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DIAGNOSTIC TEoTS OF ABILITY TO ADD IIITEGERS.
By
RUDOLPH, LIiroQUIST
THESIS
Submitted in partial satisfaction •f the requirements f«r the degree •f
llASTER OF ARTS
EDUG.VriON
in
the
GRADUATE
DIVISION
of
the
HIWIRSITY (
3F CALIFORNIA
Dec«
1922.
9-. !a/' f/^'T
Digitized by the Internet Arciiive
in 2008 with funding from
IVIicrosoft Corporation
http://www.archive.org/details/diagonstictestsoOOIindrich
PART ONE
INTKODIIGTORY STATFl/ff-^NT
Tills study la Intended to rer.ult in tho constr-uctlon
of a raeaHui'lng devlco by mcfine, of v/hlcVi the teacher will be able
to determino the particular .addition habltt, , the dovolopinent of
whlcVi reodti to be emphaslzod wltli her cIhsb.
The t^rowth of the teat raovt-iiifiit. huti ti'un a ti't)i.u;ndoua
Increase in the number of teats, but It aeoins Uiat fiuffjolont
ernphat'.j h has not been placed upon tl-ie development oi' tho&e that
are of dlatsnoatlc, rather than general value, 'i'he teacher needs
to know, not only, how i^i-t)ut la the general ability of hei- class
or of a r»r^'l-t;uiMr individual, but altio, in ceat) they are not as
goo(] aa they thould be, tVie particular habita t}u> t are undevelop-
ed.
All testa are in some sense or to aoiae degree diagnos-
tic. Even aueyi afl teat as the Courtia L>tandard Researcyi Tests
altVioui^h they werti not iaaued aa definitely diagnostic teatfi are
Bucli lo the extent to wlil ch they enable the teaclier' to cUaermine
whijch <jf tlie f(XAr IVuidamontul operations her clt'St^ has moat need
to work up(.;n. Tliey enable a teacher to know v/h ether or not her
clasfi cwi do thr-ee liy nine /uld] tlon as well as they can do two
by four iiiulti|jll cation or two by five diviaion, but they give
her no Vie].]) in duoltUng, in l.he caae; of aiMltluti, w>iother or
not it lb addition corablnatlona, as such, that tl-ie children
are hf.iV\nR troiible with or wheUier it is carr-ying, or wViether
"440 1*
.X.
' ' ■ ' ' - ■ ■• •■ ■ ' " I i ■■ .i/ ■•>. !•, , I I n III n I li I 11, I ■ il L .
Page 2.
General tests, such as Vioody-McCall Mixed i'und-
amentals, while not intended to be diagnostic, have never-
theless been interpreted as being such by many teachers.
Form One for instance, has ten addition examples scattered
throughout the test. These vary from simple examples such
as addition of the simple combinations to the addition of
a sixteen number colvunn, including decimal points. Others
of the ten examples involve the addition of broken columns,
the addition of figures written on a line connected by a
plus sign and v hich must be re-written before they are add-
ed. In fact, there are no two examples that involve exact-
ly the same addition habits. Failure to do the examples in
addition in this test could not be attributed by the teacher
to a single or to even a limited number of causes . Even
though the child had a preponderance of erx'ors in addition
examples of this test, the teacher v^ould not know whether
it \vas ignorance of combination or unfamiliarlty v/ith fuch
a fcrm as 2-f 3, or inability to cope with form such as is
found in broken column addition, or inaccuracy due to the
necessity of re-writing the example. Even though the child
should fail in the one or two examples involving one of
the above habits, the teacher could not be certain that the
Page 3
failure wan clue to a disability that required special drill
to overcome it. A test, in order to be diagnostic must not |
only include examples limited to a single ability, in so
far as it is possible to so limit them, but there must also
be sufficient Instances of each type of example so that the
child will have an opportunity to demonstrate conclusively
his ability or inability to cope with examples involving a
particular habit.
V/oody in the Vioody ■'arithmetic i^cale has devised a
test v;hich is more diagnostic than either of the tv/o that
have been discussed. But even it cannot hope to give more
than a general measur-e of power to arid, for there are not
sufficient instances of each addition habit nor are the
few of each habit that do occur conveniently segregated and
arranged. It included everything from a simple combination
to the addition of mixed numbers, with about one or two in-
stances of each addition habit. Even though a teacher could
quickly spot the failure to perform the two instances of one
operation, she could not be certain that this f£iil\are was due
to anything but chance, for t'wo examples involving any parti-
cular Aritlvnetic habit are not sufficient to measure con-
clusively a child's ability therein.
Page 4
Dr. V/alter ^. Monroe in his diagnostic tests in
Arithj-netic goes further than anyone else in meeting the
requirements of a diagnostic test namely, that, the speci-
fic habits be isolated in, so far as po&sible, anc that
each be measured under conditions that render comparison
of achievements in each test comparable with achievement
in any others.
Dr. Monroe has sought to meet the first require-
ments by providing three tests in the addition of integers:
(a) single column addition, three numbers high
(b) single column addition, thirteen numbers high
(c) foui' column addition, five nur.ibers high
The thr^ee testf-; given should show v/hether or not a
pupil's addition disability involves inaccuracies dufe to
fatigue or inaccuracies due to carrying. Beyond this it
does not seem to be diagnostic, '^here may be other and more
constantly operative causes for Inability than either of
those mentioned. Such a one miglrit be broken column form,
or inability to do the combinations themselves even when
these are isolated and the additional difficulties incident
to the use of them in actual sitviations in-oerative.
Concerning children who succeed in column addition
(five by nine) we may of course assume that they know the
combinations and that they m8.ke no mistakes in carrying but
Page 5.
concerning children *o fail in this test v- e do rfot know
to which of the above causes to ascribe their inabilitj'^
to add. Nor do we knov/ whether blank spaces in some of
the columns, such as occur in broken column addition would
not be a real obstacle to the pupil. It seems desirable,
therefore, to isolate each of these habits even more than
Dr. Monroe has done, and to measure performance in each
under comparable conditions, so that we may single out the
cause of the inaccuracy and remedy it.
