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year children in addition was such that they could add correctly
23.3 columns, such as were used in the test, with an accuracy
of 80 per cent.

The change effected by the practice of 60 minutes m the group
as a whole was an improvement in speed such that they could
add 10.7 more columns correctly in the final 15 minutes of the
practice than in the initial 15 minutes, while the change in accur-
acy was so small as not to need consideration.

Division

The experiment in division, with children in the second half
of the third year and the first half of the fourth year, was con-
ducted on the same general plan as the one in addition with
fourth-year children, except that the initial and final periods of
practice were 10 minutes each instead of 15, and that the entire
time spent in the practice was 60 minutes instead of 75. The
intervening 40 minutes between the initial and final practice-
periods were divided for different groups of classes into periods
of 20 minutes, 10 minutes and 2 minutes. This plan of time re-
quired 4 school days to complete the experiment when the inter-
vening time was divided into 20-minute periods. 6 consecutive
school days wiicn the intervening time was divided into lo-minute
periods, and 22 consecutive school days when the intervening
40 minutes were divided into periods of 2 minutes each. The
method of conducting the practice was described in Chapter I,
and a sample sheet of the material used may be found on page
<)8. It is necessary to remember that in scoring these division
combinations a credit of one was given for each combination
if the quotient and remainder were both correct. If either was
wrong, no credit was given. In other words, the score was



Improvement in the Group as a Whole 25

found by deducting one from the entire number of combina-
tions worked for each one that was incorrectly worked.

In Table VII, an exact class record is given of a class for
which the intervening 40 minutes were divided into periods of
ten minutes each. This arrangement made all the practice-
periods of this class of equal length. Such a record is presented
because it gives samples of all data to be used in this discussion
and affords the best opportunity to note the change in ability
from day to day. In all, eighteen classes took part in the division
experiment, which means that there were eighteen such records
as the one here presented, from which the data to be presented
in the following discussion were obtained. Six of these 18
records involve exactly the same number of practice-periods as
the one given in Table VII. Six others have but two practice-
periods between the initial and the final periods and so arc more
brief. The other 6 have 20 practice-periods between the initial
and final periods and hence occupy more than three times the
space occupied by the one given in Table VII. Six hundred
six third- and fourth-year children took part in the experiment,
in the course of which about 6500 papers were scored and en-
tered in these 18 records. These can not be printed here, but
they are placed on record in Teachers College where any one
may use them. The following table gives sufficient data to enable
the reader to understand the exact sources from which the
summaries that are to be presented later were ol)tained.

Table VII reads as follows, " Boy B in the initial practice-
period worked 70 combinations, 63 of which were correct, or
90 per cent of them. In the second practice-period he worked
72 combinations with 72 correct. In the third practice he
worked 80 combinations with 80 correct. In the fourth prac-
tice he worked 87 combinations with 83 correct. In the fifth
practice-period he worked 99 combinations with 92 correct. In
the final practice-period he worked 119 combinations with 112
correct, or 94 per cent of them. His gain in number of com-
binations worked correctly was 49, his gain per cent 78, and
his gain in accuracy expressed in per cent was 4 j^er cent." So
for any other individual. In the following discussion the
quantities mentioned above will be referred to by the numbers
in italics at the bottom of the columns. The mctliod of finding
the per cent of accuracy, the gain per cent in number of problems



26



Practice in the Case of School Children



TABLE VII

Record of Class VII in Six Ten-minute Practice-periods in
Division



Indi-


Initia




2nd


3


-d


4th


5


th


Final






Gain




vid-


Practice-


Practice-


Practice -


Practice -


Practice -


Practice-






uals


Tt^^inA


-parir^A


•Po^-.r^A


Po^;^,4


p«,.i«^i


p««;,^^






IT






r^er




xer




-L er




x^er


"


i:


Cll-Jl






Pai-
































x^er




Boys


S.


C. %C.


s.


C.


s.


C.


s.


C.


S.


C.


s.


C. %C.


Gross


cent


Ac.


