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An experimental comparison of two methods for alternating mathematics and science

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AN EXPERIMENTAL COMPARISON OF TWO METHODS
FOR ALTERNATING MATHEMATICS AND SCIENCE
A Thesis
Presented to
the Faculty of the School of Education
University of Southern California
In Partial Fulfillment
of the Requirements for the Degree
Master of Science in Education
by
Arthur Chesley Francis
January 1940
UMI Number: EP53814
All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
Dissertation Publishing
UMI EP53814
Published by ProQuest LLC (2014). Copyright in the Dissertation held by the Author.
Microform Edition © ProQuest LLC.
All rights reserved. This work is protected against
unauthorized copying under Title 17, United States Code
ProQuest LLC.
789 East Eisenhower Parkway
P.O. Box 1346
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<3t f t
This thesis, written under the direction of the
Chairman of the candidate’s Guidance C om m it­
tee and approved by all members of the C om ­
mittee, has been presented to and accepted by
the Faculty of the School of Education in partial
fulfillment of the requirements for the degree of
M aster of Science in Education.
D ate...
Guidance C om m ittee
0. C. Crawford
Chairman
0. H. Hull
William G. Campbell
TABLE OF CONTENTS
CHAPTER
PAGE
I.
1
THE PROBLEM..................
Statement of the p r o b l e m ....................... 2
Importance of the problem....................... 3
II.
PROCEDURE. . . . ................ ............... 4
Procedure of the experiment.................... ,4
Groups of students participating in the
e x p e r i m e n t .............. ................. 4
The school subject to be measured............. 5
Mathematics studied by thegroups.............. 6
The test used.................................. 6
Scoring the tests.............................. 7
Procedure of tabulation.......... # ............. 7
Comparability of groups...................... 11
Gains in test scores for eachgroup........... 11
Comparisons of gains for B8 and A8 groups
se p a r a t e l y ................................. 12
Comparisons of gains for combined B8 and
AQ groups................................... 13
III.
RESULTS OF THE B8 GROUPS .
.................... 15
Comparability of groups........................ 16
Gains made on tests by each B8group*........... 23
iii
CHAPTER
PAGE
Comparisons of the three pairs of groups in
terms of
g a i n s .............................
34
Comparisons of the ten-week groups, as a
single group, with the semi-weekly alter­
nation groups, as a single group, in terms
of g a i n s ....................................... 44
Statistical comparisons of B8 groups for
prediction purposes.............................47
IV.
RESULTS OF THE AB GROUPS
......................... 56
Comparability of groups...................... . . 57
Gains made
ontests by each A8 g r o u p ............ 63
Comparisons of the two pairs of groups in
terms of g a i n s ................................. 66
Comparisons of the ten-week groups, as a single
group, with the semi-weekly alternation
groups, as a single group, in terms of
gains..................... .................... 76
Statistical comparisons of A8 groups for
prediction purposes •. • •
V.
.................. 81
RESULTS OF B8 AND AQ GROUPS C O M B I N E D .............. 88
Comparisons in terms of gains of the combined
ten-week groups as a single group, with the
combined semi-weekly alternation groups as
a single g r o u p .........................
89
iv
CHAPTER
PAGE
Statistical comparison of gains of these
same two large combined groups for the
purpose of showing predictions .............
94
Statistical comparison of these two large
groups showing a second method of making
predictions. .....................
VI.
. . . . .
96
CONCLUSIONS AND RECOMMENDATIONS.................. 100
Conclusions..................................... 101
Recommendations..........
101
LIST OF TABLES
TABLE
PAGE
I. Intelligence Quotients of Groups IA and IB,
B8 Classes . . . . . . .
........................ 18
II. Intelligence Quotients of Groups IIA and IIB,
B8 C l a s s e s ............
21
III. Intelligence Quotients of Groups IIIA and IIIB,
B8 C l a s s e s ..................................... . 2 4
IV.
Gains Made by Group IA, Ten-Weeks, B8 Class. . . .
27
V. Gains Made by Group IB, Semi-Weekly Alternation,
B8 C l a s s ................
VI.
VII.
Gains Made by Group IIA, Ten-Weeks, B8 Class . . .
28
29
Gains Made by Group IIB, Semi-Weekly Alternation,
B8 C l a s s ......................................... 51
VIII.
IX.
Gains Made by Group IIIA, Ten-Weeks, B8 Class. . . 32
Gains Made by Group IIIB, Semi-Weekly Alternation,
B8 C l a s s ......................................... 33
X. Comparisons of the Two Methods in Terms of
Gains for B8 Groups............................. . 3 5
XI. Comparisons of Ten-Week Groups as a Single
Group, with Semi-Weekly Alternation Groups as
a Single Group, in Terms of Gains for B8 Classes 45
XII. Statistical Comparison of Gains of B8 Groups Show­
ing Predictions for Separate Class Groups and
for the Total Groups as Single Large Distribu­
tions. ........................................... 51
TABLE
PAGE
XIII. Statistical Comparison of B8 Groups Showing
a Second Method of Making Predictions by
Different Computations of the Means . . . . . .
54
XIV. Intelligence Quotients of Groups IVA and IYB,
A8 Classes.........................
58
XV. Intelligence Quotients of Groups VA and VB,
A8 C l a s s e s . ...............................
XVI. Gains
XVII. Gains
61
Made by Group IVA, Ten-rWeeks, A8 Class. . . 64
Made by
Group IVB, Semi-Weekly Alternation,
A8 Class......................................... 65
XVIII. Gains
XIX. Gains
Made by
Group
YA, Ten4Weeks* rA8 Class . . .67
Made by Group VB, Semi-Weekly Alternation,
A8 Class.
..................................... 68
XX. Comparisons of the Two Methods in Terms of
Gains for A8 G r o u p s ..................
69
XXI. Comparisons of Ten-Week Groups as a Single
Group, with Semi-Weekly Alternation Groups as
a Single Group, in Terms of Gains for A8
C l a s s e s ..................................... . . 77
XXII. Statistical Comparison of Gains of A8 Groups
Showing Predictions for Separate Class Groups
and for the Total Groups as Single Large Dis­
tributions................
83
vii
TABLE
PAGE
XXIII. Statistical Comparison of A8 Groups Showing
a Second Method of Making Predictions by
Different Computations ofthe M e a n s ........... 86
XXIV. Comparisons of Ten-Week Groups as a Single
Group, with Semi-Weekly Alternation Groups
as a Single Group, in Terms of Gains for the
Combined B8 and A8 Classes..................... 90
XXV. Statistical Comparison of Gains of B8 and A8
Groups Combined Showing Predictions for the
Total Ten-Week Groups and for the Total SemiWeekly Alternation Groups as Single Large
Distributions.......... ..
................. 95
XXVI. Statistical Comparison of B8 and A8 Groups
Showing a Second Method of Making Predic­
tions by Different Computations of the Means. 98
LIST OF FIGURES
FIGURE
1.
PAGE
Distribution of the Intelligence Quotients of
Group IA, Ten-Week B8 Class, and of Group IB,
Semi-Weekly Alternation
2#
B8 Class...................19
Distribution of the Intelligence Quotients of
Group IIA, Ten-Week B8 Class, and of Group IIB,
Semi-Weekly Alternation
3.
B8 Class..........
22
Distribution of the Intelligence Quotients of
Group IIIA, Ten-Week B8 Class, and of Group IIIB,
Semi-Weekly Alternation
B8 Class.
.............. 25
4. Distribution of Gains Made on Tests by the Ten-Week
B8 Group IA, and the Semi-Weekly Alternation B8
Group IB........................................... 38
5. Distribution of Gains Made on Tests by the Ten-Week
B8 Group IIA, and the Semi-Weekly Alternation
B8 Group IIB...............................
40
6. Distribution of Gains Made, on Tests by the Ten-Week
B8 Group IIIA, and the Semi-Weekly Alternation
B8 Group IIIB...................................... 43
7. Distribution of Gains Made on Tests by the Com­
bined Ten-Week B8 Groups, I, II, IIIA, and the
Combined Semi-Weekly Alternation B8 Groups, I,
II, IIIB...........................
48
FIGURE
PAGE
8. Distribution of the Intelligence Quotients
of Group IVA, Ten-Week A8 Class, and of
Group IVB, Semi-Weekly Alternation A8 Class . . 59
9. distribution of the Intelligence Quotients
of Group VA, Ten-Week A8 Class, and of Group
VB, Semi-Weekly Alternation A8 Class............ 62
10. Distribution of Gains Made on Tests by the Ten^..
Week A8 Group IVA, and the Semi-Weekly Alterna­
tion A8 Group IVB................................ 73
11. Distribution of Gains Made on Tests by the TenWeek A8 Group Vt, and the Semi-Weekly Alter­
nation A8 Group VB.
.. ......................... 75
12. Distribution of Gains Made on Tests by the Com­
bined Ten-¥/eek A8 Groups, IV,VA, and the Com­
bined Semi-Weekly Alternation A8 Groups, IV,VB. 80
13. Distribution of Gains Made on Tests by the Com­
bined Ten-Week B8 and A8 Groups, I, II, III,
IV, VA, and the Combined Semi-Weekly Alterna­
tion B8 and A8 Groups, I, II, III, IV, VB
• • 93
CHAPTER
I
THE PROBLEM
About five years ago, the subject of general scienee
was introduced into the eighth grade course of study in the
junior high schools of Los Angeles.
Up to that time its
study had been confined to grade nine.
It was the feeling
at that time that much of the content of mathematics for
grade eight was not of practical value to the student.
There­
fore, half of the time which had been allotted to mathematics
was given to the new subject of general scienee.
It was left
to each junior high school to revise its mathematics-general
science program in accordance with this new time allotment.
/
Some schools chose to teach each subject ten consecutive
weeks during the semester while others chose to teach the
two subjects on a semi-weekly alternation basis throughout
the semester.
ter method.
The Edison Junior High School adopted the lat­
But after using it for five years, the mathematics
science teachers of this school felt that a change was advis­
able.
Since one subject was taught two days and the other the
remaining three days of each week, the continuity of both was
broken and it seemed that the students were not making suf­
ficient progress in either subject.
Reports had come from teachers in other junior high
schools that the progress made by their students in mathematics and
z
science was satisfactory due to the fact that each subject
was taught ten consecutive weeks, thus giving continuity in
the presentation of subject matter*
The Edison teachers felt that these hearsay reports
were not sufficient proof of the superiority of the ten-week
method td warrant its adoption in their school.
Therefore, the mathematics-science teachers of Edison
Junior High School decided to compare these two methods of
teaching the two subjects by actually using them experiment­
ally in their own school.
It is the purpose of this thesis to show the evidence
obtained from this experiment and to draw conclusions as to
the relative merits of both these methods.
In this chapter we shall present the problem involved
in this experimental investigation, and also show the impor­
tance of its solution in making the mathematics-science pro­
gram for the eighth grade in the Edison school.
Statement of the problem.
It was the purpose of 'this
study to compare experimentally by a standardized arithmetic
test the achievement of several pairs of comparable groups.
Each pair had studied mathematics by two different methods,
each on a different time-basis, in order to determine which
method was the better.
The two different time-bases for these
methods were (1) one of the groups of each pair studied math­
ematics for ten consecutive weeks, and science the other ten,
3
and (2) the other group of each pair studied mathematics on
a half-week basis, alternating with science the other half*
Importance of the problem*
This study was of special
importance to Edison Junior High School in that the results
of the experiment determined the time-basis to be adopted
for the teaching of mathematics and science in the eighth
grade.
It is hoped that the findings will be of value in other
Los Angeles schools in selecting a time-basis for the teach­
ing of these two grade eight subjects.
Also the findings of this experiment may shed light
on the psychological element of time-space learning where
there are specified intermissions of time in the learning
process.
CHAPTER
II
PROCEDURE
In accordance with the plan of solving the problem
stated in the preceding chapter, the mathematics teachers
conducted an experiment.
The procedure which was followed
in conducting this experiment and in tabulating the result­
ing data is related in this chapter.
I.
PROCEDURE OF THE EXPERIMENT
In describing the experimental procedure, the follow­
ing points will be discussed:
groups of students participa­
ting in the experiment; the school subject to be measured;
the mathematics studied by the groups; the test used; and the
scoring of tests.
Croups of students participating in the experiment.
When students entered Edison Junior High School, they were
divided into three main groups, slow, average, and high on
the basis of their intelligence quotients.
This procedure
was followed in grouping students for each of their respective
classes.
Hence all groups in each of these three classifi­
cations were assumed to be reasonably comparable.
With this
student classification already accomplished, the mathematics
teachers decided that (1) every teacher who had two average
5
or high groups in grade eight mathematics should participate
in the experiment by having one group study on the ten-week
basis, and the other on the semi-weekly alternation plan,
(2) the slow groups should be omitted since they were doing
special remedial work.
On this basis, ten groups under five teachers were
selected for the experiment, six of which were B8, and four
were A8.
There was a total of 330 students involved, 198 of
whom were B8, and 132 were A8.
With this arrangement, it was felt that the chances
for success of the two methods were placed on an equal foot­
ing because each pair of groups was (1) reasonably compar­
able,
(2) instructed by the same teacher, and (3) given the
same amount of mathematical content.
The school subject to be measured.
The course of study
in general science for grade eight as set up for the Los
Angeles Schools was informal and allowed a choice of subjects
best adapted to each particular class.
Since all science
classes were not studying the same topics no standardized
test could be found to measure equally all content studied.
The mathematics course of study called for a definite assign­
ment however, and there were standardized tests available
which would measure the content of the prescribed course of
study.
Hence it was decided that mathematics hlone was to
be measured.
6
Mathematics studied by the groups.
In further prep­
aration for the experiment, the mathematics teachers together
with the school administrators planned the content of* work
for both the B8 and A8 groups.
The text1 adopted for all
grade eight students in Los Angeles was used.
But in order
that all B8 groups would receive instruction in the same
topics in connection with this text, an outline of work to
be covered was given to each B8 teacher.
The same procedure
was followed for the A8 groups.
The test used.