Another point Dr. Monroe has not taken into account
fully is that of reducing achievements in the different
habits to comparable units. Grade norms are, of course, in
a measure, such a unit. If eight and six examples correct
are the norms for a grade in each test, respectively, the
teacher can of course compara roughly the achievement of
her class in one test with its achievement in the other but
to the extent that a pupil deviates from the average it be-
comes increasingly difficult to make an accurate comparison
between the accomplishments of any one pupil in two tests.
If a series of tests, is to be diagnostic in a mot t helpful
way, it seems desirable that the teacher should be able to
re(iuce the achievements in the several tests, particularly
in the case of children that deviate most frojn the average,
to a common denominator.
Page 6
Standard deviation position in a normal age dis-
tribution woulr] of coTarse furnisb sucri a common iinit. The
difficulty of using such a measure in the past has been
that of providing the teacher with the means whereby raw
scores can be readily transmuted into standard deviation
values. A part of this study will be the construction of
such a table for each of the tests so that scores in each
may be quickly transmuted into a common unit of measure
and the teacher be enabled to compare directly the achieve-
ments of a child In each test with his achievements in any
other test and thus locate his particular disability with
respect to addition-integers.
The proposed tests aim to be diagnostic by virtue
of:
1. Isolating, in so far as is possible, and measuring,
those habits in addition of integers v;hich seems to be of
significance in the contribution which they make to the
child's ability to add integers, buch the following s eem
to be:
(a) "•ddition of combinations
(b) Bridging the tens
(c) Single colvunn addition
(d) Addition with carrying
(e) Cjpoken coliunn addition
2. Providing means whereby teachers may compare direct-
ly the achievements of a pupil in one test with his achieve-
ments in each of the othf;rs.
Page 1.
OUTLINE OF PI^OCKDURF:
I. Selection of examples -
By random sampllnc of single digits, and their
combinations into examples. This procedure was used in ell
tests except the first, in which case the addition combina-
tions and their reverses were used.
. II. Arrangement of examples in order of difficulty -
The relative difficulty of the examples was de-
termined by preliminary testing of grades three to six in
tv/o schools. Note ; Those examples, in the case of which a
high percentage of error was due to blurred numbers, were
placed at the end of each test. There were nine such in-
stances in the five tests, ("ae' in test II; "h ' in test III;
"g", "j" and "r" in test IV; "b", "a", "g," and "j" in test
V) pages.
III. Giving of tests to 1400 children in grades three
to six inclusive in seven schools in Berkeley, California.
Instructions to teachers and tabulation of results are con-
tained on pages 13-20.
IV. Tabulation of data -
The zero scores in test II that were due to mis-
understanding of instructions were omitted. Also, the re-
sults of the sixth grade of one school were omitted because
a re-test shov/ed first test to have been erroneovis .
V. Re-testing of the high third, high fourth, high
fifth, and high sixth grades of one school after an inter-
Page 8.
val of two weeks to get data on reliability of test.
VI. Statistical treatment of results -
1. Inter-correlation of each test v;i th every
other test.
2. Correlation of each test a second giving to
the same children after an interval of two weeks .
3. Correlation with VJoody-McGall.
4. Means and standard deviations for each dis-
tribution.
5. Computation of sigma values for each score
in each test.
Page 9
EVALUATING RESULTS
VALIDITY OF TEST - It has not seemed necessary to
gather evidence as to the validity of the test beyond that
furnished by the inter-correlations. They seem to indicate
that a single type of ability is being measured. They are
uniformly high, except in the cases of test II with other
tests. Since the correlations in which test II plays a
part are consistently lov>fer, and since this test resulted
in an undue number of zero scores, and, also, since its
self-correlation was ].ower than the other tests, it is
assumed that its validity as v/ell as its reliability is
lov/ and it is therefore thrown out as of little value.
The low correlation of test V with V/oody-McCall is inter-
preted as being due to the fact that ^''oody-McCall tests a
great many abilities other than addition.
RELIABILITY - or the degree to which it is con-
sistent in its diagnosis, would have been determined by
checking one half of the test against the other half. A
simpler met?iod however, though a somewhat less satis factorj^
one, was used, namely that of correlating one giving of the
test against another giving of the same test. These corre-
lations were uniformly high except in the case of test II.
The latter v.as discarded as unreliable.
Page 10
OBJECTIVITY - There has been no evaluation made of
the objectivity either the giving or the scoring of the test.
It was sought to increase the former as much as possible by
simplifying the instructions as much as was consistent with
clear understanding. Since the instructions are not in any
case complex it is not thought that lack of objectivity of
giving is a serious factor.
Steps taken to insure objectivity of scoring were
the preparation of an an£v;er sheet with answers so spaced
ths-t they could be placed directly under the examples and
answers checked.
In future giving of the test it is planned to have
children score their ov;n papers since the case of adminis-
tration seems of greater importance than the slight sacri-
fice in objectivity of scoring due to the fact that the
children do it. No check was made on the relative accuracy
of teachers and pupils' scoring.
SCALING - The examples were arranged in order of
difficulty by giving them to 300 children in grades tJiree
to six in two schools, each pupil having time to work all.
In the final form all examples were included, arranged in
order of difficulty without any attempt being made to select
only those tlriat represented equal increments of difficulty.
Page 11
RANGE OF APPLICABILITY - It was desired to secure
a test which could be used in grades three to six inclusive.
Reference to graphs on pages 39-43 and tables of means and
standard deviation on pages 45-46 will show that test I is
valuable in grades three to sijc inclusive though slightly
less so in the latter two as shov/n by the niunber of perfect
scores made there. Test III and IV easily measure grades
three to six. Test V while it measures grades, four, five
and six well, does not measure grade three. Ther efore , ex-
cept for test I in grade sir. and test V in grade three the
tests have a range of applicability suited to these grades.