A


55


54


98


60


58


84


82


77


75


76


75


74


71


96


17


30


— 2


B


70


63


90


72


72


80


80


87


83


99


92


119


112


94


49


78


+ 4


C


27


25


93


45


38


55


51


59


56


66


60


70


66


94


41


168


+ 1


D


16


12


75


14


14


25


24


25


24






18


17


95


5


63


+ 20


E


77


77


100


99


98


106


106


112


112


133


133


153


151


99


74


96


— 1


F


27


21


78


28


18


19


13


30


23


36


32


45


42


93


21


100


+ 15


G


31


25


81


36


31


46


43


41


35


38


31


49


42


86


17


68


+ 5


H


50


49


98


51


48


61


61


56


53


55


53


65


63


97


14


29


— 1


I


62


62


100


62


62


69


69


69


67


60


60


68


68


100


6


10





J


34


30


88


48


44


58


57


54


52


52


50


53


52


98


22


73


+ 10


K


18


17


95


36


33


54


50


49


44


77


70


63


59


94


42


40


— 1


L


30


19


63


35


25


46


33


64


58


40


36


37


35


95


16


84


+ 32


M


90


86


96










104


98


114


113


128


127


99


41


48


+ 3


N


69


68


99


62


59


83


81


78


75


77


77


92


89


98


21


31


— 1





43


38


89


53


45


69


65


61


58


56


53


63


60


95


22


58


+ 6


P


27


27


100


27


27


33


32


31


27


28


28


30


27


90








—10


Q


78


74


95


89


86


109


99


104


104


108


98


107


98


91


24


32


— 4


K


37


37


100


58


58


67


67


62


60


69


67


77


77


100


40


105





S


48


44


92






63


57


55


52


66


66


77


77


100


33


75


+ 8


T


67


66


99


88


86


103


101


102


101


99


95


112


111


99


45


68





U


54


51


94


57


54


92


68


69


67


62


62


64


64


100


13


25


+ 6


V


35


32


92






40


37


35


32


44


41


56


54


97


22


69


+ 5


w


27


25


92


27


25


43


39


38


35


36


36


39


39


100


14


56


+ 8


Girls




































A


14


11


79


29


21


37


29


31


19






38


31


82


20


55


+ 3


B


24


23


96


37


36


30


29


35


32


45


45


50


50


100


27


117


+ 4


C


39


36


93


39


38


52


51


53


49


50


48


57


53


93


17


47





D


50


46


92


59


56


69


68


62


62


51


51


81


80


99


34


74


+ 7


E


56


54


97


61


58


61


61


65


65


55


53


66


62


94


8


15


— 3


F


52


49


94


54


52


71


71


66


63


50


48


76


73


96


24


49


+ 2


G


46


41


89


49


46


51


49


50


47


49


46


66


61


91


10


24


+ 2


H


42


39


93


53


51






68


68


65


64


79


77


98


38


97


+ 5


I


76


74


98


20


19


83


83


78


77


81


79


110


107


97


33


45


— 1


J


47


46


98


50


48


69


68


55


55


55


55


66


61


97


18


37


— 1


K


10


8


81


7


2


25


24


14


7


28


26


20


13


65


5


63


—23


L


72


69


96


71


69


93


92


93


93


90


87


98


96


98


27


39


+ 2


M


45


45


100


55


54


66


62


77


76


63


54


71


67


95


22


19


— 5


N


6f)


57


95


59


59


72


72


57


56


70


70


84


83


99


26


46


+ 4


O


24


16


67


37


22


36


27


45


36


41


32


52


45


87


29


181


+ 20


P


45


45


100


39


38


59


59


64


58


62


61


71


70


99


25


56


— 1


Q


24


20


83


16


15


40


39


32


30


34


31


29


26


90


6


30


+ 7


R


42


40


95


43


42


50


50


50


49


45


44


52


49


94


9


23


— 1


S


23


20


87






29


29


36


25


33


32


37


36


98


16


80


+ 1


T


26


23


89


44


41


56


53


51


50


43


42


56


56


100


33


143


+ 11


U


30


16


46


30


16


37


29


32


23


37


32


38


36


95


30


125


+ 49


/


2


S


4


5


6


7


8


9


10


11


12


IS


H


IB


16


17


18



S.=Problenis solved.
C.=Problems correct.
%C.=Per cent of problems correct.
Ac.=Accuracy.



Improvement in the Group as a Whole



27



worked correctly, and the gain in accuracy expressed in per
cent is the same as that used in addition, Table I, to which the
reader may refer for the method if the meaning of these figures
is not clear. " S " indicates problems solved, " C " problems
solved correctly, and "Ac." accuracy.

Initial Ability
To determine the initial ability of this group of 606 children
of last half of third and first half of fourth year in giving the
results for the division combinations, two factors must be con-
sidered, — first the number of such combinations they worked cor-
rectly, and second the accuracy of their performance. The data
to determine the first factor are the number of combinations
worked correctly in the initial ten-minute period, a sample of
which is given in Table VII, column 5. The numbers of com-
binations answered correctly by the 606 children were distributed
class by class, boys and girls separate, but only the following
summary of this distribution table can be presented here. (Table
VIII.)