Since the superiority, if any, of
either method was to be determined by comparing the gains
made by the groups using one method with the gains made by
the groups of the other, it was necessary to give both a
pretest and a final test.
Therefore, on the first day of the experiment each
teacher gave a pretest to his or her two groups.
The stand­
ardized test given was the New Stanford Arithmetic Test, Form
W, for grades £-9, which because of its wide range in grades
would include most of the mathematical experiences of grade
eight.
It has five pages and is divided into two parts,
Arithmetic Reasoning, and Arithmetic Computation, the time
1
£.J. Brueckner, C.J. Anderson, and G.O. Banting,
Mathematics for the Eighth Grade. (Sacramento: California
State Printing Office, 1932)• 380 pp.
7
required for the former being 20 minutes and for the latter
30 minutes.
Both parts were given consecutively to each
class, thus requiring only one period for the entire test.
The teachers carefully adhered to the directions for admin­
istering the test as given on a separate sheet.
The test
together with the directions for administering it are given
following this page.
The students studying mathematics for ten consecutive
weeks were given a final test at the end of that time, and
those on a semi-weekly alternation basis received their final
test during the last week of the semester.
In giving Ithe
final the same form was used as had been used for the pretest.
Scoring the tests.
Each teacher scored the "answers"
of his or her groups according to the specific directions
given by the standardized test.
A careful recheck of all
tests was made by the school counselor and afterwards by the
writer.
The actual score made by each student was used in pref­
erence to either the arithmetic age or school grade because
it was more adaptable to statistical purposes.
II.
PROCEDURE OF TABULATION
In describing the procedure of tabulation, the follow­
ing points will be discussed:
comparability of groups; gains
New Stanford Arithmetic Test
e
By Truman L. Kelley, Giles M. Ruch, and Lewis M. Terman
TEST: FORM W
FOR GRADES 2-9
Grade.B oy or
Name..........................................................
•
girl......
#
A ge...................When is your next birthday ?......................... : ........... . . How old will you be then ?. . ..
Name of school.......................................................................................... D ate................................. .................
Score
Arith. School1 Score
Age
Grade
120
119
118
117
116
H5
114
113
112
111
110'
109
108
107
106
105
104
103
102
101
19-2
18-11
18-8
18-5
18-2
17-11
17-8
17-6
17-4
17-2
17-0
16-10
16-8
16-6
16-5
16-3
1 6-2
1 6-0
15-11 10.0
1 5 -9 , 9.8
100
99
98
97
96
95
94
93
92
91
90
89
88
87
86
85
84
83
82
81
Arith. School1 Score 1Arith. School1 Score
Age
Grade
Age
Grade
15-8
15-6
15-4
15-2
15-0
14-10
14—8
T4—6^
14-4
14-1
13-11
13-9
13-7
13-5
13-3
13-1
12-11
12-10
12-8
12-7
9.7
9.5
9.3
9.2
9.0
8.9
8.7
8.5
8.4
8.2
8.1
7.9
7.8
7.6
7.5
7.4
7.2
7.1
7.0
6.8
80
79
78
77
76
75
74
73.
72
71
70
69
68
67
66
65
64
63
62
61
12-6
12-4
12-3
12-2
12-0
11-11
11-10
11-9
11-8
1 1-7
11-6
1 1-5
1 1-4
11-3
11-2
11-1
11-0
10-11
10-10
10-9
6.7
6.6
6.4
6.3
6.2
6.1
6.0
5.9
5.8
5.7
5.7
5.6
5.5
5.4
5.3
5.2
5.1
5.0
4.9
4.8
60
59
58
57
56
55
54
53
52
51
50
49
48
47
46
45
44
43
42
41
Arith. School1 Score
Age
Grade
10-8
10-7
10-6
10-6
10-5
10-4
10-3
10-2
10-1
10-0
9-11
9-11
9-1 0
9 -9
9 -8
9 -7
9 -6
9 -5
9 -4
9 -3
4.7
4.6
4.6
4.5
4.4
4.4
4.3
4.3
4.2
4.1
4.1
4.0
4.0
3.9
3.9
3.8
3.7
3.6
3.6
3.5
40
39
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
Arith. School1
Age
Grade
9 -3
9 -2
9-1
9 -0
8-11
8-10
8 -9
8-8
8 -7
8-6
8 -5
8 -4
8 -3
8-2
8-1
8—0
7-11
7 -10
•7-8
7 -6
7-5
3.4
3.4
3.3
3.3
3.2
3.2
3.1
3.1
3.1
3.0
3.0
2.9
2.9
2.8
2.8
2.8
2.7
2.7 .
2.6
2.6
2.6
1 G rade defined as in th e ta b le in th e D irections for A dm inistering.
* A rith m etic ages a b o v e th is p o in t a re ex tra p o la ted values.
To
th e
E x a m in e r .
D o
not administer this test without first reading carefully the Directions for Administering.
T
S core
est
A r it h .
A ge
S chool
Grade
Arith. Reas.
Arith. Comp.
Total (Average) Arith.1
1 T h e T o ta l A rith m etic Score is th e average of th e scores on th e
tw o te sts.
Published by World Book Company, Yonkers-on-Hudson, New York, and Chicago, Illinois
Copyright 1929 by World Book Company. Copyright in Great Britain. All rights reserved, nsat : w-16
PRINTED IN U.S.A.
[l]
BSP" This te s t is copyrigh ted.
The reproduction o f any p a rt o f it b y m im eograph, hectograph, or in any o ther
w ay, w h eth er th e reprodu ctions are so ld or are fu rn ish e d fr e e fo r u se, is a violation o f th e copyright law .
N ew s ta n f. A rith .
w
9
TEST 1. ARITHM ETIC REASONING—Continued
3
21 Frances sold $156 worth of books.
She received a com m ission of 40%.
H ow much did she earn?
Answer............
32 H ow much more is earned each day
by a man w orking 6 days for $45 than
by a man w orking 6 days for $32.40?
Answer............
22 In a class of 38 pupils, only 19 had
perfect attendance records for a month.
-What per cent' of all the class had per­
fect attendance records?
Answer............
33 H ow m any dollars’ worth of mer­
chandise must a clerk sell at a com m is­
sion of 2 / % to earn a salary of $1,000
23 If a man uses 25 gallons of gasoline
in driving 375 miles, how far can he
drive on 8 gallons, assum ing that he will
obtain the same m ileage per gallon?
Answer ......
24 W hen $3 w ill buy 5 yards of g in g ­
ham, how much w ill 7 yards cost?
Answer.........
25 John buys papers for $1.80 per hun­
dred and sells them at 3 cents each. H ow
much does he make on 250 papers?
Answer............
26 H ow much cheaper is a bill of $100
worth of goods bought at a 20% dis­
count than one of the same amount
bought at tw o successive discounts of
10% and 10% ?
Answer.
27 A man’s autom obile will go 110 miles
on 10 gallons of gasoline. If gasoline
costs 22 cents per gallon, w hat is the
cost of the gasoline per mile?
Answer............
2® Ice is .92 as heavy as water. A cubic
foot of water w eighs 62.5 lbs. W hat is
the w eight of a cubic foot of ice ?
Answer............
29 If 6 men can build a house in 180
days, how long w ill it take 8 men to
build it?
Answer............
30 A broker charges $25 com m ission on
every sale, plus 5% on all over $200.
W hat would be his com m ission on a
500-dollar sale?
Answer.
31 Fred w ill sell his bicycle for $24.
That is 34 less than it cost him. W hat
did it cost him?
A
?
Answer..........
Go right on to the next column.
D IR EC TIO N S: Get the answers to these examples as quickly as you can without making mistakes. Look
carefully at each example to see what you are to do.
Begin here.
(1)
1 2
a >'ear?
5+3=
(2)
(3)
(4)
(5)
Add
0
4
Add
4
6
Subtract
5
2
Subtract
7
7
5
--- -
----
Answer............
34 A train makes a run of 159 miles in 7
hours. One trip it was delayed and made
only 63 m iles the first 4 hours. A t w hat
average rate per hour must it go the re­
mainder of the distance in order to arrive on tim e?
Answer........
N ew Stanf. A rith . w
TEST 2. ARITHM ETIC COMPUTATION
4
3
(6)
(8)
Subtract
17
3
(7)
Subtract
12
4'
Add
24
7
(9)
Subtract
89
76
(10)
(11)
(12)
(13)
2)6
Subtract
895
437
Add
87 654-2
754290
389364
(16)
(17)
7)56
8 -4 =
35 A certain house was assessed at
$5,000. T he tax on it was $125. W hat
was the tax rate?
Answer........
36 Tulip bulbs should be planted 4 to the
square foot. A square plot should be
how many feet on a side in order to hold
36 bulbs?
Answer............
4X 8 =
37 W hat actual rate of interest would
be obtained if you bought 6% preferred
stock at $75 per share? (Par value
$ 100->
Answer....:...
(14)
38 For $90 each a man bought 5 shares
of a preferred stock paying 5% . (Par
value $100.) A fter his first dividends he
sold his stock for par value. Ignoring
brokerage charges, how many dollars
did he make on the transaction ?
Answer............
6X 7 =
J 15)
M ultiply
452
4
(18)
(19)
(20)
39 If they have the same thickness, a
pancake 6 inches in diameter is how
m any tim es as large as one 3 inches in
diameter?Answer........
9)77
Subtract
Subtract
40 A house and lot were valued at $5,000.
T he taxes amounted to $60 a year. It
cost $200 annually for depreciation and
incidental expenses. For what must it
rent per m onth in order that the owner
m ay clear 8 per cent on its value?
Answer............
End of Test 1. Look over your work.
.
(21)
(22)
„
%X H =
3)14.1
1000
%
42 5
■%
(23)
(24)
(25)
Subtract
5 2%
2 7 Ys
Add
3 5%
2 4 y3
___________
37) 8 5 3 5 9
'
Go right on to the next page.
'N e w Stanf. A rith . W
5
T E ST 2 .‘A R IT H M E T IC CO M PUTATIO N— Continued
(26)
(27)
6
A X % =
A dd
%
%
,
(31)
Subtract
73%
5%
Subtract
31 %
15%
T E ST 2 . ARITHM ETICS C O M P U T A T IO N — Cohcluded
(29)
Add
7 8%
129%
(28)
Subtract *
31%
19y4
(30)
6
(32)
(47)
(48)
A dd
%
7/io
.
(49)
Add
3 T ons 1000
4 T ons 500
300
7 T ons 1750
2 T ons 800
lb.
lb.
lb.
lb.
lb.
4 . 6 5 is w hat per
cent of 1 5 . 5 0 ?-
N ew Stanf. A rith . W
(50)
‘
239)1 4 4 8 3 4
j
A nsw er =
(33)
Add
54%
2 1 %2
(51)
(52)
4 5 is 1 5 % of
w hat number?
’
(53)
A dd
7
- 9
(4 )! =
• (54)
2 5216 5 12 8
-8
(35)
(34)
Subtract
55%
18%
Subtract
32%
24%
% + 5
Number =
(37)
(36)
--------
% + % + % + %
(55)
H ow many degrees are
there in angle B A C ?
A
(38-39)
H EIG H T IN FEET O F F A M OU S W A T E R F A L L S
—
to
L
tn
0>
S
®
<P
o
O
O
§ 8 8 8 8 o
A ccording to the graph,
what is the approximate
height in feet of
iy w
O o
o o
NIAGARA
|_
UPPER Y O S EM ITE
|
N ia g a r a
..................
SHOSHONE
LOWER YELLOWSTONE
(57)
Find the value of F in the
following expression if k
equals 8 and h equals 5.
Multiply
42n
—10 n
F =
kh2
H
tarn H i
VICTORIA
36*
90°
(56)
Upper Y osem ite
1
......
F =
Answer =
(60)
(40)
65.3 2 - 4 7 . 2 =
(43)
(42)
(41)
Subtract
5 2%
46%
%+%+
6.2-3.895
(59)
(58)
.
Find the volume of
. this figure.
Simplify the following
expression:
%2
V 1 5 1 2 9
8 x + 6 y — \2 x — 3 y )
A nsw er =
(44)
(45)
M ultiply
78.94
3.04
Subtract
%
% '
.
(46)
Answer =
Sales amount to $5240.
Commission rate is 10%.
Find amount of commission.
Answ er =
Turn the page and go right on.
End of Test 2. Look over your work.
Number right
Score
1| 2| 31 4| 5| 6| 7| 8| 91 10| HI 12) 13 14| 15 16) 17) 18 191 20121) 22123| 24| 25 26| 27) 28) 29) 301
| 3 7[11|18 23 28 31 33 36 38| 4 0 1 43) 46) 491 5 2 1 551 57| 59) 601 611 63) 641 651 67| 691 711 72| 74) 75) 771 79|
10|
Number right 1 311 321 33| 34| 351 36| 371 38| 39) 40| 411 42| 43| 44| 451 46| 47| 48) 49) 50 . 51 1 52) 53) 541 55| 56| 571 58| 59) 60)
Score
181| 84| 86| 88 | 90| 92 94| 96) 98|101|103|106|109|111|112|113| 113)114|114|115|115)116)116|117)117|118|119| 1211123|125|
New Stanford Arithmetic Test
10
B y Truman L. Kelley, Giles M. Ruch, and Lewis M. Terman
DIRECTIONS FOR ADM INISTERING
The following instructions are all that are needed to
give the tests. The instructions for scoring the tests are
given with the scoring key. More detailed information
concerning the construction, validity, and reliability of
the tests; the norms; and the interpretation and uses
of the test results are given in the complete Guide for
Interpreting the New Stanford Achievement Test.
The person in charge of .the testing program for a
school or school system will need a copy of the complete
Guide for Interpreting the New Stanford Achievement Test
in order to have the test results interpreted and used
properly. (The Guide must be ordered separately.)
GENERAL DIRECTIONS
N.B.
i
The teacher should become thoroughly acquainted
with all the directions in this booklet before attempting to
give the test.
