The success or failure of the teacher in applying the principles which
have been discussed in the preceding chapters is measured by the
achievements of the children. Of course, it is also possible that the
validity of the principle which we have sought to establish may be
called in question by the same sort of measurement. We cannot be sure
that our methods of work are sound, or that we are making the best use
of the time during which we work with children, except as we discover
the results of our instruction. Teaching is after all the adaptation of
our methods to the normal development of boys and girls, and their
education can be measured only in terms of the changes which we are able
to bring about in knowledge, skill, appreciation, reasoning, and the
like.
Any attempt to measure the achievements of children should result in a
discovery of the progress which is being made from week to week, or
month to month, or year to year. It would often be found quite
advantageous to note the deficiencies as well as the achievements at one
period as compared with the work done two or three months later. It will
always be profitable to get as clearly in mind as is possible the
variation among members of the same class, and for those who are
interested in the supervision of schools, the variation from class to
class, from school to school, or from school system to school system.
For the teacher a study of the variability in achievement among the
members of his own class ought to result in special attention to those
who need special help, especially a kind of teaching which will remove
particular difficulties. There should also be offered unusual
opportunity and more than the ordinary demand be made of those who show
themselves to be more capable than the ordinary pupils.
The type of measurement which we wish to discuss is something more than
the ordinary examination. The difficulties with examinations, as we have
commonly organized them, has* been their unreliability, either from the
standpoint of discovering to us the deficiencies of children, or their
achievements. Of ten problems in arithmetic or of twenty words in
spelling given in the ordinary examination, there are very great
differences in difficulty. We do not have an adequate measure of the
achievements of children when we assign to each of the problems or words
a value of ten or of five per cent and proceed to determine the mark to
be given on the examination paper. If we are wise in setting our
examinations, we usually give one problem or one word which we expect
practically everybody to be able to get right. On the other hand, if we
really measure the achievements of children, we must give some problems
or some words that are too hard for any one to get right. Otherwise, we
do not know the limit or extent of ability possessed by the abler
pupils. It is safe to say that in many examinations one question may
actually be four or five times as hard as some other to which an equal
value is assigned.
Another difficulty that we have to meet in the ordinary examination is
the variability among teachers in marking papers. We do not commonly
assign the same values to the same result. Indeed, if a set of papers is
given to a group of capable teachers and marked as conscientiously as
may be by each of them, it is not uncommon to find a variation among the
marks assigned to the same paper which may be as great as twenty-five
per cent of the highest mark given. Even more interesting is the fact
that upon re-marking these same papers individual teachers will vary
from their own first mark by almost as great an amount.
Still another difficulty with the ordinary examination is the tendency
among teachers to derive their standards of achievement from the group
itself, rather than from any objective standard by which all are
measured. It is possible, for example, for children in English
composition to write very poorly for their grade and still to find the
teacher giving relatively high marks to those who happen to belong to
the upper group in the class. As a result of the establishment of such a
standard, the teacher may not be conscious of the fact that children
should be spurred to greater effort, and that possibly he himself should
seek to improve his methods of work.
Out of the situation described above, which includes on the one hand the
necessity for measurement as a means of testing the success of our
theories and of our practice, and on the other hand of having objective
standards, has grown the movement for measurement by means of standard
tests and scales. A standard test which has been given to some thousands
of children classified by grades or by ages, if given to another group
of children of the same grade or age group will enable the teacher to
compare the achievement of his children with that which is found
elsewhere. For example, the Courtis tests in arithmetic, which consist
of series of problems of equal difficulty in addition, subtraction,
multiplication, and division may be used to discover how far facility in
these fields has been accomplished by children of any particular group
as compared with the achievements of children in other school systems
throughout the country. In these tests each of the problems is of equal
difficulty. The measure is made by discovering how many of these
separate problems can be solved in a given number of minutes.[20]
A scale for measuring the achievements of children in the fundamental
operations of addition, subtraction, multiplication, and division has
been derived by Dr. Clifford Woody,[21] which differs from the Courtis
tests in that it affords opportunity to discover what children can
achieve from the simplest problem in each of these fields to a problem
which is in each case approximately twice as difficult as the problems
appearing on the Courtis tests. The great value of this type of test is
in discovering to teachers and to pupils, as well, their particular
difficulties. A pupil must be able to do fairly acceptable work in
addition before he can solve one problem on the Courtis tests.
