THE MEASUREMENT OF ARITHMETICAL ABILITIES
One of the best ways to clear up notions of what the functions are which
schools should develop and improve is to get measures of them. If any
given knowledge or skill or power or ideal exists, it exists in some
amount. A series of amounts of it, varying from less to more, defines
the ability itself in a way that no general verbal description can do.
Thus, a series of weights, 1 lb., 2 lb., 3 lb., 4 lb., etc., helps to
tell us what we mean by weight. By finding a series of words like
_only_, _smoke_, _another_, _pretty_, _answer_, _tailor_, _circus_,
_telephone_, _saucy_, and _beginning_, which are spelled correctly by
known and decreasing percentages of children of the same age, or of the
same school grade, we know better what we mean by 'spelling-difficulty.'
Indeed, until we can measure the efficiency and improvement of a
function, we are likely to be vague and loose in our ideas of what the
function is.
A SAMPLE MEASUREMENT OF AN ARITHMETICAL ABILITY: THE ABILITY TO ADD
INTEGERS
Consider first, as a sample, the measurement of ability to add integers.
The following were the examples used in the measurements made by Stone
['08]:
596 4695
428 872
2375 94 7948
4052 75 6786
6354 304 567
260 645 858
5041 984 9447
1543 897 7499
---- --- ----
The scoring was as follows: Credit of 1 for each column added correctly.
Stone combined measures of other abilities with this in a total score
for amount done correctly in 12 minutes. Stone also scored the
correctness of the additions in certain work in multiplication.
Courtis uses a sheet of twenty-four tasks or 'examples,' each consisting
of the addition of nine three-place numbers as shown below. Eight
minutes is allowed. He scores the amount done by the number of examples,
and also scores the number of examples done correctly, but does not
suggest any combination of these two into a general-efficiency score.
927
379
756
837
924
110
854
965
344
---
The author long ago proposed that pupils be measured also with series
like _a_ to _g_ shown below, in which the difficulty increases step by
step.
_a._ 3 2 2 3 2 2 1 2
2 3 1 2 4 5 5 1
4 2 3 3 3 2 2 2
- - - - - - - -
_b._ 21 32 12 24 34 34 22 12
23 12 52 31 33 12 23 13
24 25 15 14 32 23 43 61
-- -- -- -- -- -- -- --
_c._ 22 3 4 35 32 83 22 3
3 31 3 2 33 11 3 21
38 45 52 52 2 4 33 64
-- -- -- -- -- -- -- --
_d._ 30 20 10 22 10 20 52 12
20 50 40 43 30 4 6 22
40 17 24 13 40 23 30 44
-- -- -- -- -- -- -- --
_e._ 4 5 20 12 12 20 10
20 30 3 40 4 11 20 20
10 30 20 4 1 23 7 2
20 2 40 23 40 11 10 30
20 20 10 11 20 22 30 25
-- -- -- -- -- -- -- --
_f._ 19 9 9
14 2 19 24 9 4 13
9 14 13 12 13 13 9 14
17 23 13 15 15 34 12 25
26 29 18 19 25 28 18 39
-- -- -- -- -- -- -- --
_g._ 13
13 9 14 12 9
9 13 12 9 14 24
23 19 19 29 9 9 13 21
28 26 26 14 8 8 29 23
29 16 15 19 17 19 19 22
-- -- -- -- -- -- -- --
Woody ['16] has constructed his well-known tests on this principle,
though he uses only one example at each step of difficulty instead of
eight or ten as suggested above. His test, so far as addition of
integers goes, is:--
SERIES A. ADDITION SCALE (in part)
By Clifford Woody
(1) (2) (3) (4) (5) (6) (7) (8) (9)
2 2 17 53 72 60 3 + 1 = 2 + 5 + 1 = 20
3 4 2 45 26 37 10
-- 3 -- -- -- -- 2
-- 30
25
--
(10) (11) (12) (13) (14) (15) (16) (17) (18)
21 32 43 23 25 + 42 = 100 9 199 2563
33 59 1 25 33 24 194 1387
35 17 2 16 45 12 295 4954
-- -- 13 -- 201 15 156 2065
-- 46 19 --- ----
--- --
(19) (20) (21) (22)
$ .75 $12.50 $8.00 547
1.25 16.75 5.75 197
.49 15.75 2.33 685
----- ------ 4.16 678
.94 456
6.32 393
----- 525
240
152
---
In his original report, Woody gives no scheme for scoring an individual,
wisely assuming that, with so few samples at each degree of difficulty,
a pupil's score would be too unreliable for individual diagnosis. The
test is reliable for a class; and for a class Woody used the degree of
difficulty such that a stated fraction of the class can do the work
correctly, if twenty minutes is allowed for the thirty-eight examples of
the entire test.