C o n d it io n s o f t h e T e s t
The New Stanford Arithmetic Test can be given satis­
factorily by any teacher or principal who is willing
to follow the directions in this manual conscientiously
and who is reasonably skillful in discipline. The tester
should possess a pleasing personality and be able to speak
the necessary directions clearly and distinctly enough
that every one in the test group may hear.
Pupils in Grades 2 and 3 should be tested in ordinary
classroom groups. Pupils in Grades 4 to 9 may be
tested in ordinary classroom groups or in larger groups
of a hundred or more, if proper controls are provided.
Pupils in Grades 4 to 9 inclusive may be grouped for
testing purposes.
Good testing conditions demand that there should be
quiet throughout the testing period. Strict obedience
and attention on the part of the pupils are absolutely
necessary. No questions should be permitted after the
testing begins. There should be a spirit of rapport
between the tester and the pupils. No visitors should be
allowed. Sufficient assistants should be provided, when
large groups are being .tested, to see that every one
understands what he is to do and that he has the neces­
sary materials with which to do it, and in order to
discourage copying or giving assistance in any way.
The person administering the test' must speak dis­
tinctly and at a moderate speed. Undue stress and
levity are to be avoided. An agreeable manner, but one
suggestive of authority, is essential. Give all commands
in a quick, energetic voice distinct enough for all those
for whom it is intended to, hear. Avoid shouting. Give
all directions slowly, with careful attention to emphasis
where it is needed. Follow the directions exactly. Be
watchful, and in so far as possible prevent disturbances
within or without the room which might in any way
interfere with the work of any pupil. 'Permit no whis­
pering or copying. Distribute test booklets and admin­
ister the tests with dispatch. Systematize the work
to avoid delay in administration. Adhere carefully to
the time limits. A stop watch i s . desirable but riot
absolutely necessary. (The time limits are liberal.
Probably many pupils will finish before time is called.)
Plan your procedure in detail before attempting to ad­
minister the tests. Their administration is easy, but
it is necessary to know exactly what to do at each
moment of the testing time.
The total working time for the arithmetic test, Grades
2 to 9 inclusive, is 50 minutes. The gross time is slightly
longer. The test may be given in a single sitting, or
each of the two parts may be given in a separate sitting.
DIRECTIONS FOR ADMINISTERING
{Identical for all five forms.
To be followed verbatim)
“ Here is a test to show how much you have learned.
I will give each of you a test book. Do not write on it
or open it until I tell you to.” (See that this is obeyed.)
After all are provided with test books and pencils:
“ Now fill the blanks at the top of the first page, here.
(Hold up test book and point to the blanks.) Do it as
quickly as you can, but write plainly. On the first line
where it says N am e , write your name. (Pause.) After
the word Grade , write the number that tells what grade
you are in. (Name the grade.) After Boy or GirZ,
write the word that tells which you are. (Pause.) On
the next line, where it says Age, tell how old you are now.
(Pause.) Then tell when your next birthday will come
and how old you will be then. (Pause.) On the next
line write the name of this school. (Give the name.)
Published by World Book Company, Yonkers-on-Hudson, New York, and Chicago, Illinois
Copyright 1929 by World Book Company. Copyright in Great Britain. All rights reserved
PRINTED IN U.S.A.
New Stanford Arithm etic Test: Directions for Administering
"\
At the end of the same line write the date.” (Name
the date. Give to the younger children any necessary
assistance.)
After the blanks have been filled : “ Now, listen care­
fully and do just what I tell you to do. Do not begin
until I say Go. The very second I say Stop, you must
stop and hold your pencils up. After we have begun, you
must not ask questions. If you break your pencil, hold
up your hand and I will give you another. Do your
best and do not pay any attention to what any one else
is doing.
“ Now turn the page over to Test 1. It says Test 1
at the top of the page.”
T e s t
1.
A rith m e tic
R e a s o n in g
“ Read the directions at the top of the page: Find
all the answers as quickly as you can. Write the an­
swers on the dotted lines. Use the margins of the paper
to figure on. (Pause slightly.) The test has two pages.
As soon as you have finished the first page, go right on
to the next. Ready — Go.” , (See that pupils do not
Stop at the end of the first page.)
Allow 20 m inutes; then say : “ Stop. Turn to Test 2
on the next page. Be sure to turn just one leaf.”
(Pause and make sure that all the pupils have found the
place.)
T e s t
2.
A rith m e tic
C o m p u ta tio n
“ Read the directions at the top of the page: Get
answers to these examples as quickly as you can without
making mistakes. Look carefully at each example to see
what you are to do. (Pause slightly.) You may use the
margins of the paper to figure on if you need to. There
are three pages of this test. As soon as you have finished
the first page, go right on to the next. Ready — Go.”
(See that pupils do not stop at the end of the first or
second page.)
. Allow 30 minutes; then s a y : “ Stop. Close your
books.” Collect all books immediately.
DIRECTIONS FOR SCORING
The Directions for Scoring are given in full with the
. scoring keys enclosed in each package of tests.
DIRECTIONS FOR INTERPRETING TEST
SCORES 1
By means of the parallel rows of figures across the
bottom of each test, the pupil’s achievement on any test
is automatically given an equated value related to the
table of norms. For example, the rows of figures across
the bottom of the Arithmetic Reasoning Test in Form
V appear as follows :
Sum
.
.
0
1
2
3
S c o re , .
.
3
12
21
29
i
22
23
24
25
, 96
98
100
102
The upper row represents the number of blanks to be
filled in the test. The lower row represents the equated
value, in terms of the norms, for having satisfactorily
completed any number of those blanks. If, for instance,
a pupil successfully filled 23 blanks in the examination,
the scorer checks the number 23 in the upper row of
figures and directly under the 23 is 98 , which represents
the equated value of the actual achievement. Thus,
g8 is the score (not 23 ), and is the only score to be used to
represent the pupil’s achievement in any further refer­
ence made to it.
Age and grade norms. On the front page of the Arith­
metic Test a table of norms is given by means of which
pupils’ scores may be changed to arithmetic ages and
grades.
T
able
of
F r a c t io n a l P a r t s
of
Date of testing . . . Sept. Oct. Nov. Dec. Jan.
is
IS is IS 15
Annual promotions
Grade, a low section .
Grade, a high section
.0
.0
■S
.1
.1
.6
.2
.2
•7
•3
•3.
.8
G rades C o m pleted
Feb. Mar. April May June
is
IS iS 15 iS
■S
•4
.4 Midyear .0
.9 promotions .5
.6
.1
.6
■7
.2
•7
.8
■3
.8
•9
•4
•9
There is provided in each package of booklets a Class
Record which may be used if it is desired to bring together
on one sheet the complete record of scores of the pupils
of a grade or class for convenient reference. The names
of the pupils may be entered either alphabetically or in
order of the total scores, according to preference. A
tabular form is provided on the back of the Class Record
for tabulating the distributions of scores.
1 T h e p erson in charge of a te stin g p ro g ram for a school or school system
will need a copy of th e com plete. Guide for Interpreting the New Stanford
Achievement Test in o rd e r to h a v e th e te s t re su lts in te rp re te d a n d used p ro p e rly .
11
in test scores for each group; comparisons of the two methods
in terms of gains for B8 and A8 separately; and finally, com­
parisons of the two methods in terms of gains for these two
grades combined.
Comparability of groups.
As previously stated in this
chapter each pair of groups in the experiment was assumed to
be reasonably comparable due to the school classification of
average or high.
However, there was no proof of the extent
to which they were comparable.
Therefore, it was necessary
to compute the mean of the intelligence quotients of every
group in order to show the extent of comparability of each
pair instructed by the five teachers respectively.
The in­
telligence quotients were obtained from the school counselor’s
office.
With these data tables were made and the mean intel­
ligence quotient was computed for each group, thus enabling
the extent of comparability of each pair of groups to be shown.
This statistical evidence of comparability of each pair of
groups was supported by the construction of graphs.
Gains in test scores for each group.
Having proved
the comparability of each pair of groups, the next step was
to tabulate the gains made by the students in each group
preparatory to comparing the gains made by each pair of
groups studying the two different methods.
Therefore, a
table was made for each of the ten groups showing the pretest
12
score, the final score, and the gain of each student in the
group.
For example, a pretest score of 112 and a final score
of 120 would give a gain of 8; also a pretest score of 110
and a final score of 104 would give a gain of -6.
Comparisons of gains for B8 and AS groups separately.
In order to arrive at some conclusion regarding the superior­
ity of one method over the other, it was necessary to compare
the gains made by these different groups.
were made.
To this end, tables
In one table the gains of each pair of groups in
the B8 level were compared, and the mean, standard deviation,
and standard error of the mean of each group were calculated.
In another the gains of the B8 ten-week groups were combined
and compared with those of the combined B8 semi-weekly alter­
nation groups, and the mean, standard deviation, and standard
error of the mean of each combination were calculated.
These
resulting data were then c ompiled in still another table in
order to serve as a basis for the following calculations:
(a) the chances of each group using one method being superior
to the group using the other method; and (b) the chances of
the combined groups of one method being superior to the com­
bined groups of the other.
In a table following this, the
reliability of the results obtained in the previous one was
checked.
Not only were the gains of B8 groups compared statis­
tically but also graphically.
A graph showing this comparison
13
was made for each pair of groups.
Another one was drawn
comparing the gains of the combined ten-week groups with those
of the combined semi-weekly alternation groups.
The same procedure was followed for the A8 groups
both in the tabulation of gains and in their graphic repre­
sentation.
Comparisons of gains for combined B8 and J8 groups.
Having compared the gains of the B8 and A8 groups separately,
it was then necessary to combine these two major groups in
order to compare the gains of the entire ten-week groups in
the experiment with those of the entire semi-weekly alter­
nation groups.
This was very important as the results of the
comparison would show the solution to the problem as to which
method was the better of the two*
In order to accomplish this, tables were made.
First,
the gains of the combined B8 and A8 ten-week groups were com­
pared with those of the combined B8 and A8 semi-weekly alter­
nation groups, and the mean, standard deviation, and standard
error of the mean of each combination were calculated.
These
resulting data were then compiled in another table in order
to serve as a basis for the calculation of what chances the
combined groups using one method had of being superior to
the combined groups using the other method.
This was followed
by still another table which checked the reliability of the
14
results obtained in the previous one.
Also a graph was drawn showing a comparison of the
gains of the combined ten-week groups with those of the com­
bined semi-weekly alternation groups.
All these statistical results and graphic representa­
tions will be shovm in the next three chapters.
CHAPTER
III
RESULTS OF THE B8 GROUPS
Having described the method of procedure of this ex­
periment and also the method of tabulation, we shall now state
and analyze in detail the results obtained in the B8 groups.
This is only part of the total results of the experiment.
The
\two chapters following will deal with the remainder, namely,
the results in A8 and the results of the combined B8 and A8
groups, respectively.
These results, on which depend the relative merits of
the two teaching methods, will be based upon the gains made by
the students on two tests given, one a pretest at the begin­
ning, the other a final test at the end of the experiment.
It was stated in the preceding chapter that there were
six groups of students in B8.
This means that there were
three pairs of groups, each pair representing the two methods
of Reaching and instructed by the same teacher.
In reporting the B8 results, the three pairs of groups
will be designated as IA, IB; IIA, IIB; and IIIA, IIIB.
The
A-groups were those who had studied mathematics on the tenweek basis, and the B-groups those who had studied the same
subject on the semi-weekly alternation basis*
In this chapter we shall discuss the B8 results in the
following order:
comparability of groups; gains made on tests
16
by each
of the six groups; comparisons of the gains of these
six groups
arrangedin pairs; comparisons of the ten-week
groups,
as a single group, with the semi-weekly alternation
groups,
as a single group, in terms of gains; and finally,
comparisons of all groups statistically for prediction pur­
poses in order to determine the relative merits of both
teaching methods for the B8*s.
All statistical computations used in this study follow
the formulas given in Statistics for Teachers. by Tiegs and
Crawford.^Comparability of groups♦
One of the necessary require­
ments for the successful completion of the experiment was
that each pair of groups studying the two methods be compar­
able in every possible way, leaving only the variable of the
two different time-bases.
They were comparable in that each
pair of groups was taught by the same teacher, was scheduled
for an equal length of class period, and had the same number
of periods of elass instruction in mathematics during the
semester.
The final and very necessary comparability would
have to be that of ability or intelligence.
Although, as
stated in the preceding chapter, each pair of groups tapght
by the same teacher was assumed to be comparable because they
^I.¥l. Tiegs and C, C. Crawford, Statistics for Teachers.
(Boston: Houghton Mifflin Co., 1930). 212 pp.
17
were either average or high according to the school classifi­
cation, it was necessary to prove this comparability in order
to carry out the experiment successfully.
To this end the
intelligence quotient of each individual in all six B8
groups was obtained from the school counselor’s office and
compiled in tables, one for each pair of groups.
In these
tables, the students were listed alphabetically but for ob­
vious reasons the names have been omitted in this report.
Table I shows the intelligence quotients of groups
IA and IB.
It is seen
that the size of the groups is reason­
ably comparable due totheir respective
33.
membership of 35 and
The derived mean intelligence quotients of 114.50 and
112.65, respectively, prove a high degree of comparability
in intelligence.
Figure I was constructed to show a graphic comparison
of the intelligence quotients of these same groups.
Although
the groups were not equal in number, the slight difference of
two students did not call for the construction of the graph
on a percentage basis.
In this graphic presentation, large
class intervals of 20 were used
for the reason that a smaller
class interval presented the difficulty of too many ’’saw-teeth”
which would lessen the visual effect of comparability in the
frequency polygon.
This figure shows at a glance that both
groups have the same range since they meet the base line in
the same class intervals.
The areas are practically the same.
18
TABU
I
INTELLIGENCE QUOTIENTS OF GROUPS IA AND IB,
B8 CLASSES
Pupil
no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
]$ean
Group
IA
Group
IB
131
115
114
105
148
106
103
102
116
117
130
111
108
133
108
116
99
115
102
135
108
105
111
110
99
118
122
109
118
105
133
121
89
101
131
116
110
108
116
109
123
112
102
112
138
125
99
114
124
117
110
118
131
127
98
97
113
103
123
98
107
101
125
106
107
98
106
122
114,50
112.65
imimi&sMWm:
M
61
20
The slight difference which occurs in the height is partly
due to the fact that group IA,1represented by the solid line,
has two more students than group IB,
Hence the figure shows
the two groups to be highly comparable in intelligence.