TABLE VIII



Number of


Division Combinations Answered Correctly
Initial Ten-minute Period


IN


THE




Number of

combinations



to
4


5
to
9


10
to
14


15
to
19


20
to
24


25
to
29


30
to
34


35
to
39


40
to
44


45

to
49

44


50
to
54

38


55

to
59

42


60

to
64

28

4.6


65
to
69


Individuals





23


33


65


56


52


51


57


45


15




Per cent




3.8


5.4


10.7


9.2


8.6


8.4


9.4


7.4


7.3


6.3


6.9


2.5









TABLE VIII-


—Continued












Number of

combinations ....


70
to
74


75

to
79


80
to

84


85
to
89


90
to
94


95
to
99


100
to
104


105
to
109


110
to
114


115
to
119


120
to
124


125
to
129


Total


Individuals


13


16


3


9


5


5





3


1





1


1


606


Per cent


2.1


2.6


.5


1.5


.8


.8





.5


.2





.2


.2


100



Median


34 . 5 combinations


25 Percentile


21.7


75 Percentile


52.5


P.E.


15.4


P-E- t.-obt. Av.


.63



28 Practice in the Case of School Children

Table VI 11 shows a distribution (in groups of 5) of the num-
ber of combinations done correctly in the initial 10 minutes. It
shows that 23 children (or 3.8 per cent of the entire group)
did from 5 to 9 combinations, 33 children (or 5.4 per cent of the
entire group) did from 10 to 14 com])inations, etc. The range
in number of ccjmjjinations done correctly is from 5 to 126. The
median number is 34.5 combinations. This means that there
were just as many children who worked 34.5 combinations or
more correctly in 10 minutes as there were who worked 34.5
coml)inations or less. The upper 25 per cent of the group did
53 combinations or more while the lower 25 per cent did 22 com-
binations or less. The percentages in the table show that 74
per cent of the group did from 15 to 59 combinations. The
variability of the group is also shown by the P. E., 15.4. The
(lata presented here give 34.5 combinations as the most likely
true median.

Acctiracy in Dnnsion

With what degree of accuracy did this group work the com-
binations which they attempted in the initial to minutes, is the
next (|uestion for consideration. The source of the data used
in answering this question can best be seen in Table VII, column
./. The per cents of accuracy of which those in column ./ are
I)art were distributed class by class, boys and girls separate. The
following table (Table IX) shows a summary from this larger
tabic.

Table IX reads as follows from the right side: " Two hundred
and thirty-seven children (or 39.1 per cent of the entire group)
worked 96 to lOO per cent of their combinations correctly, etc."
The ]>er cents are given in groups of 5. The range is from 26
per cent of accuracy to 100 per cent. The figures show almost
a right-angle distribution. However, had the per cents been
scaled more finely at the upper end, the distribution would then
have shown a very decided skewness toward the lower end. The
median i)cr cent of accuracy is 93, that is, just as many children
worked with an accuracy of 93 to 100 per cent as with an
accuracy of 26 to 93 per cent. The upper 25 per cent of the
children worked with :m accuracy of 97 per cent or more. The
lower 25 per cent worked with an accuracy of 85 per cent or less.



Improvement in the Group as a Whole 29

TABLE IX
The Per Cent of Coukect Answeks to the Division Combina-
tions IN THE Initial Ten-minute Pehiods



Per cent of correct answers


26
to
30


31
to
35


36
to
40


41
to
45


46
to
50


51
to
55


56
to
60


61
to
65


Individuals


1





1


3


7


5


7


12


Per cent


.2





.2


.5


1.2


.8


1.2


2







TABLE IX-


—Continued












Per cent of correct answers


66
to
70


71
to
75


76
to

80


81
to

85


86
to
90


91
to
95


96
to
100


Total


Individuals


11


22


31


53


70


146


237


606






Per cent


1.8


3.6


5.1


8.7


11.6


24.1


39.1


100







Median 93 per cent

25 l^ercentile 85

75 Percentile 97
P.E. 6

The upper 63 per cent of these children worked with an accuracy
of 91 per cent or more.

With the accuracy of the group determined, we can now
define the initial ability of the group more exactly by saying
that it was such that the median number of columns worked
correctly by the group as a whole was 34.5 combinations, and
according to the median per cent of accuracy 34.5 combinations
were 93 per cent of the median number of combinations at-
tempted. The chances are that one child out of two in the group
taken at random would do 34.5 combinations correctly in 10
minutes with an accuracy of 93 per cent.