NORlvIS - Norms for each test, in to far as the test-
ing of 1400 children in seven schools of one school system,
can be said to be a basis for norms, are stated in th(
following forms :
1. Means for each grade three to six inclusive.
2. Means for each age eight to twelve inclusive.
3. Each of the above expressed in terms of a
Sigma index (score -t- sigma) reduced to a
scale of 1-200.
Tables 21-30 inclusive give value of accomplishment
in each test in terms of sigma. In the last mentioned table
a score which gives an equivalent of 100 is the norm for the
age or grade classification into v.hich the pupil falls.
These norms are accurate only to the extent that the follow-
ing conditions hold:
Page 12
1. That each dit tribution approximates a normal proba-
bility curve.
2. That the reliability of each test approaches unity
as measured by r-epeatin^^ the tests; self correlation .8 or
over for all except test II.
3. That the ran£,e of the tests is such as to makE them
applicable to the group to be measured. Reference to distri-
bution tables 7-16 inclusive shows that except for test I in
grades six and test V in grade three the tests do measure the
children.
4. That the inter-correlation approach ninity. (Accept
for correlations with test II these are all plus 0.8 or higher.
USE OF TABLC OF SIGlJlA
INDEX VALUES
1. Norm for any sge or grade is alvyays such a score as
will give a Sigma Index of 100. If a child's accomplishment
in each test is such as to give him a Sigma Index of IOC in
each, his scores are satisfactory for his age, assuming that
he is of average intelligence. If, however, he has a sigma
score of 80 in test IV and IOC in test III his accomplishment
in the farmer is not as much as we have a right to expect
from him. Concerning each of the tests in wlriich a child gets
a Sigma Index of less than 100 v;e may say, v/ithin the limits
of the accuracy of the test that he is not doing as well as
he could do in it.
Page 14.
So, while this administration of the test is not for the
purpose of determininc the abilities of our own pupils in
addition but rather as a step iril the construction of t. test,
still I believe that it v/ill be possible for you to tell some-
thing from the results as to the type of difficulty which
should be vorked upon with your own class or with particular
pupils. Do not expect too much in this respect, however,
from this preliminary test.
DIRECTIONS FOR GIVING
Give the tests in the order indicated above.
1. Distribute the first test, printed side down on the
desk of each child.
2. Read the following directions to the children:
"This is a test to see how well you can work addition
combinations. V/ork as rapidly as you can until you have fin-
ished all of them. I'll give you all time to finish. V/hen
you have finished turn your paper over, printed side c'own,
and sit quietly until the others have finished. Now, turn
your pe.pers over, v/rite your name, grade and date at the
top of the paper and go to work."
For each of the following tests, say:
The next is a test in (insert name), ""rite your
name, grade and date at the top of the sheet. V/ork all of
the examples and v/hen you have done so turn yoior pape r over
and wait quietly for the others to finish.
3. '-'ive no help in the working of the examples except
to make clear numbers that may be blurred or to make clear
the directions above.
4. Note carefully in minutes and seconds how long it
takes each of tlie first five and the last one to finish.
Page 13,
Thousand Oaks School.
Berkeley, California.
October 23, 1922.
II'iSTRUCTlOHS TO T5ACHERS FOR GIVING PRISLIMIKARY TESTS
Teachers of Grades 3,4,5 and 6.
Please give the following tests today, preferably
during the regular Ai'ithmetic period and in place of the
Arithmetic Lesson.
1. Addition Combinations.
2. Bridging the Tens.
3. Column Addition without carrying.
4. Column Addition with carrying.
5. Broken column Addition.
This is a preliminary test for the purpose of determin-
ing the relative difficulty of the items in each test. There
is, therefore, to be no time limit to the test, but each cViild
is to work all the examples. When the relative difficulty of
the elements has been determined these will be re-arranged and
the final test psppared.
The purpose of the test, in it's completed form, is to
help diagnose audition difficulties. It should be pos?ible
by the use of these tests to determine .just why a class or an
individual pupil is weak in addition so that steps may be taken
to remedy the difficulty. Later this same step may be taicen
with respect to difficulties in the other fundamentals.
Page 15.
ADDITIOri COMINATIOHS
Name Grade Date Right:
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Page 16,
■ade
Date
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9 •
SINGLE COLUMN ApDITlON
Name
^ Grade •,
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Right c
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Page
17.
•
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BROKEN
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Name
ADDiTIOK 0,'ITH CARRYING)
Grade Date
Rights
a
b
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427
892
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457'
108
295
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Page 18,
T4BLE 1.
RSSULTS 03? PRELIMIMARY TEST IMG- -AJ^DITION C OMBIH.AT I ONS
Ex. Errors "^x. Srrora Ex. Errors Ex. "Plrrors
a
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by
9
z
2
az
5
bz
8
Pat:
:!e ^^.
TAhLE 2
INSULTS G¥ PR-aiMIKARY TEST liiG— BRIDGING TTIT! imiS
EX.
No. of
Terrors
FX,
Ho. of
Errors
Ex.
No. of
Hrrore
Hx.
No. of
"Hlrrors
a
30
i£
19
u
28
ae
38
b
29
1
20
V
30
af
21
c
18
m
18
w
16
e^
31
d
i;l
n
17
X
21
ah
22
e
10
d
12
y
21
ni
21
f
iS
p
26
z
' 25
aj
15
£
1 r>
4. iil
si
24
fxU.
28
ai:
25
h
24
r
?7
sb
21
i
s
afc
19
J
4L
t
50
ad
20
Ex.
N . of
:"rr or £
^x.
uO
T aB"
Ko.of
"terrors
c
1.
■x.
SlliGLE COLinai ATriTIOi^
No. of No. of
"rrors 'FSc. -Errors
a
27
h
70
17
V
31
u
54
i
4y
P
b3
w
29.
c
53
J
43
Q
53
X
52
d
33
k
36
r
39
y
46
e
17
1
25
s
52
f
;i7
m
2 b
t
39
. g.