Table II shows the intelligence quotients of groups
IIA and IIB with 39 and 33 students, respectively.
Although
there are six more students in the former groups, yet the de­
rived mean intelligence quotients of 110.32 and 108.41, re­
spectively are very close and prove the comparability of these
groups in intelligence.
Figure 2 presents the graphic distribution of the
intelligence quotients of these two groups.
The difference
of six between the numbers of the two groups did not warrant
the construction of the graph on a percentage basis.
It is
seen that the same large class intervals of 20 were used here
and for the same reason as in the preceding graph.
This figure
shows that both groups have the same range since they meet
the base line in the same class intervals.
The area of the
polygon of group IIA, represented by the solid line, is
slightly larger than that of group IIB.
This is probably due
to the fact that group IIA has six more students than group
IIB.
By studying the figure closely, we see that group IIA
exceeds group IIB
by two students in the 120-140 interval$
three students in the 100-120 interval and one student in the
80-100 interval.
If these particular students were the six
21
TABLE
II
INTELLIGENCE QUOTIENTS OF GROUPS IIA.AND IIB*
B8 CLASSES
Pupil
no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
Mean
Group
\ IIA
Group
IIB
118
99
108
100
115
125
104
107
109
102
104
130
107
111
138
106
115
104
120
105
95
108
106
114
104
110
89
125
120
103
109
92
121
123
93
107
118
108
120
118
110
98
87
116
111
107
112
109
108
107
113
102
119
116
106
106
98
97
114
109
118
109
103
104
123
96
101
92
117
106
110
94
110.32
108.41
m
58^
23
to be taken out to make the groups.equal in number, the poly­
gons would be identical*
Hence it is reasonable to conclude
that the two groups are highly comparable.
fable III shows the intelligence quotients of group
IIIA with 28 pupils and of group IIIB with 30.
The computed
mean intelligence quotients of 99.29 and 101.00 prove the
first phase of their comparability in intelligence.
second phase is shown graphically in Figure 3.
The
As in Figure
1 , the only difference between the two groups is two students,
which fact did not require the graph to be constructed on a
percentage basis.
This fugure shows that although the areas
of the polygons are approximately equal, the intelligence
quotients of group IIIB, the broken line, are slightly higher
than those of group IIIA.
However, this difference is not
sufficient to prevent them from being considered comparable
and being entered in this experiment.
The same class inter­
vals of 20 were used here as in preceding figures.
These tables and graphs with their statistical and
visual proof of comparability in each pair of groups of the
B8 class leaves only, as has been stated previously, the
variable of the two time-bases which are the distinguishing
elements of the two teaching methods.
Gains made on tests by each B8 group.
Each group was
given a pretest at the beginning of the experiment and a final
test at the end.
Obviously the gain of each student was his
24
TABLE
III
INTELLIGENCE QUOTIENTS OF GROUPS IIIA AND IIIB,
B8 CLASSES
Pupil
no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Mean
Group
IIIA
Group
IIIB
99
100
98
108
96
107
96
102
92
102
108
84
103
111
109
104
87
92
97
96
80
95
92
118
82
105
106
97
96
112
97
103
97
98
;9i
98
101
100
101
102
100
98
93
93
94
101
103
114
111
104
94
112
102
99
101
104
102
104
99.29
101.00
26
final score minus that of his pretest.
Since the relative -
merits of the two methods of teaching were to he determined
by the comparison of the gains made by the students on the
two tests, it was first necessary to tabulate the gains for
each group in order that they might serve as a statistical
basis for these later comparisons.
In these tables, the
scores are placed, not from the highest to the lowest in
consecutive order, but according to the names of the students
arranged alphabetically.
The names are omitted as in the
former tables.
Table 17 shows the pretest score, the final score, and
the gain of each student in group IA using the ten-week method*
The range of the pretest score is from 76 to 111, of the final
score from 73 to 117, and of the gains from -5 to 23.
^here
were five students who made losses which are expressed as
negative gains in the table.
Table V shows the pretest score, the final score, and
the gain of each student in group IB using the semi-weekly
alternation method.
This group and IA were taught by the
same teacher. - The range of the pretest score is from 68 to
111, of the final score from 70 to 112, and of the gains
from -3 to 15.
In this group there were only two students
who made losses.
Table 71 shows the pretest score, the final score, and
the gain of each student in group IIA using the ten-week
27
TABLE
IV
GAINS MADE- BY GROUP IA, T1N-WEEKS*
B8 CLASS
Pupil
no.
1
a
3
4
5
6
7
8
9
10
11
ia
13
14
15
16
17
18
19
ao
81
22
83
ft4
85
86
87
88
89
30
31
38
33
34
35
Pre­
test
score
Final
test
score
110
88
93
80
109
94
87
79
90
100
88
76
76
103
94
107
107
96
98
86
99
88
89
93
87
100
97
79
97
91
103
108
88
99
111
113
99
110
94
117
99
97
88
104
105
96
89
73
110
109
110
104
105
105
109
107
100
106
106
88
111
103
89
109
89
103
108
90
96
113
Ga in
3
11
17
14
8
5
10
9
14
5
8
13
- 3
7
15
3
- 3
9
7
83
8
18
17
13
- 5
11
6
10
IS
- 8
0
6
8
- 3
8
28
TABLE. 7
GAINS MADE BY GROUP IB* SMI-WEEKLY ALTERNATION,
B8 CLASS
Pupil
no.
Pre­
test
score
Einal
test
score
Gain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
94
73
97
96
98
82
102
90
76
102
94
68
100
90
97
79
111
98
104
75
87
89
86
93
77
92
77
93
93
81
92
97
103
85
108
105
109
88
104
97
87
112
100
70
101
100
97
90
112
107
108
82
98
88
88
99
81
95
81
97
108
88
89
105
9
12
11
9
11
6
2
7
11
10
6
2
1
10
0
11
1
9
4
7
11
- 1
2
6
4
3
4
4
15
7
- 3
8
29
TABLE
VI
GAINS MADE BY GKOUP IIA, TEN-WEEKS,
B8 CLASS
Pupil
no.
Pre­
test
score
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
318
32
33
34
35
36
37
38
39
96
94
88
83
78
94
83
883
81
85
84
91
92
101
107J
103
91
93
91
91
90
91
100
89
91
83
89
103
104
94
107
68
96
106
75
107
87
96
107
Final
test
score
108
103
96
86
81
100
98
86
92
81
97
93
98
95
111
107
99
104
96
98
93
97
102
93
99
88
91
• 101
107
103
112
81
94
107
76
107
105
103
10*7
Gain
12
9
8
3
3
6
15
3
11
- 4
13
2
6
- 6
4
4
8
11
5
7
3
6
2
4
8
5
2
- 2
3
9
5
13
- 2
1
1
0
18
7
0
30
method.
The range of the pretest score is from 75 to 107,
of the final score from 76 to 112, and of the gains from -6
to 18.
In this group there were four students who made
losses.
Table VII shows
the pretest score, the final score, and
the gain of each student in group IIB using the semi-weekly
alternation method.
second teacher.
This group and IIA were taught by the
The range of the pretest score is from 53-to
104, of the final score from 62 to 107, and of the gains from
-11 to 20.
There were five students who made losses in this
group.
Table VIII shows the pretest seore, the final score,
and the gain of each student in group IIIA using the ten-week
method.
The range of the pretest score is from 63 to 106,
of the final score froiji 64 to 104, and of the gains from -7
to 12.
There were seven students who made losses in this group.
Table IX shows the pretest score, the final score, and
the gain of each student in group IIIB using the semi-weekly
alternation method.
third teacher.
This group and IIIA were taught by the
The range of the pretest score is from 71 to
104, of the final score from 73 to 107, and of the gains from
-4 to 11.
group.
There were ten students who made losses in this
31
TABLE
711
GAINS MADE BY GROUP IIB, SEMI-WEEKLY ALTERNATION,
B8 CLASS/
;
Pupil
no.
Pre­
test
score
Pinalr
test
score
Gain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
86
89
80
89
87
93
86
91
87
95
93
81
77
96
76
75
91
80
82
96
89
96
80
86
53
104
70
89
99
102
94
94
91
106
83
81
78
96
103
93
100
99
104
105
86
86
104
89
79
96
81
81
102
95
107
83
96
62
106
71
91
97
105
94
105
87
20
- 6
1
-11
9
10
7
9
12
9
12
5
9
8
13
4
5
1
- 1
6
6
11
3
10
9
2
1
2
- 2
3
0
11
- 4
32
TABLE
VIII
GAINS MADE BY GROUP IIIA, TIN-WEEKS,
B8 CLASS
Pupil
no.
Pre­
test
score
Final
test
score
Gain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
87
93
98
77
78
86
75
93
95
80
97
95
81
72
79
74
89
75
68
63'
86
75
82
106
70
81
72
70
80
104
102
86
75
92
76
98
100
88
100
92
79
84
88
78
94
76
77
71
82
87
85
102
71
83
75
64
- 7
11
4
9
- 3
6
1
5
5
8
3
- 3
- 2
12
9
4
5
1
9
8
- 4
12
3
- 4
1
2
3
- 6
33
TABLE
IX
GAINS MADE BY GROUP IIIB, SEMI-WEEKLY ALTERNATION,
B8 CLASS
Pupil
no,
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Pretest;
score
100
88
86
99
88
75
71
74
99
86
91
104
78
72
79
87
75
81
80
93
89
84
81
98
81
74
80
. 75
81
101
Final
test
score
Gain
99
85
82
95
89
75
73
80
96
85
93
107
79
76
78
91
81
84
88
90
100
95
77
98
88
84
79
•76
82
104
•
r-
1
3
4
4
1
0
2
6
- 3
- 1
2
3
1
4
- 1
4
6
3
8
- 3
11
11
- 4
0
7
10
- 1
1
1
3
34
Comparisons of the three pairs of groups in terms of
gains*
Having tabulated the gains of each of these six groups,
the next step was to make comparisons in terms of gainst of
these six groups arranged in pairs, each pair representing
both methods of teaching and taught by the same teacher*
To
this end Tgble X was made*
The three pairs of groups are arranged in three para­
llel columns.
The students of each group are arranged in a
frequency distribution according to the number making each
amount of gain as shown in the corresponding class interval*
The gains arranged consecutively in class intervals of two
range from (-12)-(-11) to 22-23.
The reasons for arranging the groups in pairs were:
(a) that the number of students in one group making each
amount of gain might be compared with the number of students
in the corresponding group making the same amount of gain;
(b) that the mean, standard deviation, and standard error of
the mean might be calculated from each frequency distribution;
(c) that the obtained means of the corresponding groups could
be compared; and (d) that the mean, standard deviation, and
standard error of the mean might be used as bases in statis­
tical comparisons to be made later.
The means of groups IA and IB being 8*14 and 6.70,
respectively, show a difference of 1.44.
This is equivalent
to each student in group IA making a gain of 1.44 more than
35
TABLE
X
COMPARISONS OF THE TWO METHODS IN TERMS OF GAINS
FOR B8 GROUPS
Gains in
Number of pupils showing each amount of ga in
class
Ten
weeks
Semi­
weekly
Ten
weeks
Groups
intervals
IA
22-23
20 - 21
18 - 19
16 - 17
14 - 15
12 - 13
10 - 11
8 - 9
6 - 7
4 - 5
2 - 3
0 - 1
(- 2} (- ,1)
(- 4) - (- 3)
C- 6) - (- 5)
C- 8} - (-7)
(-10) - ( -9)
(-12) - (-11)
1
Number
Mean
S.D.
S . E Jvl.
IB
Semi­
weekly
Ten
weeks
Groups
IIA
IIB
Semi­
weekly
Groups
IIIA
IIIB
1
i
2
3
3
4
5
4
2
4
1
1
3
1
1
1
1
7
4
7
4
4
3
1
1
1
3
2
5
5
6
8
4
2
1
1
3
4
6
3
3
4
4
2
1
1
2
1
5
1
5
4
3
1
4
1
1
3
1
3
2
5
6
4
6
1
35
8.14
6.66
1.13
39
33
6.70
5.72
4.99
4.18
•73 '
.80
33
5.73
6.17
1.07
28
3.71
5.35
1.01
30
2.40
4.39
.80-
S.D. is Standard Deviation, and S.E. M. is the Standard
Error of the Mean.
IA and IB are the groups taught by one teacher; IIA and
IIB are the groups taught by another teacher; and IIIA and
IIIB are taught by still another teacher.
36
each student in group IB*
This obtained difference is only
slightly in favor of group IA representing the ten-week
method•
Although the standard deviation and the standard
error of the mean of each group was calculated for the chief
purpose of serving as a basis for statistical comparisons to *.
be made later, we shall at this time point out their signifi­
cance in relation to the means of these two groups.
We shall
do the same for the other groups when we arrive at a discussion
of theif results.
Group IA has a standard deviation of 6.66.
This sig­
nifies that approximately two-thirds of all the gains, or S3
gains, would be included between the limits of 8.14 (the mean)
plus 6.66, or 14.8G, and 8.14 minus 6.66, or 1.48.
This is
verified by observing the distribution itself.
The standard error of the mean of group IA is 1.13.
This signifies that if we gave this group a large number of
tests and calculated the gains, two-thirds of the time our
means would not deviate more than 1.13 from the observed
mean of 8 ‘
.14.
That is, two-thirds of all our future means
of this group would lie between the gains of 7.01 and 9.S7.
In group IB, with a standard deviation of 4.18, approx­
imately two-thirds of all the gains, or 22 gains, would lie
between the limits 6.70 (the mean) plus 4.18i, or 10.88, and
6.70 minus 4.18, or 2.52.
37
The standard error of the mean of group IB is ,73.
This means that if we gave this group a large number of tests
-and calculated the gains, two-thirds of the time their means
would not deviate more than .73 from the mean of 6.70.
