablet cost together? in all?
3. There were 12 berries on a straw berry plant. 6. Mother put a pan of 1(; cookies in the oven.
Along carnme some birds, and soon only 5 berries The fire was too hot and 7 conkies got burned. IHow
were left. How many berries had the birds eaten? many cookies did not get burned ?