39
n
40
u
57
Page 20 <
TABL^i: 4.
RBbULTb 01' PRSLlMli4AKY TE^TIMG— APDITIOli vVITK CARKYIIjG
BX.
Wo. of
ifirrors
44
Bx.
IJo.of
Srrors
Ex.
Ho .of
iSrrors
lux.
K o . of
Srrors
43
k
36
40
^
34
35
22
59
m
£^0
81
46
25
n
50
36
52
73
TABLE 5.
BR0KSI3 CCLUKN /J)i;iTiOH
^:x.
No. of
"Errors
■^.
Ko.of
^x.
KG. of
I'lrrors
■Rx.
NO. Of
^'rrors
£.-■
iOl
f
Te-
yr
92
P
58
b
95
e
ll?
1
77
1
79
c
96
h
68
m
51
r
fiB
d
bi
i
y4
n
51
8
uS
e
77
J
135
60
t
73
TABLE 6 .
i!il>j»..B'".iii OF PUriLS T>J\II>G ~ACH 7S3T
GRADE
Test I
Teet II
Teet III
Test I\
r Test V
TKIRD
78
SO
80
21
21
FOURTH
74
68
74
73
73
FI7Tn
77
74
77
79
80
SIXTH
87
89
88
87
R8
TOTAL
316
311
319
260
262
#
#
DIAGNOSTIC TESTS IN ADDITION
Directions f-or giving.
It is very important that the following directions be care-
fully followed if we are to get from the results the hel|> we
hope to get in guiding children.
1. Distribut§ tests, face downward. Warn children not to
turn papers over until told to do so.
2. Have them fill m blanks on the back of the test paper.
Give whatever help is necessary to insure an accurate record of
each of the items.
3.- Read " Instruct ionsTIQ ^e f^upils" aloud, the children
following silently.
4. Note carefully the beginning time. It is well to jot
it down on a £iece of paper.
5. Allow time as follows :-
Test 1 Addition Combinations 2 minutes
Test 2 Bridging the tens 2 "
Test 3 Single column addition 2 "
Test 4 Addition with carrying 3 "
Test 5 Broken column addition 3 "
6. When time is up say, "Stop. Turn papers over."
7. Collect papers.
SCORING PAPERS
1. Count "Number Attempted."
Jk. . This can be quickly done, since the examples are ar-
ranged in rows of 10 each. Take as the number, the last one
worked, disregarding those skipped.
2. Check with a t^ all answers that are incorrect.
Ansv;er sheets should be cut or folded so th?.t they may be placed on
the paper directly under the examples.
3. Enter the "Number CORRECT" in the blank after RIGHTS on
the back of the test paper.
4. Arrange the papers of each test in order of the size of
score, the hSigVjhettsooDce on top.
5. Fasten the papers of each test together by a clip or
a rubber band and turn them in to the Principals office.
Page 22.
11^310135759
2 2 5 7 2 51 1 1
5627231212
517 2 339384
3
2
5
4
7
a
2
1
■ 4
4
7
2
5
3
2
4
2_
4
I-
1
3
6
5
4
5
3
9
6
6
6
3
4
2
3
7
2
5
7
8
3
6
4
3
2
4
3
4 ■
8
a
6
4
1
6
1
3
5
2
3
8
1
1
6
4
6
7
9_
3 •
4
9
2
2
7
8
4
7
9
1
9
5
4
4
8
3
9
9
3
4
3
2
6
7
£
7
6
9
7
5
9
8
9
6
8
:* 9
S
6
9
3
5
5
9
8
6
6
7
7
8
8
9
5
5
7
8
7
5
9
6
7
8
7
8
8
9
?f)^e as.
22
(reverse side of page }
AGE YSAKS kOHTHS
KO. ATTEAIPTKD _110. RIGHT
GRADE.
NAME^
DATE
SCHOOL
INSTRUCTIONS TO PUPILS .
Today we are to take five short tests to see how well we
can work the different forms of addition. This first test is
in addition combinations.
When I say "BEGlli'' turn your paper over, and work as many
as you can, being careful to get them right. Stop work prompt-
ly when i tell you to do so. Ready — Begin.
TIMB TWO KIHUTES.
Test 1.
Addition combinations.
Page 24.
1^ 4 2
19 * 3
18 f 3
19 I 5
17 + 5
IQ I 4
16 r+ 4
17 I 4
17+7
15 + 6
14 + ■?■
18+5
12 + 8
15+9
19+7
19 + 6
17 + 6
13+7
l4 + 8
17 + 8
19 + 9
16 + 6
16 + 9
15+7
12+9
16+7
13 + 8
IS + 8
17 + 9
19 + a
14 ^ ^ ■"
14 + 9 '^
16 + 9 =
19 + 8 =
15 + 8 =
10 + 5 =
13 + 9 i:
8
3
4
1
4
1
2
4-
4
2
1
9
5
6
3
4
5
2
1
6
r^
eJ
4
5
2
6
7
7
1
7
6
4
6
1
3
6
2
5
6
1
6
5
7
4
7
2
3
1
7
7
1
1
1
8
6
6
7
1
4
6
8
5
8
8
4
5
2
1
6
8
3
8
2
4
2
7
X
8
6
7
9
8
3
1
8
6
7
8
9
5
2
4
6
6
5
1
6
9
3
3
5
8
.6
6
8
5
5
7
3
6
2
3
7
9
5
6
5
9
5
7
2
9
4
1
9
3
Pe(T« 25,
(reverse side of page 24 )
AGE ^YSARS liCNTHS.
GRADS
NASitE
DATE
SCHOOL
NO. ATTEMPTED
HIGHT
IIJSTRJCTIONS TO PUPILS .
This is a test to see how well you can work another form
of addition. READY— BEGIN .
TT5ST 2. BRiDGlKG THE TEUS .
AGE Y3 - BS MOiiTIiS_
GRADE
KAltS
DATE
SCHOOL
SO ATTEMPTED
RIGHT
IKSTRUCTIOHS TC PUPILS .