That
is, two-thirds of all the future means of this group would
lie between the gains of 5.97 and 7.43.
Figure 4 shows a graphic comparison of the gains made
by these same two groups.
Here we see that group IA of the
ten-week method has a greater range of gains than group IB.
If we add the excess high gains and the excess low gains of
group IA, the result would offset the excess height of the
polygon of group IB.
Thus we can reasonably conclude that
the areas of the two polygons are approximately equal.
The
difference of two students between the two groups affects
the comparison of the areas but little.
The figure also shows
that in the two extremes of the gains made by group IA, the
excess of high gains was greater than the excess of low gains.
This gives the slight advantage to the group of the ten-week
method in this case.
The small difference of .01 between the means of groups
IIA and IIB is to be ignored.
Consequently, their means are
considered to be equal and neither of the two methods had any
advantage.
Group IIA has a standard deviation of 4.99.
Two-thirds
of all the gains, then, would be included between the limits
39
of 5.72 (the mean) plus 4.99, or 10.71, and 5.72 minus 4.99,
or .73.
Approximately 26 gains would be found in this range.
The standard error of the mean of group IIA being .80
means
that if we gave this group a large
number of tests and
calculated the gains, two-thirds of the time our means would
not deviate more than .80 from the mean of 5.73.
That is,
two-thirds of all our future means of this group would lie
between the gains of 4.93 and 6.53.
Group IIB with a standard deviation of 6.17 signifies
that approximately two-thirds of all the gains, or 22 gains,
would be included between the limits 5.73 (the mean) plus
6.17, or 11.90, and 5.73 minus 6.17, or -.44.
The standard error of the mean of group IIB being 1.07
means
that if we gave this group a large
number of tests and
calculated the gains, two-thirds of the time our means would
not deviate more than 1.07 from the mean of 5.73.
That is,
two-thirds of all our future means of this group would lie
between the gains of 4.66 and 6.80.
Figure 5 compares the gains of groups IIA and IIB in
a graphic way.
Class intervals of 4 were used for the gains
as in the preceding figure.
A^ a glance we see that the
greater range of gains was made by group IIB, a semi-weekly
alternation group, the broken line.
Group IIA has a larger
number of students centering around the mean as shown by the
height of its graph, the solid line.
This is partly due to
BIS
jshe fact that group IIA has six more students than the other
group*
If these six students were excluded from its polygon
we could reasonably conclude that the resulting excess height
of the polygon of group IIA would be offset by the sum of the
excess high gains and the excess low gains of group IIA*
the areas of the polygons would be approximately equal*
Thus
It
also shows that in the two extremes of the range of gains made
by group IIB, the excess high gains approximately nullify the
excess low gains, thus giving neither of the two methods the
advantage.
Table X shows the means of groups IIIA and IIIB to be
5.71 and 2.40, respectively.
The difference of 1.31 is equiv­
alent to each student in group IIIA making a gain of 1.31 more
than each student in group IIIB.
This obtained difference
slightly favors the ten-week method represented by group IIIA.
Group IIIA has a standard deviation of 5.35.
This means
that approximately two-thirds of all the gains, or 19 gains,
would be included between the limits 3.71 (the mean) plus
5.35, or 9.06, and 3.71 minus 5.35, or -1.64.
The standard error of the mean of group IIIA is 1.01.
This signifies that if we gave this group a large number of
tests and calculated the gains, two-thirds of the time our
means would not deviate more than 1.01 from the mean of
That is, two-thirds of all our future means of this group
would lie between the gains of 2*70 and 4.72.
3.71.
Group IIIB has a standard deviation of 4.39, which
means that approximately two-thirds of all the gains, or 20
gains, would be included within.the limits 2.40 (the mean)
plus 4.39, or 6.79, and 2.40 minus 4.39, or -1.99.
The standard error of the mean of group IIIB is .80
signifying that if we gave this group a large number of tests
and calculated the gains, two-thirds of the time our means
would not deviate more than .80 from the mean of 2.40.
That
is, two-thirds of all our future means of this group would
lie between the gains of 1.60 and 3.20.
Figure 6 is a graphic presentation of the gains of
groups IIIA and IIIB.
Class intervals of 4 were used for the
gains as in the two preceding figures.
A first glance at this
figure might indicate that the areas of the polygons are un­
equal.
By studying it more closely, however, we see that if
we add the excess high gains and the excess low gains of
group IIIA, the result would about equal the excess height of
the polygon of group IIIB.
Consequently, the areas of the
polygons are approximately equal.
The difference of two
students between the two groups affects the areas so slight­
ly that it can be practically ignored.
greater range of gains.
Group IIIA has the
The excess of high gains is greater
than the excess of low gains giving this group, representing
the ten-week method, the slight advantage.
44
Comparisons of the ten-week groups, as ja single group,
with the semi-weekly alternation groups. as £ single group, in
terms of gains.
Having made comparisons of the six groups
arranged in pairs, the next step was to put the same three
ten-week groups together as a single group and compare their
gains with those of the same three semi-weekly alternation
groups placed together as a single group.
The reason for do­
ing this was that, according to statistics, the greater the
numbers of cases examined the higher the reliability of the
results.
To this end, Table XI was made.
The main point was to calculate the mean, the standard
deviation, and the standard error of the mean of each of the
combined groups.
The purpose of this was to*see in particular
if the means came within close range of each other.
If they
should, then there would be very little difference between the
two methods but if not, then a superiority of one method over
the other would be the result.
The derived mean, standard deviation, and standard
error of the mean will not only be used in the present compar­
ison but will also serve as some of the statistical bases for
further comparisons to be made later.
The two combined groups are arranged in this table in
two parallel columns.
The students of each combined group are
arranged in a frequency distribution according to the number
making each amount of gain as shown in the corresponding class
45
TABLE
XI
COMPARISONS OF TEN-WEEK GROUPS AS A SINGLE
GROUP, WITH SM I-WEEKLY ALTERNATION GROUPS
AS A SINGLE GROUP, IN TERMS OF GAINS
FOR B8 CLASSES
Gains in
Number of pupils
showing each
amount of gain
class
Ten*
week
groups
Semi­
weekly
groups
intervals
I,II,
IIIA
I,II,
IIIB
22
20
18
16
14
12
10
8
6
4
2
0
( 4 (4 4)
C- 6)
C- 8)
(-10)
(-12)
2
-
Number
Mean
S*D.
S.E.M.
23
21
19
17
15
13
11
9
7
55
3
1
) c- i)
f- 3)
(- 5}
(- 7)
(- 9)
(-11)
1
1
2
2
4
8
7
15
10
13
16
8
4
8
3
1
1
4
14
11
13
9
13
13
7
8
1
1
102
6.00
5.97
.59
96
5.02
5.33
.54
S.D. is the Standard Deviation, and S.E.M. is the
Standard Error of the Mean*
46
interval.
The gains arranged consecutively in class intervals
of two and range from (-12)-(-11) to 22-23.
It is seen also
that the number of students is practically equal for both
methods, that of the ten-week method-being 102 and that of
the senond method being 96.
The means of the ten-week method and the semi-weekly
alternation method are 6.00 and 5.02, respectively.
This
signifies that they are within close range of each other.
The small difference of .98 is slightly in favor of the tenweek method.
We notice that the derived standard deviations of the
two methods are within close range of each other, as are also
the standard errors of the means.
The standard deviation of the ten-wefek group being
5.97 signifies that approximately two-thirds of all the gains,
or 68 gains, would be included between the limits 6.00 (the
mean) plus 5.97, or 11.97, and 6.00 minus 5.97, or .03.
The standard error of the mean of the ten-week group
being .59 means that if we gave this large group an infinite
number of tests and calculated the gains, two-thirds of the
time our means would not deviate more than .59 from the mean
of 6.00.
That is, two-thirds of all our future means of this
group would lie between the gains of 5.41 and 6.59.
The semi-weekly group having a standard deviation of
47
5.35 signifies that approximately two-thirds of all the gains,
or 64 gains, would be included between the limits 5.02 (the
mean) plus 5.33, or 10.35, and 5.02 minus 5.33, or -.31.
The standard error of the mean of the semi-weekly group
being
.54 means that if we gave this group alarge number of.
tests
and calculated the gains, two-thirds of the time our
means would not deviate more than .54 from the mean of 5.02.
That is, two-thirds of all our future means of this group
would
liebetween the gains of 4.48 and 5.56.
Figure 7 shows a graphic comparison of the gains of
these two large groups.
At a glance we see that one polygon
is practically superimposed upon the other, the distribution
of gains of one group following closely that of the other.
Hence we can very reasonably say that the areas of the poly­
gons are equal.
Observing the graph more closely, we see that
the ten-week group has more high gains and fewer low gains
than the semi-weekly group.
This indicates a slight advantage
in favor of the ten-week method.
Statistical comparisons of B8 groups for prediction
purposes.
In the first place, we shall compare (a) the three
pairs of groups, IA and IB, IIA and IIB, and IIIA and IIIB
in order to determine for each pair the chances that the group
of one teaching method would have of being superior to the
group of the other teaching method; and (b) the three tenweek groups I, II, IIIA, as a single group, with the three
II
49
semi-weekly groups, I, IIJ,IIIB, also as a single group, in
order to determine the chance that one of these combinations
would have of being superior to the other.
The pairs of groups and the two combinations will be
seen in the same table.
From the means and standard errors
of the means of each pair and also of each combination will
be derived other statistical data in order that the chances
of superiority may be determined.
In the second place, we shall compare the same three
ten-week groups, I, II, IIIA, as a single group, with the
same three semi-weekly groups, I, II, IIIB, as a single group,
not only for the purpose of prediction, but also to serve as
a check on the results of the first comparison.
The means used here as a basis for comparing the com­
bined groups are very different from those used in the first
comparison.
The means used in the first comparison were those
obtained as a result of making a frequency distribution of each
of the combined groups as a single separate group, while here
the original means of each of the groups, I, II, IIIA, the tenweek groups, are simply added and used as a basis of comparison,
the same being done with the original means of each of the
groups, I, II, IIIB.
In addition to these two sums of the
means, other data will be calculated in order that the chances
of superiority of one combined group over the other may be
shown.
50
Having stated the two ways in which these comparisons
will be made, we shall now discuss the first with the use of
Table XXI.
This table shows each of the three pairs of groups
and each of the combined groups as previously mentioned.
The table indicates that the obtained difference be­
tween the two means of groups IA and IB is 1.44.
The signif­
icance of this difference is determined by dividing 1.44
by the standard error of the difference, or 1.35, which yields
1.07, or the ratio.
By referring to a standard-error table,1
this ratio was found to signify that the chances are only 6
to 1 that group IA, the ten-week method, would be superior to
group IB, the semi-weekly method.
In order to be practically
certain that there is a real difference in merit between
these two groups, the difference between the means should be
at least three times as great as the standard error of the
o
difference.
In the present case, the difference, 1.44, is
just a shade larger than the standard error of the difference,
1.35.
On the basis of these data (as revealed by the table
referred to) we can say that the chances are very slight that
group IA, the ten-week method, would actually make higher
^ . W . Tiegs and C.C. Crawford. Statistics for Teachers,
(Boston: Houghton Mifflin Co., 1930). p. 137.
2
H.W. Tiegs, Tests and Measurements for Teachers,
(Boston: Hpughton Mifflin Co., 1 9 3 1 ) . p . 234.
TABLE
XII
STATISTICAL COMPARISON OP GAINS OF B8 GROUPS SHOWING PREDICTIONS
FOR SEPARATE CLASS GROUPS AND FOR THE TOTAL
GROUPS AS SINGLE LARGE DISTRIBUTIONS
Groups
compared
Means
(Ten
weeks}
Means
(Semi­
weekly)
S.E.M.
(Ten
weeks)
S.E.M.
(Semiweekly)
Diff.
EP
Ratio
i
1.07
Chancei
i
IA
vs.
IB
8.14
6.70
1.13
.73
1.44
1.35
IIA
vs. I IB
5.72
5.73
.80
1.07
.01
1.34
IIIA vs.IIIB
3.71
2.40
1.01
.80
1.31
1.29
1.02
5.5to 1
All students in
groups I,11,IIIA
6.00
.98
.80
1.23
8.2 to 1
1 to 1
.59
vs.
All students in
groups I,IIfIIIB
.007
6 to 1
5.02
.54
IA,IIA, and IIIA are the ten-week groups,
IB,IIB# and IIIB are the semi­
weekly alternation groups. S.E.M. is the Standard Error of the Mean, and
is
the Standard Error of the difference.
5a
gains in mathematics than group IB, the semi-weekly method*
The difference between the two means of groups IIA and
IIB was *01*
The significance of this difference was deter­
mined by dividing *01 by the standard error of the difference,
or 1.34, which yields .007, or the ratio.
In the standard-
error table we find this small ratio to signify that the
chances are 1 to 1 that the semi-weekly group IIB (the group
whose mean was the larger by .01) is superior to group IIA, the
other method.
On the basis of these data we can say that the
chances are even that one of these two groups would make high­
er gains in mathematics than the other.
The difference between the means of IIIA and IIIB was
1.31.
By dividing this difference by the standard error of
the difference, or 1.29, we obtain the ratio, or 1.02.
The
standard-error table reveals this ratio to indicate that the
chances are 5.5 to 1 that the ten-week group IIIA is superior
to the semi-weekly group IIIB.
On the basis of these data we
can say that the chances are very slight that group IIIA would
actually make higher gains in mathematics than group IIIB.
Finally, in this table the three ten-week groups, as
a single group, were compared with the three semi-weekly groups,
as a single group.
The difference between the means of these
two large groups is .98.
By dividing this difference of .98
by .80, the standard error of the difference, we obtain the
ratio of 1.23.
The standard-error table shows this ratio to
53
signify that the chances are 8,2 to 1 that the I* II. IIIA
combination is superior to the I, II, IIIB combination.
On the basis of these data we can say that the chances
are very slight that the ten-week combination would make high­
er gainsin mathematics than the semi-weekly combination.
As a result
of combining the ten-week groups and also
the semi-weekly groups, we see that the chances of superiority
of the ten-week groups were slightly increased.