The measurement of even so simple a matter as the efficiency of a
pupil's responses to these tests in adding integers is really rather
complex. There is first of all the problem of combining speed and
accuracy into some single estimate. Stone gives no credit for a column
unless it is correctly added. Courtis evades the difficulty by reporting
both number done and number correct. The author's scheme, which gives
specified weights to speed and accuracy at each step of the series,
involves a rather intricate computation.
This difficulty of equating speed and accuracy in adding means precisely
that we have inadequate notions of what the ability is that the
elementary school should improve. Until, for example, we have decided
whether, for a given group of pupils, fifteen Courtis attempts with ten
right, is or is not a better achievement than ten Courtis attempts with
nine right, we have not decided just what the business of the teacher of
addition is, in the case of that group of pupils.
There is also the difficulty of comparing results when short and long
columns are used. Correctness with a short column, say of five figures,
testifies to knowledge of the process and to the power to do four
successive single additions without error. Correctness with a long
column, say of ten digits, testifies to knowledge of the process and to
the power to do nine successive single additions without error. Now if a
pupil's precision was such that on the average he made one mistake in
eight single additions, he would get about half of his five-digit
columns right and almost none of his ten-digit columns right. (He would
do this, that is, if he added in the customary way. If he were taught to
check results by repeated addition, by adding in half-columns and the
like, his percentages of accurate answers might be greatly increased in
both cases and be made approximately equal.) Length of column in a test
of addition under ordinary conditions thus automatically overweights
precision in the single additions as compared with knowledge of the
process, and ability at carrying.
Further, in the case of a column of whatever size, the result as
ordinarily scored does not distinguish between one, two, three, or more
(up to the limit) errors in the single additions. Yet, obviously, a
pupil who, adding with ten-digit columns, has half of his answer-figures
wrong, probably often makes two or more errors within a column, whereas
a pupil who has only one column-answer in ten wrong, probably almost
never makes more than one error within a column. A short-column test is
then advisable as a means of interpreting the results of a long-column
test.
Finally, the choice of a short-column or of a long-column test is
indicative of the measurer's notion of the kind of efficiency the world
properly demands of the school. Twenty years ago the author would have
been readier to accept a long-column test than he now is. In the world
at large, long-column addition is being more and more done by machine,
though it persists still in great frequency in the bookkeeping of weekly
and monthly accounts in local groceries, butcher shops, and the like.
The search for a measure of ability to add thus puts the problem of
speed _versus_ precision, and of short-column _versus_ long-column
additions clearly before us. The latter problem has hardly been
realized at all by the ordinary definitions of ability to add.
It may be said further that the measurement of ability to add gives the
scientific student a shock by the lack of precision found everywhere in
schools. Of what value is it to a graduate of the elementary school to
be able to add with examples like those of the Courtis test, getting
only eight out of ten right? Nobody would pay a computer for that
ability. The pupil could not keep his own accounts with it. The supposed
disciplinary value of habits of precision runs the risk of turning
negative in such a case. It appears, at least to the author, imperative
that checking should be taught and required until a pupil can add single
columns of ten digits with not over one wrong answer in twenty columns.
Speed is useful, especially indirectly as an indication of control of
the separate higher-decade additions, but the social demand for addition
below a certain standard of precision is _nil_, and its disciplinary
value is _nil_ or negative. This will be made a matter of further study
later.
MEASUREMENTS OF ABILITIES IN COMPUTATION
Measurements of these abilities may be of two sorts--(1) of the speed
and accuracy shown in doing one same sort of task, as illustrated by the
Courtis test for addition shown on page 28; and (2) of how hard a task
can be done perfectly (or with some specified precision) within a
certain assigned time or less, as illustrated by the author's rough test
for addition shown on pages 28 and 29, and by the Woody tests, when
extended to include alternative forms.
The Courtis tests, originated as an improvement on the Stone tests and
since elaborated by the persistent devotion of their author, are a
standard instrument of the first sort for measuring the so-called
'fundamental' arithmetical abilities with integers. They are shown on
this and the following page.
Tests of the second sort are the Woody tests, which include operations
with integers, common and decimal fractions, and denominate numbers, the
Ballou test for common fractions ['16], and the "Ladder" exercises of
the Thorndike arithmetics. Some of these are shown on pages 36 to 41.