This is a test to see how well you c; n work still another
form of addition. READY — BEGIN.
TEST 3. SIKGLB COLUMH ADDITION
TIME TWO lilNJTES
Ps&e 26.
803
101
512
632
221
lae
534
855
296
42'^
247
136
875
563
268
i94
041
14 7
857
158
3 92
426
710
204
589
774
659
775
352
173
564
4&7
8<.6
892
511
24 9
375
432
717
501
573
205
780
992
651
789
S87
202
954
875
902
289
666
359
898
119
959
359
200
356
48
530
552
430
140
£5
5
75
S6
802
9
88
28
75
918
63
68
842
8
1
217
5
752
8
92
8
2
3
794
57
62
29
7
340
2
562
757
656
4
467
21
3
89
56
135
55
69
27
347
44
60
ik^
7
3
572
83
39
52
8
995
3
95
82
977
73
983
1
602
66
435
793
379
382
4 5
830
79
563
395
48
90
42
89
372
79
79
99
83
4
987
396
379
72
'21Q
527
33
68
63
380
701
962
16
AGS
Y^ARS
GRADH.
DATE
SCHOOL
Page 27,
(reverse side of page 26 )
MONTHS
1^0. ATTBMPTEH
NO. RIGHT
INSTRUCTIONS TO PUPILS.
This is a test to see how well you can worK another form
of addition. READY— BEGIN .
TO T^ACII'CR Nate Change in time.
TBST 4 . ADI'ITION WITH CARRYING
Time Tt'iree minutes
AGS
Y15ARS
MONTHS
NO. ATTISMPTED
NO. RIGHT
GR.U)3_
NALE__
DATS
SCHOOL
INSTRUCTIONS TO PUPILS .
THIS is a test to see how well you csn woric still lanotber
fiorm of addition. READY— JSITG IN.
TaST 5. BROKEN COLUMN ADDITION
TIME THREE MINUTES
- LIaGImOSTIC 7H
ADDITIOInI .
pB^;e 28.
Tefct i.
pombi
nations.
4
5
6
8
5
10
8
6
10
10
7
9
9
10
6
10
5
9
6
7
9
7
9
10
10
4
3
5
5
6
7
9
9
9
6
8
10
11
5
•
11
16
11
12
8
4
8
1
7
8
11
11
12
13
3
8
7
12
10
14.
18
4
13
2
7
10
11
8
3
16
14
■ 9
13
14
12
13
14
15
16
15
13
12
16
14
15
17
12
13
15
17
Test
3. Single
Column Addit
ion
20
29
19
19
21
17
20
20
20
21
26
20
27
25
25
23
27
28
29
27
31
23
30
28
31
Test 4.
Addit
ion
With 1
Garryin
1078
1376
i
2134
1777
1504
1442
663
2097
1399
758
1839
951
2292
2243
2060
IIST
1
1621
993
1871
1731
Test 5.
Broke
n Coluran .
Addition
357
655
1428
919
1287
753
901
1613
1239
1371
17
1777
845
1370
1137
1622
1307
1066
1754
2071
193:
Test 2.
Bridg
21
22
22
23
22
21
24
2X
infi
the Ti
21
23
20
24
.336
23
25
22
25
ens .
28
22
27
22
21
23
21
■ fee
26
27
20
23
25
26
23
15
22
i::
<■ r
' • r
ts
Page 29.
TABLW. 7.
DISTRIBUTION OF SCORES iiY GRAD15— ADDITION COMBINATION
No*Correct
* 3d (irade
4tVi Grafie
5th Grade
6th Grade
U-o
a
4-7
2
1
1
8-11
3
12-15
1 '
16-19
20-23
5
i 2
24-27
14
1
23-31
IB
2
1
32-35
38
6
36-3y
45
18
4
1
40-4'5
74
28
4
2
44-47
36
23
13
%
48-51
41
34
12
4
52-55
28
67
23
12
56-59
15
51
20
11
60-63
7
51
33
24
'54-67
7
3?
37
P4
5S-71
7
29
36
33
72-75
3
?.5
51
43
76-79
18
38
29
80-83
9
24
40
84-87
8
15
14
88-89
5
17
22
1 90
7
23
40
L
343
417
3&4
5«j3
Page SO,
TABLH 8
DISTRIBUTIOH OF SCORES BY GRADE— BRIDGING TII^ TTSNS
Iffo. Correct
3rd Grade
4th Grade
5'th Grade
6th Grade
0-1
?2
39
9
4
2-3
14
8
4-5
10
2
3
1
6-7
26
5
9
1
8-9
29
8
2
10-11
33
8
8
B
12-13
40
20
8
4
14-15
38
35
13
4
16-17
32
37
21
11
18-19
34
45
12
18
2U-21
21
53
26
16
22-23
9
32
28
21
, 24-25
3
27
36
20
£6-27
1
21
31
23
28-29
2
14
34
32
30-^1
13
27
15
32-33
(>
9
8
25
34-35
7
23
18
3e-37
5
15
19
38-39
13
40
43
&
?24
401
351
286
k
Pa^je 31,
?ABLE 9
DISTRIBUTION OF SCORES BY GRAD3--SINai^ COLUMN ADDITION
No. Correct
3rd Grade
4th Grade
5th Grade
6th Grade
13
1
1
9
1
2
13
1
3
19
4
17
4
5
30
14
1
6
25
5
1
1
7
28
6
9
1
8
33
16
5
2
9
31
30
6
1
10
37
38
13
6
11
29
40
13
13
12
32
59
35
16
13
10
56
■d'd
iikJ
14
11
36
27
24
15
4
33
34
30
16
2
24
27
23
17
1
17
33
30
18
13
35
25
19
1
10
22
26
20
5
23
17
21
2
15
9
22
1
12
13
23
2
8
15
24
3
13
17
25
5
13
12
K
345
422
367
303
'J; -I ■».. - iVl I ^i J,
I t
Pac« '^2,
-TABLE IC
DISTRIBUTION OF SCORES BY GRAIR-- ADDITION WITH CARRYING
No. correct
3cl Grade
4th Grade
5th Grade
' 6th Grade
26
10
1
1
14
1
2
25
2
3
26
4
2
2
"
4
33
10
7
1
5
63
15
1
4
6
43
33
7
2
7
34
42
15
13
8
37
58 j 23
16
9
24
52 i 32
25
10
8
36
32
28
11
3
45 i 54
36
12
7
38 1 49
38
13
1
1
26 42
39
14
12 19
32
15
1
18 18
24
16
3
17
19
17
4
8
11
18
10
7
19
3
7
11
20
5
5
11
^1 ^
344
417
345
319 ^
.:[
CI
ex
li^
!t:
5
SI
Page 33,
TABLE
i>ISIHI£uTIOB 0? SCCK^a 2Y GH.'O}'^ — BPOKT^N COLUM? ADDITICM
Jbio. Correct
34 Grade
4th Gradg
5tb Grsade
6tb Grade
116
la
2
1
48
13
2
2
Z
46
25
8
2
3
45
SO
8
8
4
44
58
25
17
5
24
57
31
27
12
65
^y
37
1
4
54
54
50
8
43
58
41
9
4i
5k;
10
16
37
32
ii
21
19
12
<0
11
16
13
1
11
9
14
J ^
4
14
15
1
1
13
16
3
12
17
1
2
^
18
1
2
4
19
4
20
3
If
341
415
360
365
Pa^.;-?? M.