The ratio of
the combined groups, namely, 8.2 to 1, is slightly larger than
each of the ratios obtained in the comparison of the respective
pairs.
The second method of comparing the three combined tenweek groups with the three combined semi-weekly groups is
shown in Table XIII.
As was stated previously, the means of
IA,
IIA, and IIIA, respectively, were added as were those of
IB,
IIB, and IIIB.
vThe obtained sums of the means then be­
came the basis on which further calculations were made in order
to arrive at the chances of superiority of one combined group
over the other.
This table shows the obtained difference between the
sums of the means to be 2.74.
Dividing this quantity by the
standard error of the difference between the sums of the means,
or 2.30, we obtain the ratio of 1.19.
This ratio signifies
that the chances are 7.6 to 1 that the I, II, IIIA combination
is superior to the I, II, IIIB combination.
With these data
54
TABLE
XIII
STATISTICAL COMPARISON OF B8 GROUPS SHOWING
A SECOND METHOD OF MAKING PREDICTIONS BY
DIFFERENT COMPUTATIONS OF THE MEANS
B8 groups
Ten-week groups
Items
Mean
%
Sum of means
groups A and B
E of sum of
means, groups
A and B
Semi-weekly groups
IA
IIA
IIIA
IB
IIB
IIIB
3.14
1.13
5.72
.80
3.71
1.01
6.70
.73
5.73
1.07
2.40
.80
17.57
14.83
1.71
1.52
Diff. between
sums of means,
group A minus
groups B
2.74
E of diff. be­
tween sums of
means of groups
A and B
2.30
Eatio
1.19
Chances
D/ED
7. 6 to 1
E|£ is the Standard Error of the Mean; E is the Standard
Error; D is the difference, and ED.the Standard Error of the
difference.
55
as a basis, we ean say that the 'chances are again very slight
that the ten-week combination would make higher gains in
mathematics than the semi-weekly combination.
In both these ways of comparing the two combined groups,
we found that the results obtained were practically the same,
as the ratios were 8.S to 1, and 7.6 to 1, respectively, and
both gave the ten-week method the advantage, but of no great
significance.
CHAPTER
IV
RESULTS OF TEE A8 GROUPS
Having shown the results obtained in the B8 groups,
we shall now state the results of the A8 groups both in tab­
ular and- in graphic form pointing out the significant features
of such as we proceed.
As in the case of the B 8 ’s, so these A8 results will
be based upon the gains made by the students on the two tests
given, one a pretest at the beginning, the other a final test
at the end of the experiment.
These gains will determine the
relative merits of the two different methods.
It was stated in Chapter II that there were four groups
of students in A8,
This means that there were two pairs of
groups, each pair representing the two different time-bases of
teaching and instructed by the same teacher.
In relating the A8 results, the two pairs of groups
will be designated as IVA, IVB, and VA, VB.
The A-groups
were those who had studied mathematics on the ten-week basis,
and the B-groups those who had studied the same subject on
the semi-weekly alternation basis.
We shall show the A8 results in this chapter in the
following order:
comparability of groups; gains made on tests
by each of the four groups; comparisons of the gains of these
four groups arranged in pairs; comparisons of the ten-week
57
groups, as a single group, with the semi-weekly alternation
groups, as a single group, in terms of gains; and finally,
comparisons of all groups statistically for prediction pur­
poses in order to determine the relative merits of both teach­
ing methods for the A 8 ’s.
Comparability of groups.
The comparability of the
48 groups was obtained in the same manner as that of the B 8 ’s.
The A8 groups were comparable in that each pair of groups was
taught by the same teacher, was scheduled for an equal length
of class period, and had the same number of periods of class
instruction in mathematics during the semester leaving only
the comparability of intelligence to be proved.
To this end
the intelligence quotient of each individual in all four A8
groups was obtained from the school counselor’s office and
compiled in tables, one for each pair of groups.
In these
tables, the students were listed alphabetically but for obvious
reasons the names haVe been omitted in this report.
Table XIV shows the intelligence quotients of groups
IVA and IVB.
Although there are five more students in the
former group, yet the derived mean intelligence quotients of
99.07 and 94.83, respectively, are reasonably close and prove
the comparability of these two groups in intelligence.
Figure 8 is a graphic comparison of the intelligence
quotients of these same two groups.
Large class intervals of
20 were used here for the same reason as stated in the
58
TABLE
XIV
INTELLIGENCE QUOTIENTS OF GROUPS IVA AND IVB,
A8 CLASSES
Pupil
no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Mean
Group
IVA
Group
IVB
96
89
85
112
104
110
111
101
93
100
95
97
82
117
116
92
96
91
113
107
95
90
92
105
77
86
92
80
97
110
99
110
87
118
88
79
80
95
101
95
103
84
87
83
80
98
106
97
89
81
94
107
85
101
81
103
82
91
88'
113
106
90
113
106
105
99.07
94.83
59
:H:E
60
description of the corresponding B8 groups.
The difference
of five students between the two groups did not warrant the
drawing of the graph on a percentage basis.
It is seen at
a glance that both groups have the same range in intelligence
since they meet the base line in the same class intervals.
The polygons are very similar.
The slight difference between
the areas is due to the fact that group TVA has five more stu­
dents than group IVB.
Hence the figure shows the two groups
to be highly comparable in intelligence.
Table XV shows the intelligence quotients of groups VA
and VB.
The size of the groups is reasonably comparable due
to their respective memberships of 35 and 32.
The derived
mean intelligence quotients of 1G5.21 and 108.75, respective­
ly, prove a high degree of comparability in intelligence.
Figure 9 shows a graphic distribution of the intelli­
gence quotients of these same two groups.
The slight differ­
ence of three students did not call for the construction of
the graph on a percentage basis.
Large class intervals of
20 were used as in the preceding graph.
The fact that both
polygons meet the base line in the s^ame class intervals shows
that both groups have the same range in intelligence.
The
slight difference in area is principally due to the fact that
group VA, represented by the solid line, has three more *
students than group VB.
Hence it is reasonable to conclude
from this graph that both groups are highly comparable in
61
TABLE
XV
INTELLIGENCE Q U O T I M T S OF GROUPS VA AND VB,
A8 CLASSES
Pupil
no.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
Mean
Group
VA
Group
VB
103
113
108
103
98
113
101
112
96
114
109
112
92
111
101
104
87
106
107
103
116
98
108
87
114
112
98
107
122
106
108
101
112
93
104
103
99
102
101
110
95
84
106
101
109
117
114
118
107
119
99
121
116
109
103
114
112
125
112
117
95
116
116
112
102
103
111
105,21
108.75
63
intelligence.
These tables and graphs with their proof of compara­
bility in each pair of groups of the A8 class leave only the
variable of the two time-bases which are the distinguishing
elements of the two teaching methods.
Gains made on tests by each A8 group.
Following the
same procedure used with the B8 classes, each A8 group was
given a pretest at the beginning of the experiment and a final
test at the end.
This was done in order
made by each student.
to show the gain
It was then necessary to tabulate the
gains made by each group as they were the determinants of the
relative merits of the two different methods of teaching.
In
these tables, the scores are placed according to the names of
the students arranged alphabetically and not from the highest
to the lowest in consecutive order. . For obvious reasons the
names of the students are omitted.
Table XVI shows the pretest score, the final score, and
the gain of each student in group TVA using the ten-week meth­
od.
The range of the pretest score is from 73 to 107, of the
final score from 75 to 113, and of the gains from -7 to 24.
There were seven students who made losses which are expressed
as negative gains in the table.
Table XVII shows the pretest score, the final score,
and the gain of each student in group IVB using the semi-weekly
64
TABLE
XVI
GAINS MADE BY GBOUP IVA, TEN-?/EEKE,
A8 CLASS
Pupil
no*
Pretest
score
Final
test
score
Gain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
91
96
89
100
93
87
86
89
80
92
101
107
79
90
102
81
90
82
87
99
92
85
98
87
73
75
96
79
84
94
104
83
94
95
98
98
108
94
102
95
90
96
100
89
97
97
112
80
107
100
96
93
83
92
98
88
98
113
100
75
77
97
84
108
97
97
80
105
100
97
7
12
5
2
2
3
10
11
9
5
- 4
5
1
17
- 2
15
.3
1
5
- 1
- 4
13
15
13
2
2
’ 1
5
24
3
- 7
- 3
11
5
- 1
65
TABLE
2711
GAINS MADE BY GROUP IVB, SEMI-WEEKLY ALTERNATION,
A8 CLASS
Pupil
no.
pre­
test
score
Final
test
score
Gain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
73
102
80
96
86
73
74
77
88
74
93
93
91
95
75
97
97
79
•83
88
89
86
75
92
103
90
94
109
83
101
82
100 *
81
196
90
84
83
81
80
73
98
91
102
96
85
93
106
94
92
86
109
96
75
96
103
92
99
109
94
104
9
- 2
1
10
4
11
9
4
- 8
- 1
5
- 2
11
1
10
- 4
9
15
9
- 2
20
10
0
4
0
2
5
0
11
3
66
alternation method.
same (teacher.
This group and IVA were taught by the
The range of the pretest score happens to be
the same as that of the final score, namely, from 73 to 109.
The gains range from -8 to £0.
In this group of 30 students
there were three who showed no gain, and sis who made losses.
Table XVIII shows the pretest score, the final score,
and the gain of each student in group VA using the ten-week
method.
The range of the pretest score is from 65 to 113,
of the final score from 68 to 115, and of the gains from -10
to 10.
In this group of 35 students there were three who
showed no gain, and ten who made losses.
Table XIX shows the pretest score, the final score,
and the gain of each student in group VB using the semi-weekly
alternation method.
teacher.
This group and VA were taught by the same
The range of the pretest score is from 66 to 110,
of the final score from 70 to 110, and of the gains from -8
to 1£.
In this group of 3£ students there were two who show­
ed no gain, and ten who made losses.
Comparisons of the two pairs of groups in terms of
gains.
Having tabulated the gains of each of these four
groups, the next step was to arrange these four groups in
pairs in order to make comparisons in terms of gains.
pair represented both methods of teaching
the same teacher.
Each
and was taught by
Table XX shows this arrangement.
67
TABLE
XVIII
GAINS MADE BY GROUP VA, TEN-WEEKS,
A8 CLASS
Pupil
no.
Pre­
test
score
Final
test
score
Gain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
94
91
107
102
68
106
108
96
80
102
65
72
87
97
90
89
85
105
82
96
113
"'96
90
80
97
97
105
104
90
90
90
95
92
90
80
89
93
110
109
68
112
98
94
86
105
70
77
83
97
98
90
94
108
89
91
115
102
84
73
106
103
105
97
100
100
91
89
97
86
88
- 5
2
3
7
0
6
-10
- 2
6
3
5
5
- 4
0
8
1
9
3
7
- 5
2
6
- 6
- 7
9
6
0
- 7
10
10
1
- 6
5
- 4
8
68
TABLE
XIX
GAINS MADE BY GROUP VB, 8 M M W 5 B K X Y ALTERNATION,
A8 CLASS
Pupil
no.
Pre­
test
score
Einal
test
score
Gain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
84
110
102
85
110
87
85
101
103
106
99
84
88
89
66
79
93
98
90
100
92
68
89
109
90
90
94
78
107
95
89
87
86
103
99
87
110
84
77
104
100
110
108
88
94
90
75
82
101
110
93
98
96
70
100
102
95
83
98
73
107
93
90
90
2
- 7
- 3
2
0
- 3
5 8
3
- 3
4
9
4
6
1
9
3
8
12
3
- 2
4
2
11
- 7
5
- 7
4
- 5
0
- 2
1
3
69
TABLE
IX
COMPARISONS OP THE TWO METHODS IN TERMS OP GAINS
FOR A8 GROUPS
Gains in
Number of pupils showing
each amount of gain
class
Ten
Semiweeks weekly
Groups
IVA
IVB
intervals
24
22
20
18
16
14
12
10
8
6
4
2
0
(- 2)
(- 4)
(- 6)
(- 8)
(-10)
Number
Mean
S.D.
SiE.M.
-
25
23
21
19
17
15
13
11
9
7
5
.3
1
(((((-
Ten
weeks
Semi­
weekly
Groups
VB
VA
1
1
1}
3)
5)
7}
9)
1
2
3
2
1
1
6
7
3
3
3
5
2
5
4
1
1
1
35
5.57
6.70
1.13
1
6
4
30
5.33
5.98
1.09
2
4
7
3
5
5
1
2
4
1
1
35
2.83
5.35
.90
1
1
3
1
5
7
4
2
3
1
4
32
2.00
5.29
.94
S.D. is Standard Deviation, end S.E.M. is the
Standard Error of the Mean#
IVA and IVB are the groups taught by one
teacher; VA and VB are the groups taught by another
teacher.
70
In this table the two pairs of groups are shown in two
parallel columns.
The students of each group are arranged in
a frequency distribution according to the number making each
amount of gain as shown in the corresponding class interval#
The gains are arranged consecutively in class intervals of
two; the class intervals range from (-10)-(-9) to 24-25.
The arrangement of the groups in pairs in this table
were for the following purposes:
(a) that the number of
students in one group making each amount of gain might be com­
pared with the number of students in the corresponding group
making the same amount of gain;
(b) that the mean, standard
deviation, and standard error of the mean might he calculated
from each frequency distribution;
(c) that the obtained means
of the corresponding groups could be compared; and (d) that
the mean, standard deviation, and standard error of the mean
might be used as bases in statistical comparisons to be made
later•
The means of groups IVA snd IVB being 5.57 and 5.33,
respectively, show a difference of .24.
This is equivalent
to each student in group IVA making a gain of .24 more than
each student in group IVB.
This obtained difference, however,
is very small and is slightly in favor of group IVA represent­
ing the ten-week method.
Although the standard deviation and the standard error
of the mean of each group was calculated to serve as a basis
71
for statistical comparisons to be made later, we shall at this
time point out their significance in relation to the means of
these two groups.
We shall do the same for the other groups
when we arrive at a discussion of their results.