Courtis Test
Arithmetic. Test No. 1. Addition
Series B
You will be given eight minutes to find the answers to as many
of these addition examples as possible. Write the answers on this
paper directly underneath the examples. You are not expected
to be able to do them all. You will be marked for both speed and
accuracy, but it is more important to have your answers right than
to try a great many examples.
927 297 136 486 384 176 277 837
379 925 340 765 477 783 445 882
756 473 988 524 881 697 682 959
837 983 386 140 266 200 594 603
924 315 353 812 679 366 481 118
110 661 904 466 241 851 778 781
854 794 547 355 796 535 849 756
965 177 192 834 850 323 157 222
344 124 439 567 733 229 953 525
--- --- --- --- --- --- --- ---
and sixteen more addition examples of nine three-place numbers.
Courtis Test
Arithmetic. Test No. 2. Subtraction
Series B
You will be given four minutes to find the answers to as many
of these subtraction examples as possible. Write the answers
on this paper directly underneath the examples. You are not
expected to be able to do them all. You will be marked for both
speed and accuracy, but it is more important to have your answers
right than to try a great many examples.
107795491 75088824 91500053 87939983
77197029 57406394 19901563 72207316
--------- -------- -------- --------
and twenty more tasks of the same sort.
Courtis Test
Arithmetic. Test No. 3. Multiplication
Series B
You will be given six minutes to work as many of these multiplication
examples as possible. You are not expected to be able to do them all.
Do your work directly on this paper; use no other. You will be marked
for both speed and accuracy, but it is more important to get correct
answers than to try a large number of examples.
8246 7843 4837 3478 6482
29 702 83 15 46
---- ---- ---- ---- ----
and twenty more multiplication examples of the same sort.
Courtis Test
Arithmetic. Test No. 4. Division
Series B
You will be given eight minutes to work as many of these division
examples as possible. You are not expected to be able to do them all.
Do your work directly on this paper; use no other. You will be marked
for both speed and accuracy, but it is more important to get correct
answers than to try a large number of examples.
_____ ______ _____ ______
25)6775 94)85352 37)9990 86)80066
and twenty more division examples of the same sort.
SERIES B. MULTIPLICATION SCALE
By Clifford Woody
(1) (3) (4) (5)
3 x 7 = 2 x 3 = 4 x 8 = 23
3
--
(8) (9) (11) (12)
50 254 1036 5096
3 6 8 6
-- --- ---- ----
(13) (16) (18) (20)
8754 7898 24 287
8 9 234 .05
---- ---- --- ---
(24) (26) (27) (29)
16 9742 6.25 1/8 x 2 =
2-5/8 59 3.2
--- ---- ----
(33) (35) (37) (38)
2-1/2 x 3-1/2 = 987-3/4 2-1/4 x 4-1/2 x 1-1/2 = .0963-1/8
25 .084
---- -----
SERIES B. DIVISION SCALE
By Clifford Woody
(1) (2) (7) (8)
__ ___ ___
3)6 9)27 4 / 2 = 9)0
(11) (14) (15) (17)
___ _____
2)13 8)5856 1/4 of 128 = 50 / 7 =
(19) (23) (27) (28)
____ ______
248 / 7 = 23)469 7/8 of 624 = .003).0936
(30) (34) (36)
______________
3/4 / 5 = 62.50 / 1-1/4 = 9)69 lbs. 9 oz.
Ballou Test
Addition of Fractions
_Test 1_ _Test 2_
(1) 1/4 (2) 3/14 (1) 1/3 (2) 2/7
1/4 1/14 1/6 3/14
--- ---- --- ----
_Test 3_ _Test 4_
(1) 3/5 (2) 5/6 (1) 1/7 (2) 7/9
11/15 1/2 9/10 1/4
----- --- ---- ---
_Test 5_ _Test 6_
(1) 1/10 (2) 4/9 (1) 1/6 (2) 5/6
1/6 5/12 9/10 3/8
---- ---- ---- ---
An Addition Ladder [Thorndike, '17, III, 5]
Begin at the bottom of the ladder. See if you can climb to the top
without making a mistake. Be sure to copy the numbers correctly.
#Step 6.#
_a._ Add 1-1/3 yd., 7/8 yd., 1-1/4 yd., 3/4 yd., 7/8 yd.,
and 1-1/2 yd.
_b._ Add 62-1/2c, 66-2/3c, 56-1/4c, 60c, and 62-1/2c.
_c._ Add 1-5/16, 1-9/32, 1-3/8, 1-11/32, and 1-7/16.
_d._ Add 1-1/3 yd., 1-1/4 yd., 1-1/2 yd., 2 yd., 3/4 yd.,
and 2/3 yd.