TABLE
DISTRIBUTION OF SCORES BY AOfH— ADDITION
CCM.BI1U7I0K
1
Wo.correct
. 7
6
■ 9
10
11
12
13
1
1
1
1-4
2
1
1
5-9
1
1
10-14
1
1
1
15-19
20-24
1
4
1
1
25-29
8
8
2
30-34
3
18
13
8
1
35-39
4
31
ii6
4
6
1
40-44
3
50
1
40 ! 14
7
2
45-49
2
28
29
16
10
o
2
50-54
1
2S
44
55
22
10
1
55-59
1
14
37
34
23
7
1
60-64
5
9
30
3G
37
13
1
65-69
1
5
19
42
41
15
4
70-74
10
22
55
45
21
5
75-79
1
7
17
38
32
17
7
30-84
2
12
18
26
13
9
85-09 •
11
20
23
IS
9
90
7
17
25
25
7
S
S2
^17
319
363
300
145
46
J
TABLE 13.
Ti « ,^ o '* *%
DISTRIBUTION OF SCORTi;S BY AGTS— BRIDGING THE T^.UQ
-AGE-
Ixo. Correct
7 r 8
9
10
11
12
13
10
24
16
9
2
1
3
3
1
ii-3
8
8
3
4-5
3
5
3
5
e.7
1
13
14
2
3
1
8.9
1
19
6
5
2
1
lu-ll
4
14
15
11
11
1
12-i-?
2
26
17
8
11
1
.
14-15
1
24
21
17
9
4
'
16^17
4
25
21
27
lb
2
1
itt-iy
21
32
19
■^l
6
>0-?l
3
19
33
31
10
7
3
22-23
11
17
22
33
10
'
24-25
1
21
31
18
9
4
iJ6-27
1
3
12
ki5
23
a
3
p.a-29
4
12
25
S9
11
29
•■^0-31
1
5
21
18
6
3
32-33
2
9
12
18
5
2
34-35
1
10
13
15
7
1 ^
36
2
5
8
5
7
3
37
1
15
34
33
27
12
H
19
211 1
303
333
288
113
i 72
^"i
I /
#
\^^3 36.
TABLTi: 14,
UibTRliiUTlOll Oi"- SCORES BY AGE— o.l.GLH COL^r-i: ATT^T.CK
-
So. Correct
7
b
i#
xu
11
13
a
5
1
1
to
5
1
2-3
16
tt
3
4*5
k,
ii4
ly
D
<•
6-7
4
50
13
13
2
3-9
2
4«;
oy
15
13
4
iU-11
b
^9
(iw
42
23
7
12-13
i
.^'J
65
81
44
16
5
?.4.15
1
17
47
57
51
16
9
ir,-i7
4
>]
48
54
5
16-3 9
££:
44
38
24
7
20-21
1
9
24
r.l
13
■»
t
i:;j-ir3
1
5
11
14
17
7
24
y
^0
8
C
2
2b
1
16
11
7
£i
17
^j:;6
v.i;to
5 5
296
io5
48
-?)
PatTe ^B,
TABLE 16.
DISTRIBUTION OF SCORES BY AGS— BR0KJ5N COLUMN ADDITION
-AG
S-
No. Correct
7
a
9
10
11
12
iS
5
58
39
11
2
1
4
25
8
7
5
2
2-3
4
58
45
25
14
6
2
4-5
5
44
64
83
46
15
8
6-7
1
20
74
91
75
28
9
8.9
12
49
73
81
30
7
10-11
2
18
43
43
19
7
12-13
1
4
13
13
16
4
14-15
1
i
4
5
12
7
4
16-17
1
6
2
7
4
18-19
2
3
3 1
4
20
\
2 ,
2
H ^-
19
221
308
360
298
134
47
'
?«ge *S9.
ADDITIOM COMBIl^ATIONS.
uH ■
3^
/&
6th Grade.
^S
?i^
/(^
^^
>^2J
^T-
/4
■
31
5th Grade.
/ r
4th Grade.
^
I >
3rd Grade.
" '\ J' '» ^C' — ^t" — i ' »^" ""» T' — t" ■■■ ■% ) r- — -1 ]i r r — t h • ■ >»' • ' J
rn.
if/, /V 'Y X3 Z7 3/ 3s 3
1 f3 *7 -i/ -i'S -ij 6 J 0? ?/ To' "7/ ^3 8-f ■*o
4. 3
J3R1DGOG THE T3NS.