Group IVA has a standard deviation of 6,70,
This sig­
nifies that approximately two-thirds of all the gains, or 23
gains, would be included between the limits of 5,57 (the mean)
plus 6,70, or 12*27, and 5,57 minus 6,70, or -1,13,
By ob­
serving the distribution closely we verify this.
The standard error of the mean of group IVA is 1,13,
This signifies that if we gave this group a large number of
tests and calculated the gains, two-thirds of the time our
means would not deviate more than 1.13 from the observed mean
of 5.57.
That is, two-thirds of all the future means of this
group would lie between the gains of 4,44 and 6.70#
In group IVB, with a standard deviation of 5.98, approx­
imately two-thirds of all the gains, or 20 gains, would lie
between the limits of 5.33 (the mean) plus 5.98, or 11.31,
and 5.33 minus 5.98, or -.65.
The standard error of the mean of group IVB is 1.09.
This means that if we gave this group a large number of tests
and calculated the gains, two-thirds of the time their means
would not deviate more than 1.09 from the mean of 5.33.
That
is, two-thirds of all the future means of this group would
lie between the gains of 4.24 and 6.42.
72
Figure 10 is a graphic presentation of the gains of
groups IVA and IVB#
The base line shows that class intervals
of 4 were used for the gains.
than group IVB#
Group IVA has five more students
If these five students were excluded from
group IVA, represented by the solid line polygon, and then
the polygons placed one upon the other in such a way that
the right side of one would fall on the left side of the other,
we would find that there would be quite a similarity in shape,
thus rendering the eye to see more clearly that the areas of
both polygons are approximately equal.
In connection with
the range of gains, we find that the low gains or losses of
one group are exactly the same as those of the other as both
polygons meet the base line in the same class interval on the
left side of the graph; there is a small excess of high gains,
however, on the right side of the graph as the solid line repre­
senting the ten-week method extends slightly farther than the
broken line.
This graph shows that there is a slight advantage
in favor of the group of the ten-week method#
This table also shows the means of groups VA and VB to
be 2#83 and 2s
#00, respectively.
The differnce of .83 is
equivalent to each student in group VA making a gain of .83
more than each student in group VB#
This obtained difference
slightly favors the ten-week method represented by group VA.
Group VA has a standard deviation of 5.35.
This means
that approximately two-thirds of all the gains, or 23 gains,
m
74
would be included between the limits 2*83 (the mean) plus
5*35, or 8.18, and 2.83 minus 5.35, or -2.52.
The standard error of the mean of
group VA is
.90.
This signifies that if we gave this groupa large number of
tests and calculated the gains, two-thirds of the time our
means would not deviate more than .90 from the mean of 2.83.
That is to say that two-thirds of all our future means of this
group would lie between the gains of 1.93 and 3.73.
Group VB has a standard deviation of 5.29.
Two-thirds
of all the gains, or 21 gains, would be included between the
limits of 2.00 (the mean) plus 5.29, or 7.29, and 2.00 minus
5.29, or -3.29.
The standard error of the mean of group VB is .94,
signifying that if we gave this group a large number of tests
and calculated the gains, two-thirds of the time our means
would not deviate more than .94 from the mean of 2.00.
is, two-thirds of all the future means of
this group
That
would lie
between the gains of 1.06 and 2.94.
Figure 11 shows a graphic comparison of the gains made
by groups VA and VB.
Class intervals of 4 were used for the
gains as in the preceding figure.
students than group VB.
Group VA has three more
If we exclude these three students
from group VA, represented by the solid line polygon, then we
can reasonably say that the graph shows the areas of the two
polygons to be approximately equal.
Group VA has the greater
li
!«■!
76
range.of gains.
The excess of high gains is greater than the
excess of low gains giving this group, representing the tenweek method, the slight advantage.
Comparisons of the ten-week groups« as a single group,
with the semi-weekly alternation groups, as a single group,
in terms of gains.
Having made comparisons of the four groups
arranged in pairs, the next step was to put the same two tenweek groups together as a single group and compare their gains
with those of the same two semi-weekly alternation groups placed
together as a single group.
The purpose for doing this was
that, according to statistics, the reliability of the results
is higher when the number of eases is increased.
To this end,
Table XXX was made.
In this table the two ten-week groups, TVA and VA, are
combined and placed in one column, and parallel to it in another
column is a combination of the two semi-weekly alternation
groups, IVB and VB.
The students of each combined group are
arranged in a frequency distribution according to the number
making each amount of gain as shown in the corresponding class
interval.
The gains are arranged consecutively in class inter­
vals of two; the class intervals range from (-10)-(-9) to 24-35.
It is seen that the number of students is practically
equal for both methods, that of the ten-week method being 70
and that of the second method being 62.
77
TABLE
XXI
COMPARISONS OF TEN-WEEK GROUPS AS A SINGLE
GROUP, WITH SEMI-WEEKLY ALTERNATION GROUPS
AS A SINGLE GROUP, IN TERMS OF GAINS
FOR A8 CLASSES
Gains in
Number of pupils
showing each
amount of ga in
class
Tenweek
groups
Semi­
weekly
groups
intervals
IV, VA
IV,VB
24— -25
22 - 23
20 - 21
18 - 19
16 - 17
14 - 15
12 - 13
10 - 11
8 - 9
6 - 7
4 - 5
2 - 3
0 - 1
U-M>) - ((- 4) - ((- a) - ((- 8) - ((-10) - (Number
Mean
S.D.
S.E.M.
1
1)
3)
5)
7)
9)
1
2
3
5
5
8
9
12
8
4
5
4
2
1
1
1
7
7
1
10
9
9
6
4
1
5
70
4.20
6.22
.74
62
3.61
5.88
.75
S.D, is the Standard Deviation, and S.E,M. is
the Standard Error of the Mean.
78
First, the means of both groups were derived.
means play a very important part as statistical data.
The
If they
came within close range of each other then there would be very
little difference between the two methods but if not, then a
superiority of one method over the other would be shown.
The
table shows the derived means of the ten-week method and the
semi-weekly alternation method to be 4.20 and 3.61, respective­
ly.
This signifies that they are within close range of each
other.
The small difference of .59 is slightly in favor of
the ten-week method.
Second, the standard deviation and the standard error
of the mean of each of the two groups were derived.
The table
shows the standard deviations of the two groups to be within
close range of each other, as are also the standard errors of
the means.
These calculations, namely, the mean, the standard
deviation, and the standard error of the mean, will not only
be used in the present comparison but will also serve as some
of the statistical bases for further comparisons to be made
later.
We shall now point out the present significance of the
standard deviation and the standard error of the mean of each
of the two groups.
The standard deviation of the combined ten-week group
being 6.22 signifies that approximately two-thirds of all the
79
gains, or 47 gains, would be included between the limits 4.20
(the mean) plus
6.22, or 10.42, and 4.20 minus 6.22, or -2.02.
The standard error of the mean of this ten-week group
being .74 means that if we gave this group a large number of
tests and calculated the gains, two-thirds of the time our
means would not deviate more than .74 from the mean of 4.20.
That is, two-thirds of all our future means of this group
would lie between the gains of 3.46 and 4.94.
The combined semi-weekly group having a standard devia­
tion of 5.88 signifies that approximately two-thirds of all
the gains, or 41 gains, would be included between the limits
3.61 (the mean) plus 5.88, or 9.49, and 3.61 minus 5.88, or
-2.27.
The standard error of the mean of the same semi-weekly
group being .75 means that if we gave this group a large number
of tests and calculated the gains, two-thirds of the time our
means would not deviate more than .75 from the mean of 3.61.
That is, two-thirds of all our future means of this group
would lie between the gains of 2.86 and 4.36.
Figure 12 shows the gains of these two large groups in
a graphic way.
Class intervals of 4 were used here as in the
two preceding graphs in this chapter.
The two combined ten-
week groups, represented by the solid line, has eight more
students than the two semi-weekly combined groups.
If these
eight students were excluded, we can see at a glance that the
81
polygons would be equal in area as the distribution of the
gains
of one group followsapproximately that of the other.
The losses or low gains of both large groups are about the
same.
The combined ten-week group, however, has more hi*gh
gains
than the other group but not tola large extent.
gives
the ten-week method a slight advantage.
This
Statistical comparisons of A8 groups for prediction
purposes.
In the first jblace, we shall compare (a) the two
pairs of groups, IVA, IVB, -and VA, VB, in order to determine
for each pair the chances that the group of one teaching meth­
od would have of being superior to the other; and (b) the two
ten-week groups, IVA, VA, as a single group, with the two
semi-weekly groups, IVB, VB, also as a single group, in order
to determine the chances that one of these combinations would
have of being superior to the other.
The means used in this first comparison will be the
original means obtained for each group of the two pairs and
for each of the combined groups.
We shall elucidate this more
fully later.
In the second place, we shall eompare the same two tenweek groups, IVA, VA, as a single group, with the same two
semi-weekly groups, IVB, VB, as a single group, not only for
the purpose of prediction but also to serve as a check on the
results of the first comparison.
The means used here as a basis for comparing the com­
bined groups are very different from those used in the first
comparison.
The means used for the combined groups in the
first comparison were obtained by making a frequency distri­
bution of the gains of each of the combined groups as a single
separate group, while here the original means of each of the
two groups, IVA and VA, are simply added together to be used
as a basis of comparison, the same being done with the original
means of each of the two groups, IVB and VB*
Having stated the two ways in which these comparisons
will be made, we shall now discuss the first with the use of
Table XXII*
This table shows each of the two pairs of groups
and each of the combined groups as previously mentioned.
The means used as the basis in this comparison, as we
have already pointed out, are the original means obtained by
having made a frequency distribution for each group of the
two pairs, and for each of the combined groups, as a single
group; the means of each of the pairs of groups are taken
from Table XX, and those of the combined groups are taken from
Table XXI.
From the means and standard error of the means of each
pair and also of each combination of groups will be derived
other statistical data in order that the chances of superiority
may be determined.
Table XXII shows the obtained difference between the
TABLE
XXII
STATISTICAL COMPARISON OF GAINS OF A8 GROUPS SHOWING PREDICTIONS
FOR SEPARATE CLASS GROUPS AND FOR THE TOTAL
GROUPS AS SINGLE LARGE DISTRIBUTIONS
Biff.
Groups
compared
i/feans
(Ten
fireeks)
.Means
(Semiweekly)
S.l.M.
(Ten
weeks)
IVA vs, IVB
5.57
'5.33
1.13
1.09
1.24
1.57
.15
1.3 to 1
VA
S.83
2.00
.90
.94
.83
1,30
.64
2.8 to 1
.59
1.05
.56
2.4 to 1
vs.
VB
All students in
groups IV,VA
4.20
S.E.M.
(Semi­
weekly) .
Ratio
Chances
.74
vs.
All studdnts in
groups IV,VB
SD
3.61
.75
IVA and VA are the ten-week groups; IVB and VB are the semi-weekly alternation
groups, S.E.M. is the Standard Error of the Mean, and Ep is the Standard Error of
the difference.
84
two means of groups IVA and IVB to be .24.
The signifieance
of this difference is determined by dividing .24 by the stand­
ard error of the difference, or 1.57, which yields .15, or the
ratio.
By referring to the same standard-error table mentioned
in Chapter III, this ratio was found to signify that the
chances are only 1.3 to 1 that group IVA, the ten-week method,
would be superior to group IVB, the semi-weekly method.
Befer-
ence was also made in the last chapter to the statement that
in order to be practically certain that there is a real differ­
ence in merit between two groups, the difference between the
means should be at least three times as great as the standard
error of the difference.
In the present case, the difference
.24 is much smaller than the standard error of the difference
1.57.
On the basis of these data we can say that the chances
are indeed very slight that group IVA, the ten-week method,
would actually make higher gains in mathemamatics than group
IVB, the semi-weekly method.
As a matter of fact the chances
are about even for both groups since the ratio is 1.3 to 1.
The difference between the means of Va and VB was
.83.
By dividing this difference by the standard error of the dif­
ference, or 1.30, we obtain the ratio, or .64.
The standard-
error table reveals this ratio to indicate that the chances
are 2.8 to 1 that the ten-week group VA is superior to the
semi-weekly group VB.
On the basis of these data we can say
that the chances are slight that group VA would actually make
85
higher gains in mathematics than group VB.
And finally in this table the two ten-week groups, as
a single group, were compared with the two semi-weekly groups,
as a single group.
The difference between the means of these
two large groups.is .59.
By dividing this difference of .59
by 1.05, the standard error of the difference, we obtain the
ratio of .56.
The standard-error table shows this ratio to
signify that the chances are 2.4 to 1 that the IT, VA combi­
nation is superior to the IV, VB combination.
On the basis
of these data we can say that the chances are very slight
that the ten-week combination would make higher gains in math­
ematics than the semi-weekly combination.
As a result of combining the ten-week groups and also
the semi-weekly groups, we see that the chances of superiority
of the combined ten-week groups were slightly greater than
those of the pair of groups IVA and IVB and slightly fewer
than those of the pair VA and VB.
Ihble XKIII shows the second method of comparing the
two combined ten-week groups with the two combined semi-weekly
groups.
As was stated previously, the original means of XVA
and VA, respectively, were added together as were those of
IVB and VB.
The obtained sums of the means then became the
basis on which further calculations were made in order to ar­
rive at the chances of superiority of one combined group over
the other.
86
TABLE
XXIII
STATISTICAL COMPARISON OF A8 GROUPS SHOWING
A SECOND METHOD OF MAKING PREDICTIONS BY
DIFFERENT COMPUTATIONS OF THE MEANS
A8 groups
Items
Mean
%
Ten-•week groups
Semi-weekly groups
TVA
~IVB
VA'
5.57
1.13
2.83
.90
VB
5.33
1.09
2.00
.94
Sum of means,
groups A and B
8.40
7.33
E of sum of
means, groups
A and B
1.44
1.44
i
Diff. between
sums of means,
groups A minus
groups B
1.07
E of diff. be­
tween sums of
means of groups
A and B
2.04
Ratio
Chances
D/ED
.52
2.3 to 1
Ej$ is the Standard Error of the mean; E is the Stand­
ard Error; D is the difference, and ED the standard Error of
the difference.