Considerable facility can be measured on the Woody tests before an
ability sufficient to be registered on the Courtis tests has been
acquired. In his monograph on the derivation of these tests Mr. Woody
gives results which will enable the teacher to compare his class with
children already tested in other school systems. In the case of all of
these standard tests, school surveys and superintendents' reports are
available which will make it possible to institute comparisons among
different classes and different school systems. One form of the Woody
tests is as follows:
* * * * *
SERIES A
ADDITION SCALE
BY CLIFFORD WOODY
Name......................
When is your next birthday?...... How old will you be?.....
Are you a boy or girl?....... In what grade are you?......
(1) (3) (5) (7) (9) (10) (11) (12) (13) (15) (16)
2 17 72 3+1= 20 21 32 43 23 100 9
3 2 26 10 33 59 1 25 33 24
-- -- -- 2 35 17 2 16 45 12
30 -- -- 13 -- 201 15
(2) (4) (6) (8) 25 -- 46 19
2 53 60 2+5+1= -- --- --
4 45 37 (14)
3 -- -- 25+42=
--
(17) (19) (21) (22) (23) (26) (29)
199 $ .75 $8.00 547 1/3+1/3= 121/2 4 3/4
194 1.25 5.75 197 621/2 2 1/4
295 .49 2.33 685 (24) 121/2 5 1/4
156 ----- 4.16 678 4.0125 371/2 -----
--- .94 456 1.5907 ---
6.32 393 4.10 (30)
(18) (20) ----- 525 8.673 (27) 2 1/2
2563 $12.50 240 ------ 1/8+1/4+1/2= 6 3/8
1387 16.75 152 3 3/4
4954 15.75 --- -----
2065 ------ (25) (28)
---- 3/8+5/8+7/8+1/8= 3/4+1/4=
(31) (33) (34) (35) (36) (37)
113.46 .49 1/6+3/8= 2ft. 6in. 2yr. 5mo. 16 1/3
49.6097 .28 3ft. 5in. 3yr. 6mo. 12 1/8
19.9 .63 4ft. 9in. 4yr. 9mo. 21 1/2
9.87 .95 --------- 5yr. 2mo. 32 3/4
.0086 1.69 6yr. 7mo. ------
18.253 .22 ---------
6.04 .33
-------- .36 (38)
1.01 25.091+100.4+25+98.28+19.3614=
(32) .56
3/4+1/2+1/4= .88
.75
.56
1.10
.18
.56
----
* * * * *
SERIES A
SUBTRACTION SCALE
BY CLIFFORD WOODY
Name......................
When is your next birthday?......How old will you be?.....
Are you a boy or girl?.......In what grade are you?.......
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
8 6 2 9 4 11 13 59 78 7-4= 76
5 0 1 3 4 7 8 12 37 60
-- -- -- -- -- -- -- -- -- --
(12) (13) (14) (15) (16) (17) (18) (19) (20)
27 16 50 21 270 393 1000 567482 2 3/4-1=
3 9 25 9 190 178 537 106493
-- -- -- -- --- --- ---- ------
(21) (22) (23) (24) (25) (26)
10.00 3 1/2-1/2= 80836465 8 7/8 27 4yd. 1ft. 6in.
3.49 49178036 5 3/4 12 5/8 2yd. 2ft. 3in.
----- -------- ----- ------ --------------
(27) (28) (29) (30)
5yd. 1ft. 4in. 10-6.25 75 3/4 9.8063-9.019=
2yd. 2ft. 8in. 52 1/4
-------------- ------
(31) (32) (33) (34) (35)
7.3-3.00081= 1912 6mo. 8da. 5/12-2/10= 6 1/8 3 7/8-1 5/8=
1910 7mo. 15da. 2 7/8
--------------- -----
* * * * *
SERIES A
DIVISION SCALE
BY CLIFFORD WOODY
Name...............................