Pai;e 40.
6th Grade.
J — '-
Sth Grade.
n
4th Grade.
^^
J?6
3rd Grade.
/ 3 3r 7 ^ ^7 o O- /> /^ 2/ ^J ;^J' 2/ 2? 3/ 3J' 37 3^
.i>
J<o
X'f-
/2-
34.
2Vf
IX.
3i-
II- -
Pa^e 41.
SINGLE COLUMN ADDITION.
6th Grade.
._J
5th Grade.
n
Lr
4th Grade.
r
n
j4
a-/- ■
12.
3rd Grade.
1 1 — T II I « ^ « • i' f T !1 J—'. ! S ' ,
■J- 3 "/ >> 6 7 5^ ? ^o // ^1- '3 'V /J- y^ /> 'f /9 2<j 3y 22. 23 2^ :i
^ r r^
3<"
-J-7
ADDITION WITH CARRYING
6th Grade.
r,t;e 42.
A /
3- J ^ J t
7 S <f /o // , >_ li y/ 'i' /(, '? 'i' ''> yjiJ
-^4
91'
1%
H^
^
ra^e 43.
Jt
1
BROKEN COLIMN ADDITION.
>-J
si
6th Grade.
f
r
-^
I — ' >— 1
^
_n
i/y
-
36
.
' 1
5th Grade.
%1
-
WV^I«.
/f
'
1 1
r
I
' S_-^_
1
.>^
1 s-^
tt--^'
-
Jt
•
4th Grade.
7''
-
. x
/X
— .
'
f
1
—
J
L-u-j—i , .
p
« . ,
» V
3 ^
^ s
- C
7
J^
^
/
D
'■ /-
(, ^5 . V 1 y / fe /
-> ' > > f i. •
3rd Grade.
1 » r
/ ;?. 3 y. or (6 7 F f /<• 'f t i, «i t)^ /-J^ /fe /y /* /y ^<
Page 45.
CO
«
to
CM
oo
«
CM
o
CM
s
^
CO
o
E-'
cq
H
O
I
I
CO
2;
o
M
I
CO
CO
)Si
t
CO
00
CO
CO
C3
in
CM
00
C5»
to
«
to
o
to
CM
f-4
10
to
to
CO
r-l
U3
CM
•
CO
to
•
CM
10
M
H
CM
CO
<o
en
CM
CJ»
m
o
CM
to
i
to
C3»
00
'if
to
CM
to
CM
0»
vO
O
CJ»
«o
to
o
10
H
CM
in
CO
CM
in
00
to
CO
CM
•
CO
lO
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CM
"00"
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at
CM
to
Page 47.
TABLl? 21 ,
SIGMA II^DEX VALinSS Y'On SCO'llSS I*i ?y;ST I.
•^
- Aaii;-
'''•■-
1
B6ore
8 yrs
^ .-— . » ...
9 yrs
10 yrs
11 yra
f ' ■■
12 yrs
5
36
37
21
9
10
46
43
28
16
7
15
53
50 ;
34
•dS
15
20
60
56
41
30
22
25
68
62
48
37
29
30
75
68
54
44
36
35
83
74
61
51
44
40
90
80
67
h8
51
45
97
87
74
65
58
50
105
93
81
71
ee
55
112
99
87
78
73
60
120
105
94
85
80
65
127
HI
101
92
88
70
135
117
107
99
95
75
142
124
114
106
102
80
149
130
120
113
110
65
157
136
1.27
120
117
90
164
142
134
127
124
1.482
1.224
1.324
1.334
1.462
liOTE - to find Sigaa Index Values for intermediate points
add amount on line (#) to the value of the next lower score
above for each unit of score. 3X. 36= 83 plus 1.482 or 84.482
37= 83 plus 2x1.482 or 85.96
Pe^e 48.
T.^iiLE 22.
SIGJ».A IhHr X V A^ UIl;£ iCn scons J^ ':'EST II.
r -AGS-
Score
8 yrs.
9 yrs.
10 yrs.
11 yre.
12 yrs.
2
66
67
55
51
29
4
71
70
59
55
35
6
77
75
64
eo
40
8 •
82
7:,
68
64
45
10
a 7
S3
72
68
51
12
93
86
76
73
56
14
m
90
80
77
61
16
—
103
94
85
81
66
18
109
96
89
86
72
20 114
102
93
90
77
22 119
106
97
94
82
24 125
110
1-01
99
88
26 130
114
135
103
93
23
13G
110
110
107
98
30
141
122
114
112
104
32
146
126
113
116
110
~34
36
152
157
130
122
120
114
134
126
125
120
38
162
13B
131
129
125
if
2.688
1.984
2.096
2.164
2.659
NCTS - to find Sigma Index Valu«8 for intermediate points add
atiount on line (#) to the value of the next lower score in t'le
first column above. ;x. Score 27 = 120 dIus 2.688*132.688 or
133.
f^e,*^ 49.
TABLS 23 .
SIQMA lj;i3gX VAlUaS FOR SCOKIa Ilj TEST III
-Aa.g-
Scor«
8 yrsi
68
9 yrs.
56
1 yrs^
45
11 yra.
30
1£ yrs,
oo
77
64
o4
39
»3o
6
86
73
62
4G
47
6
95
81
70
57
55
10
104
89
76
64
12
113
97
66
(V
14
122
105
94
84
81
16
131
113
103
94
90
18
140
122
111
103
99
20
149
130
119
112
107
22
158
136
127
121
116
24
167
146
135
628
4.081
4.081
4.540
4.O'*0
aOT-'^ .. bo find Sifc.flia Index Values for interiiiedii.te pointj
add amount on line {Ir) to the value of the next lower score
in the rirst cclusji ^ccve. £x. ii^covfi 23- 15o piu& 4.C2G
or 162.628
Pa.;e 50.
TaBLH; 24.
SIGLA IIJIIDC VALU'-S T 01^ SCOI'J^S IK TTEST IV
1
Score B yrs. 9 yrs.