8?
This table shows the obtained difference between the
sums of the means to be 1.07.
Dividing this quantity by the
standard error of the difference between the sums of the means,
or 2.04, we obtain the ratio of .52.
According to the standard
error tabic this ratio signifies that the chances are 2.3 to 1
that the IV, VA combination is superior to the IV, VB combina­
tion.
With these data as a basis, we can say that the chances
are again very slight that the ten-week combination would make
higher gains in mathematics than the semi-weekly combination.
In both these comparisons of the two combined groups,
we found that the results obtained were almost identically the
same, as the ratios were 2.4 to 1, and 2.3 to 1, respectively,
and both gave the ten-week method the advantage, but of very
little significance.
CHAPTER
V
RESULTS OF THE B8 M D A8 GROUPS COMBINED
In the two preceding chapters we discussed and analyzed
the results of the B8 and A8 groups, respectively.
One of the
important items shown in the discussion of the B8 results was
the comparison in terms of gains of the three combined tenweek groups, as a single group, with the three combined semi­
weekly groups, as a single group.
Likewise in the A8 results
we showed a comparison of the two combined ten-week groups,
as a single group, with the two combined semi-weekly groups,
as a single group.
Now we shall compare two larger combinations of, groups
in terms of gains.
combined as
All the B8 and A8 ten-week groups will be
a large single group and compared with all semi­
weekly groups also combined as a large single group of the
same
two grades.
Since these two large combinations include all the stu­
dents who participated in the experiment, the results of the
comparison should show more clearly the relative merits of the
teaching methods than when the B8 or the A8 combinations of
groups were compared separately.
in this chapter we shall discuss the results of the
combined B8 and A8 groups in the following order:
comparisons
in terms of gains of the combined ten-week groups as a single
89
group, with the combined semi-weekly alternation groups as
a single group; statistical comparison of gains of these same
two large combined groups for the purpose of showing predic­
tions; and finally, statistical comparison of these two large
groups showing a second method of making predictions.
Comparisons in terms of gains of the combined ten-week
groups as a single group, with the combined semi-weekly alter­
nation groups as a single group.
The five ten-week groups to
be combined as a single groups are IA, IIA, IIIA, IVA, and YJL,
and the five semi-weekly groups to be combined as a single
group are IB, IIB, IIIB, XVB, and VB,
In each of these two
combinations of groups, the first three are B8*s, and the
remaining two are A 8 fs.
Table XXXV shows the comparison in terms of gains of
these two large combinations of groups.
In this table the
five ten-week groups, as a single group, are placed in one
column, and parallel to it in another column are the five
semi-weekly alternation groups, as a single group.
The stu­
dents of each combined group are arranged in a frequency dis­
tribution according to the number making each amount of gain
as shown in the corresponding class interval.
The gains are
arranged consecutively in class intervals of two; the class
intervals range from (-IS)-(-11) to 24-25.
It is seen that the number of students for the tenweek method is 172, and for the semi-weekly method is 158.
90
TABLE
XXIV
COMPARISONS OF TEN-WEEK GROUPS AS A SINGLE
GROUP, WITH SEMI-WEEKLY ALTERNATION GROUPS
AS A SINGLE GROUP, IN TERMS OF GAINS FOR
THE COMBINED B8 AND A8 CLASSES
Gains in
Number of %>upils
showing ejach
amount of gain
class
Ten- /
week
groups
intervals
24 - 25
22-23
20 - 21
18 - 19
16 - 17
14 - 15
12 - 13
10 - 11
8 - 9
6 - 7
4 - 5
2 - ..-3
0 - 1
(- 2) - (- 1)
t- 4) - (- 3}
(- 6) - (- 5)
(- 8) - (- 7)
(-10) - (- 9)
(-12) (-11)
Number
Mean
S.D.
S.E.M.
I,II,III,
IV,VA
Semi­
weekly
groups
I,II,III,
IV,VB
1
1
2
2
3
6
11
12
20
18
22
28
16
8
13
7
3
1
2
5
21
18
14
19
22
22
13
12
2
5
1
172
5.27
6.14
.47
158
4.47
5.59
.45
S.D. is the Standard Deviation, and S.E.M. is
the Standard Error of the Mean.
91
First, the means of these two large groups were derived.
The table shows the means of the ten-week method and the semiweekly alternation method to be 5.27 and 4.47, respectively,
indicating that they are within close range of each other.
This signifies that as far as the means are concerned there
is very little difference between the two methods.
The small
difference of .80 is slightly in favor of the ten-week method.
Second, the standard deviation and the standard error
of the mean of each of the two groups were derived.
The table
shows the standard deviations of the two large groups to be
within close range of each other.
The standard errors of the
means show a still closer range.
These calculations, namely, the mean, the standard
deviation, and the standard error of the mean, will not only
be used in the present comparison but also will serve as some
of the statistical bases for further comparisons to be made
later in this chapter.
We shall now point out the present significance of the
standard deviation and the standard error of the mean of each
of the two groups.
The standard deviation of the combined ten-week group
being 6.14 signifies that approximately two-thirds of all the
gains, or 115 gains, would be included between the limits 5.27
(the mean) plus 6.14, or 11.41, and 5.27 minus 6.14, or -.87.
92
The standard error of the mean of this ten-week com­
bined group being .47 signifies that if we gave this group
a large number of tests and calculated the gains, two-thirds
of the time our means would not deviate more than .47 from
the mean of 5.27.
That is, two-thirds of all our future means
of this combined group would lie between the gains of 4.80
and 5.74*
The combined semi-weekly group having a standard devia­
tion of 5.59 signifies that approximately tto-thirds of all
the gains, or 105 gains, would be included between the limits
4.47 (the mean) plus 5.59, or 10.06, and 4*47 minus 5.59, or
-1 .12 .
The standard error of the mean of the same semi-weekly
group being .45 signifies that if we gave this group a large
number of tests and calculated the gains, two-thirds of the
time our means would not deviate more than .45 from the mean
of 4.47.
That is, two-thirds of all our future means of this
group would lie between the gains of 4.02 and 4.92.
Figure 13 shows a graphic distribution of the gains of
these same two large groups.
The base line shows that class
intervals of 4 were used for the gains.
At a glance we see
that one polygon is practically superimposed upon the other.
Hence we can very reasonably say that the areas of the poly­
gons are equal.
Observing the graph very closely, we can
see that both groups have almost identical losses (negative
94
gains) but that the ten-week group, represented by the solid
line, has a few more high gains than the semi-weekly group.
This indicates a slight advantage in favor of the ten-week
method.
Statistical comparison of gains of these same two large
combined groups for the purpose of showing predictions.
We
shall now compare the five ten-week groups, IA, IIA, IIIA,
IVA, and VA, as a single group, with the five semi-weekly
groups IB, IIB, IIIB, IVB, and VB, as a single group, in
order to determine the chances that one of these combinations
would have of being superior to the other.
The means used as a basis in this comparison are the
original means obtained by having made a frequency distribu­
tion for each of the combined groups, as a single group. These
means are taken from Table XXIV, as are also the standard
errors of the means.
From the means and standard error of the means of each
combination of groups will be derived other statistical data
in order that the chances of superiority may be determined.
These data are shown in Table XXV.
In this table we see the statistical comparison in
terms of gains of the combined ten-week groups as a single
group, and of the combined semi-weekly alternation groups as
a single group.
Examining the data in this table we find that the
TABUS XXV
STATISTICAL COMPARISON OF GAINS OF B8 AND A8 GROUPS. COMBINED- SHOWING
PREDICTIONS FOR THE TOTAL TEN-WEEK GROUPS AND FOR THE TOTAL SEMIWEEKLY ALTERNATION GROUPS AS SINGLE LARGE DISTRIBUTIONS
Groups
compared
Means
All students in
groups I,II,III,
IV,VA
5.27
(Ten weeks)•
S .E .M .
E jj
Ratio
Chances
.47
•
o
go
- vs.
All students in
groups I,II,III,
IV, VB
4.47
(Semi-weekly)
Diff.
•65
1.23
8.2 to 1
.45
I,II,III,IV,VA are the ten-week groups, and I,II,III,IT,VB are the
semi-weekly alternation groups. S.E.M. is the Standard Error of the
Mean, and Ep is the Standard Error of the difference.
96
obtained difference between the means of these two large
groups is .80*
The significance of this difference is deter­
mined by dividing .80 by the standard error of the difference,
or .65, which yields 1.23, or the ratio.
By referring to the
same standard-error table mentioned in Chapters III and IT,
this ratio was found to signify the chances are only 8.2 to
1 that the combined ten-week group would be superior to the
combined semi-weekly group.
Reference was also made in the
last two chapters to the statement that in order to be prac­
tically certain that there is a real difference in merit be­
tween two groups, the difference between the means should be
at least three times as great as the standard error of the
difference.
In the present case, the difference between the
means, or .80, is just a shade larger than the standard error
of the difference, or .65.
On the basis of these data we can say that the chances
*
are very slight that the combined ten-week group, I, II, III,
IV, VA, would actually make higher gains in mathematics than
the combined semi-weekly group, I, II, III, IV, VB.
Statistical comparison of these two large groups show­
ing a second method of making predictions.
The reason for
this second comparison is to check the reliability of the re­
sults of the first.
The means used as a basis in this second
comparison of the combined groups are very different from
those used in the first.
The means used in the first com­
parison were obtained by making a frequency distribution of
97
the gains of each of the combined groups as one single sep­
arate group*
In this second comparison, however, the
orig­
inal means of each of the five ten-week groups, IA, IIA, IIIA,
IVA, and VA, are simply added together, and this sum used as
a basis.
The same is done with the original means of each
of the five semi-weekly groups, IB, IIB, IIIB, IVB, and VB*
From the sum of the means and standard error of the
sum of the means of each combination of groups will be de­
rived other statistical data in order that the chances of
superiority may be determined.
These data are shown in Table
XXVI.
In this table we see the statistical comparison in
terms of gains of the combined B8 and A8 ten-week groups as
a single group, and of the combined B8 and A8 semi-weekly
alternation groups as a single group.
The respective means
of the five ten-week groups are shown, as is also the sum of
the means.
The same items are shown for the semi-weekly groups.
Examining the data in this table we find that the ob­
tained difference between the sums of the means of the two
large groups to be 3.81.
Dividing this quantity by the stand­
ard error of the difference between the sums of ^he means, or
3.07, we obtain the ratio of 1.24.
According to the standard
error table this ratio signifies that the chances are 8.3 to
1 that the ten-week combination is superior to the semi-weekly
combination.
TABLE
XXVI
STATISTICAL COMPARISON OF B8 AM) A8 GROUPS SHOWING A SECOND METHOD
OF MAKING PREDICTIONS BY DIFFERENT COMPUTATIONS OF THE MEANS
Ten-week B8 and A8 groups
!
Semi-weekly B8 and A8 groups
Items
IA
IIA
IIIA \
IVA
VA
IB
IIB
IIIB
Mean
%
3.14
L. 13
5.72
.80
3.71
1.01
5.57
1.13
2.83
.90
6.70
.73
5.73
1.07
2.40
5.33
.80 ‘ 1.09
Sum of means,
groups A and B
E of sums of
means, groups
A and B
Diff. of sums o f
means, groups
A-B
E of diff. between sums of
means of groups
A and B
Ratio
Chances
U/ED
25.97
22.16
2.24
2.10
IVB
VB
2.00
.94
3.81
3.07
1.24
8.3 to 1
Ew is the Standard Error of the Mean; E is the Standard Error; D is the differ­
ence, and ED the Standard Error of the difference. The B8 groups are I,II,IIIB.
The A8 groups are IV,VA and IV,VB.
to
CD
99
With these data as a basis, we can say that the chances
are again very slight that the ten-week groups would make high­
er gains in mathematics than the semi-weekly groups.
In both these comparisons of the two large groups, we
found that the results obtained were almost identical, as the
ratios were 8.2 to 1, and 8.3 to 1, respectively.
Both com­
parisons gave the ten-week method the advantage, but of very
little significance.
CHAPTER
VI
CONCLUSIONS AND R1COMMENDATIONS
As stated in the problem chapter, it was the purpose
of this study to compare experimentally by a standardized
arithmetic test the achievement of several pairs of compar­
able groups in the eighth grade.
Each pair had studied math­
ematics by two different methods, each on a different timebasis, in order to determine which method was the better.
The two different time-bases for these methods were (1) one
of the groups of each pair studied mathematics for ten con­
secutive weeks, and science the other ten, and (S) the other
group of each pair studied mathematics on a half-week basis,
alternating with science the other half.
Science was not included in the measurement for reasons
stated in the procedure chapter.
As previously stated the standardized test was given
both at the beginning and at the end of the experiment in
order to determine the gains made by the students.
Chapters III, IV, and V show the results of this ex­
periment in terms of gains both in a statistical and graphic
manner.
It is the purpose of this chapter to state the con­
clusions and recommendations resulting from this study.
101
Conclusions.
The statistical and graphic results of
the experiment lead to the following conclusions:
fl) That for ©11 practical purposes either of the two
methods could be used with the same expectancy of results.
The
data were practically tied throughout.
(2) That the continuity of the ten-week method of
studying mathematics in grade eight had no particular advan­
tage, as shown by the experiment, over the "space-learning”
or half-weekly method.
Recommendations.
In the light of these conclusions,
the following recommendations are made:
(1) So far as this experiment is able to reveal facts,
it is recommended that Edison Junior High School may use either
method for the teaching of these two subjects.
(2) Since science alternates with mathematics in grade
eight in this school, it is recommended that the science teach­
ers prepare a test based upon the science course of study for
this grade and conduct an experiment in this subject parallel
to the mathematics experiment in order to determine which
would be the better method for the teaching of science.
(3) That other junior high schools conduct an experi­
ment in grade eight mathematics similar to that of this study
not only to check the results obtained in this experiment but
also to help standardize a method for the teaching of mathe­
matics and science in this grade throughout the city school
system.
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