Gross Gain in Number of Combinations Worked Correctly
What gain in ability did this group of 606 third- and fourth-
year children make in the course of 60 minutes of practice, is
the next question to be answered.^ This gain has been measured

" While there were 60 minutes of practice the gain was measured for
only so minutes. The initial practice-period and the final practice-period
were 10 minutes each. But the record for each of these periods gives
the adding rate at the middle of each period. Hence the amount of
practice whose eflfect is measured is from the middle of the initial
period to the middle of the final period, or 50 minutes.







Practice in the Case of School Children



both absolutely and relatively. (Xir ])rc.sent consideration is the
gnjss fjain, wliicii was found for each individual by subtracting
the number (^f combinations worked correctly in the first lo min-
utes of practice from the number worked correctly in the final
lo minutes of practice.

Jveferring to Table VII, column 16, the reader will sec that
boy A J^^•lined 17 combinations, boy 15, 49 combinations, etc. The
j:,Mins made by the OoC) cliildren were distributed class by class,
boys and girls separate, but for our discussion a summary of
this large distribution table must suffice. 'J'his summary appears
in Table X.

'IWIilJO X

Thbi Guumh Gain in Numiiku ok (Jomiiinationh Anhwkiikd ('ourkctlv,
TBOM Firrv Minutkh of J*itA(,"ncio



Conil)in;i(i()riH gniiiod .


-19

to


-14

10


-9

to
5


-4
lo
U


1

to
5



to
10


11

to

ir,


16

to
20


21

to
25


26

to
30


31

to
35


36

to

40;


41
to
45


IndividuiilH


2


5


3


15


20


49


55


GO


G9


()0


53


41


39






Per ceuta


.3


.8


.6


2.5


3,3


8.1


9.1


9.9


11.4


9.9


8.7


0.8


6.4



TABLE 'X.—Continwd



Cuiitbinutiuuu gained.


to
50


51
to
55


50
to
(50


Gl
to
G5


G()
to
70


71
to
75


7()
to
80


81
to
85


8G

to
90


91
to
95


to
100


Totiil


IndividualB


31


18


25


17


16


10


5


5


4


2


2


606






Per centB


6.1


3


4.1


2.8


2.6


1.7


.8


.8


.7


.3


.3


100







Median


27.6 combinations


25 I'ercontilo


10.7


75 I'crcoiitilo


43.4


iM<;.


IG 3


iM':-,.,.,.< A„


.t>6



In I'ablo X the gioss gains are distributed in groups of 5.
beginning at the end of the tabic one sees that 2 children (or
.3 per cent of the entire group) worked correctly from 96 to
100 cond)inations more in the final ten-minute i)eriod than in the
initial ten-minute i)eriod ; 2 children (or .3 per cent) worked
from ()i to ()5 more, etc. The distribution is somewhat skewed



Jmprovcmenl, in the (irouj^ (is a Whole 3 r

toward the Iii^di end. The range is from a loss of 19 com-
l)inations to a j^^ain of 100 combinations. 'J'hc median j^ain is
27.5 combinations. That is, the g'roui) as a whole, measured
by the median gain, i)rorited by the practice of fifty minutes to
the extent that it worked correctly in 10 minutes at the end of
practice 27.5 more cf)mbinations than it worked in 10 minutes
at the beginning of the ])ractice. This median gain means that
there were as many children who gained 27.5 combinations or
more as there were who gained 27.5 combinations or less. The
U])per 25 ])er cent of the class profited by the i)ractice to the
extent that they worked c(jrrectly in the final 10 minutes 43.4
combinations (jr more in excess of the number wf>rked in the
im'tial 10 minutes. 'i1ie lower 25 ])er cent i)rofited to the extent
of 10.7 combinations or more, f^nly 25 of the 606 children
failed to ])rorit by the i)ractice.

Relative Cain in Division

What relation did the gross gain from 50 nn'nutes of i)ractice
bear to the initial ability is our next question for consideration.
Or, putting it in other terms, what gain per cent in ability to
make the division associations involved in this experiment re-
sulted from 50 minutes of practice? This gain for the entire
grou]) was found from the individual gains made by the 606
children.

Table VII, column //, gives these gain per cents for one class.
P)oy A gained 30 per cent, boy I> gained 78 per cent, boy C,
168 per cent, etc. The same points that were made in discussing
the gain ])er cents in addition obtain here. These individual
gain per cents were distributed class by class, boys and girls
separate, but only the smnmary of this large table is given in
Table XI, which shows the form of the distribution, the central
tendency, and variability of the group.