10 yrE.
11 yrs
12 yrs.
1
67
59
47
20
36
73
64
52
36
41
3
78
69
57
42
.4fi
4
t4
74
63
48
fiS
5
90
79
63
54
fi7
6
95
83
73
61
f^9
7
101
88
78
67
fi7
S
106
93
S4
73
7a
9
112
98
£9
79
7fl
10
lis
103
94
86
R.-^
11
i;i5
103
';;9
02
rtH
12
li39
113
105
94
13
134
117
110
104
Q9
14
14^
1^-:
115
111
104
15
146
127
120
117
ino
16
151
132
126
123
lift
17
157
137
131
129
ion
13
162
142
136
136
iy.fi
19
icS
14c
142
142
20
174
151
147
148
T 'J /-•
#
3.58
4.12
3.8
3.2
— 6»a6 i
Fa^:» ?>1.
TABLS 25
SIGaA INET^i VALJt-L: _?<!!> SC^: PrS IK ?5ST Y.
Score
8 yrfei
9 /rs
10 vrs,
11 yrg,
la yrf
2
A
6
8
10
11
12
13
- 14
15
16
le
Id
20
M^
92
Jti'JL
ice
X'iU
1^8
1&6
164
172
IflO
LIS
I9e
S.QO
65
100
106
liJ
116
124
iro
IZG
14;
14 6
It a
U5
IVl
1''7
183
5.S82
65
7X
7<>
83
31.
92
9B
103
108
114
iiy
li-4
i.'^.y
I'b
141
146
IM
157
If.?
168
5.376
57
63
69
74
30
36
38
104
110
llf
igs
127
l?,y<
\7,9
145
151
157
16?
166
5.882
60
66
71
76
ei
S7
QO
97
102
107
113
118
1J:3
129
1.^)4
139
145
l^C
•155 •
5.262
Pag4 52,
7ABLB 26.
SIGKA INDISX VM-U^'IS FOR SrQRBS IN TCST I.
-GHADT5-
Score
Third
Fourth
:"ifth
i:ixth
<
30
20
11
10
39
28
18
15
48
35
26
4
20
57
43
32
12
25
67
50
39
20
30
76
57
46
28
35
85
65
63
36
40
95
72
60
44
45
104
79
66
52
50
113
87
73
60
55
122
94
80
68
60
132
101
87
76
65
141
109
94
84
70
150
116
101
92
75
159
123
108
100
80
169
131
115
108
85
178
138
122
116
90
187
145
129
124
i
1.851
1-470
■ " •' ■
1,388
1,6
hOTK - to find Sigma index Values for intermediate points add
amount on line (#) to the value oi the m-xt lower score in thft
first column above. ICx. ii^ore 27= 67 plus (2 x 1.851) or 70.702
4,
-->
i c
Page 55,
TABLE 27,
SIGMA INDEX VALUTAS POR SCOR-RS IfJ Tp^K? II.
-GRADE-
Score
Third
Fourth
Fifth
bixth
2
67
63
52
37
4
73
67
56
42
6
80
72
60
47
8
86
76
64
52
10
92
80
68
56
12
99
85
72
61
14
105
89
76
66
16
111
93
80
71
18
118
98
84
76
20
124
102
88
81
22
130
106
92
86
24
137
111
96
91
26
143
115
100
96
28
150
119
104
101
30
156
124
108
106
32
162
128
112
111
34
169
132
116
116
36
175
137
120
121
#
-.,.3.i74
iLill^
2<02 ,
2.4«a
IilOTS - to find Sigma index Values for intermediate pointe add
amount on line (#) to the value of the n-xt lower score in the
first column above, ifix. Score 27-143 plus 3.174 or 146.174
Pai^o 54.
TaBLB 28.
SIGliA IhlQ-'^X VaLuKS FOB SCORi;S IK TKSY IH
-GRAX*'^-
Score
Third
yourth
Firth
Bixtta
1
62
36
'^<■J
15
2
67
44
33
20
3
72
49
38
25
4
77
54
42
30
S
83
59
47
35
6
88
64
51
40
7
93
69
56
45
8
99
74
61
50
^
104
7y
66
55
10
- 109
84
70
61
11
114
89
74
66
12
120
94
79
71
13
125
99
83
76
14
130
104
88
81
15
135
110
92
36
16
141
115
97
91
17
146
120
102
96
16
151
125
106
101
19
156
130
111
106
20
162
135
115
111
21
167
140
120
116
5.263
5.076
4.558
5.063
TABLE 29.
Page 55,
SIGMA IHDEX VA in?;S FOP SCORES IN TI^ST IV.
T
-GRAD15-
Score
Third
Fourth
Fifth
Sixth
1
64
50
33
31
2
71
56
39
37
3
79
61
45
43
4
87
67
51
48
5
95
72
57
54
6
102
78
63
60
7
110
84
69
66
6
118
89
75
71
9
125
95
80
77
10
133
101
86
83
11
141
106
92
89
12
149
112
98
94
13
156
118
104
100
14
164
123
110
106
15
172
129
116
112
16
179
134
122
117
17
187
140
128
123
18
195
146
134
129
19
151
139
135
20
157
145
141
, #
7.718
5.632
5.898
5.762
P8m:;s 56.
TABLE 30.
SIGMA Il'JDBX VALUl^S FOR SCCRSS Hi T^ST V .
-GRADE-
Scor«
Third
Fourth
i'ifth
Sixth
1
85
63
51
55
2
95
70
58
ei
3
105
77
65
66
4
115
84
71
71
5
125
91
78
76
6
135
99
85
32
7
145
106
92
87
8
154
113
99
92
9
164
120
105
97
10
174
127
112
103
11
154
134
119
108
12
194
141
126
113
13
149
133
118
14
157
140
124
15
163
146
1£9
16
170
153
134
17
177
160
139
18
184
166
145
19
191
174
150
20
199
181
155
#<
9.92
7.142
6.826
5.262
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