When is your next birthday?....... How old will you be?......
Are you a boy or girl?.......... In what grade are you?......
(1) (2) (3) (4) (5) (6)
__ ___ ___ __ ___ ___
3)6 9)27 4)28 1)5 9)36 3)39
(7) (8) (9) (10) (11) (12)
4 ÷ 2 = __ __ 6 × __ = 30 ___ 2 ÷ 2 =
9)0 1)1 2)13
(13) (14) (15) (16) (17)
______________ _____ 1/4 of 128= _____ 50 ÷ 7 =
4)24 lbs. 8 oz. 8)5856 68)2108
(18) (19) (20) (21) (22)
______ 248 ÷ 7 = _____ _____ ______
13)65065 2.1)25.2 25)9750 2)13.50
(23) (24) (25) (26)
____ ________ _______ _____
23)469 75)2250300 2400)504000 12)2.76
(27) (28) (29) (30)
7/8 of 624 = ______ 3 1/2 ÷ 9 = 3/4 ÷ 5 =
.003).0936
(31) (32) (33)
5/4 ÷ 3/5 = 9 5/8 ÷ 3 3/4 = _____
52)3756
(34) (35) (36)
62.50 ÷ 1 1/4 = ______ ______________
531)37722 9)69 lbs. 9 oz.
* * * * *
SERIES A
MULTIPLICATION SCALE
BY CLIFFORD WOODY
Name......................
When is your next birthday?...... How old will you be?.....
Are you a boy or girl?....... In what grade are you?.......
(1) (2) (3) (4) (5) (6) (7)
3 × 7 = 5 × 1 = 2 × 3 = 4 × 8 = 23 310 7 × 9 =
3 4
-- ---
(8) (9) (10) (11) (12) (13) (14) (15)
50 254 623 1036 5096 8754 165 235
3 6 7 8 6 8 40 23
-- --- --- ---- ---- ---- --- ---
(16) (17) (18) (19) (20) (21) (22)
7898 145 24 9.6 287 24 8 × 53/4
9 206 234 4 .05 21/2
---- --- --- --- --- --
(23) (24) (25) (26) (27) (28) (29)
11/4 × 8 = 16 7/8 × 3/4 = 9742 6.25 .0123 1/8 × 2 =
2 5/8 59 3.2 9.8
------ ---- ---- -----
(30) (31) (32) (33) (34)
2.49 12 15 6 dollars 49 cents 2-1/2 × 3-1/2 = 1/2 × 1/2 =
36 -- × -- 8
---- 25 32 ------------------
(35) (36) (37) (38) (39)
9873/4 3ft. 5in. 21/4 × 41/2 × 11/2 = .0963 1/8 8ft. 91/2in.
25 5 .084 9
---- --------- --------- ----------
* * * * *
A series of problems in reasoning in arithmetic which were given in
twenty-six school systems by Dr. C.W. Stone furnish a valuable test in
this field, as well as an opportunity for comparison with other schools
in which these problems have been used.[22] A list of problems follows.
Solve as many of the following problems as you have time for;
work them in order as numbered:
1. If you buy 2 tablets at 7 cents each and a book for 65
cents, how much change should you receive from a two-dollar
bill?
2. John sold 4 Saturday Evening Posts at 5 cents each. He kept
1/2 the money and with the other 1/2 he bought Sunday papers
at 2 cents each. How many did he buy?
3. If James had 4 times as much money as George, he would have
$16. How much money has George?
4. How many pencils can you buy for 50 cents at the rate of 2
for 5 cents?
5. The uniforms for a baseball nine cost $2.50 each. The shoes
cost $2 a pair. What was the total cost of uniforms and shoes
for the nine?
6. In the schools of a certain city there are 2200 pupils; 1/2
are in the primary grades, 1/4 in the grammar grades, 1/8 in
the High School, and the rest in the night school. How many
pupils are there in the night school?