In Table XI, the gain ])er cents in division combinations cor-
rectly answered resulting from 50 mimites of ])ractice are dis-
tributed in groui)s of 15. The figures show that the distribution
is skewed t(jward the high end. A wide range, from a loss of
74 ])er cent to a gain of 400 per cent, is seen. The mode is in
the grou]) 6t to 75 per cent. The median is 75 ])er cent. That
is, there were just as many children who gained 75 per cent or



32



Practice in the Case of School Children



more as there were who gained 75 per cent or less. The upper

25 per cent of the class gained 116 per cent or more, while the

lower 25 per cent gained 47 per cent or less. The figures in

the table show that 96 per cent of the children profited by the

practice.

TABLE XI

Gain Pek Cent in Division Combinations Correctly Answered
FROM Fifty Minutes of Puactice





-74


-59


-44


-29


-14


1


16


31


46


61


76


Gain, per cent ....


to


to


to


to


to


to


to


to


to


to


to




-60


-45


-30


-15





15


30


45


60


75


90


Individuals


1


3


2


5


14


16


42


59


77


85


76


Per cents


.2


.5


.3


.8


2.3


2.6


6.9


9.7


12.7


14


12.5



Gain, per cent ....


91
to
105


106

to

120


121
to
135


136

to

150


137
to
165


166

to
180


181
to
195


196
to
210


211

to
225


226
to
240


241

to
255


Individuals


50


33


30


22


9


10


8


8


10


11


4


Per cents


8.3


5.4


5


3.6


1.5


1.7


1.3


1.3


1.7


1.8


.7



Gain, per cent ....


256
to
270


271
to

285


286
to
300


301
to
315


316
to
330


331
to
345


340
to
360


361
to
375


376
to
340


341
to
405


Total


Individuals


4


5


2


2


3


2


3


4


2


4


606


Per cents


.7


.8


.3


.3


.5


.3


.5


.7


.3


.7


100



Median


75 per cent


25 Percentile


47


75 Percentile


116


P.E.


34.5



Gross Gain in Accuracy

To know the complete effect of the practice we must measure
the change in accuracy of making the associations as well as the
gain in speed in making them. The gain in accuracy is given
as a gross amount, but is expressed in per cents. Table VII,
column 18, gives these gross gains for one class. They were
found by subtracting the numbers in column 4 from the corre-
sponding numbers in. column 75, Boy A lost 2 per cent, boy
B gained 4 per cent, etc. The gains made by the 606 children
were distributed class by class, girls and boys separate, but only
a summary of the large table is given here, in Table XII, in



Improvement in the Group as a Whole



?,2>



which the form of the distribution, the central tendency and the
variabiHty of the group are clearly shown.

TABLE XII

Gross Gain in Accuracy in Division, Expressed in Per Cents of
Answers that Were Correct, from Fifty Minutes of Practice



Per cent gained


-45
to
-41


-40
to
-36


-35
to
-31


-30
to
-26


-25

to
-21


-20
to
-16


-15
to
-11


-10
to
-6


-5

to
-1




to




4


Individuals


1


1





2


3


3


11


37


113


211


Per cents


.2


.2





.3


.5


.5


1.8


6.1


18.6


34 8






TABLE 'Xll— Continued


Per cent gained


5

to
9


10
to
14


15

to
19


20
to
24


25

to
29


30
to
34


35

to
39


40
to
44


45

to
49


50
to




54


Individuals


102


52


26


13


11


8


2


5


4


1


Per cents


16.8


8.6


4.3


2.1


1.8


1.2


.3


.8


.7


2







Median
25 Percentile
75 Percentile
P.E.



2.6 per cent

1.4

8.

4.5



Table XII shows a distribution of the gross gains in accuracy
in division expressed in per cents of answers that were correct
resulting from the 6o minutes of practice. The gains are given
in groups of 5. The figures show a distribution conforming
closely to the normal distribution curve. The range is from 45
per cent lost to 54 per cent gained. The median gain is 2.6 per
cent which shows that there were as many children who gained
2.6 per cent or more in accuracy as there were who gained 2.6
per cent or less. The figures show that 435 children made the
associations with equal or greater accuracy at the end of the
practice than at the beginning, while 171 made them with less


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Online LibraryThomas Joseph KirbyPractice in the case of school children → online text (page 3 of 9)