#Step 5.#
_a._ Add 4 ft. 6-1/2 in., 53-1/4 in., 5 ft. 1/2 in., 56-3/4 in.,
and 5 ft.
_b._ Add 7 lb., 6 lb. 11 oz., 7-1/2 lb., 6 lb. 4-1/2 oz.,
and 8-1/2 lb.
_c._ Add 1 hr. 6 min. 20 sec., 58 min. 15 sec., 1 hr. 4 min.,
and 55 min.
_d._ Add 7 dollars, 13 half dollars, 21 quarters, 17 dimes,
and 19 nickels.
#Step 4.#
_a._ Add .05-1/2, .06, .04-3/4, .02-3/4, and .05-1/4.
_b._ Add .33-1/3, .12-1/2, .18, .16-2/3, .08-1/3 and .15.
_c._ Add .08-1/3, .06-1/4, .21, .03-3/4, and .16-2/3.
_d._ Add .62, .64-1/2, .66-2/3, .10-1/4, and .68.
#Step 3.#
_a._ Add 7-1/4, 6-1/2, 8-3/8, 5-3/4, 9-5/8 and 3-7/8.
_b._ Add 4-5/8, 12, 7-1/2, 8-3/4, 6 and 5-1/4.
_c._ Add 9-3/4, 5-7/8, 4-1/8, 6-1/2, 7, 3-5/8.
_d._ Add 12, 8-1/2, 7-1/3, 5, 6-2/3, and 9-1/2.
#Step 2.#
_a._ Add 12.04, .96, 4.7, 9.625, 3.25, and 20.
_b._ Add .58, 6.03, .079, 4.206, 2.75, and 10.4.
_c._ Add 52, 29.8, 41.07, 1.913, 2.6, and 110.
_d._ Add 29.7, 315, 26.75, 19.004, 8.793, and 20.05.
#Step 1.#
_a._ Add 10-3/5, 11-1/5, 10-4/5, 11, 11-2/5, 10-3/5, and 11.
_b._ Add 7-3/8, 6-5/8, 8, 9-1/8, 7-7/8, 5-3/8, and 8-1/8.
_c._ Add 21-1/2, 18-3/4, 31-1/2, 19-1/4, 17-1/4, 22, and 16-1/2.
_d._ Add 14-5/12, 12-7/12, 9-11/12, 6-1/12, and 5.
A Subtraction Ladder [Thorndike, '17, III, 11]
#Step 9.#
_a._ 2.16 mi. - 1-3/4 mi.
_b._ 5.72 ft. - 5 ft. 3 in.
_c._ 2 min. 10-1/2 sec. - 93.4 sec.
_d._ 30.28 A. - 10-1/5 A.
_e._ 10 gal. 2-1/2 qt. - 4.623 gal.
#Step 8.#
_a_ _b_ _c_ _d_ _e_
25-7/12 10-1/4 9-5/16 5-7/16 4-2/3
12-3/4 7-1/3 6-3/8 2-3/4 1-3/4
------- ------ ------ ------ -----
#Step 7.#
_a_ _b_ _c_ _d_ _e_
28-3/4 40-1/2 10-1/4 24-1/3 37-1/2
16-1/8 14-3/8 6-1/2 11-1/2 14-3/4
------ ------ ------ ------ ------
#Step 6.#
_a_ _b_ _c_ _d_ _e_
10-1/3 7-1/4 15-1/8 12-1/5 4-1/16
4-2/3 2-3/4 6-3/8 11-4/5 2-7/16
------ ----- ------ ------ ------
#Step 5.#
_a_ _b_ _c_ _d_ _e_
58-4/5 66-2/3 28-7/8 62-1/2 9-7/12
52-1/5 33-1/3 7-5/8 37-1/2 4-5/12
------ ------ ------ ------ ------
#Step 4.#
_a._ 4 hr. - 2 hr. 17 min.
_b._ 4 lb. 7 oz. - 2 lb. 11 oz.
_c._ 1 lb. 5 oz. - 13 oz.
_d._ 7 ft. - 2 ft. 8 in.
_e._ 1 bu. - 1 pk.
#Step 3.#
_a_ _b_ _c_ _d_ _e_
92 mi. 6735 mi. $3 - 89c 28.4 mi. $508.40
84.15 mi. 6689 mi. 18.04 mi. 208.62
--------- -------- -------- --------- -------
#Step 2.#
_a_ _b_ _c_ _d_ _e_
$25.00 $100.00 $750.00 6124 sq. mi. 7846 sq. mi.
9.36 71.28 736.50 2494 sq. mi. 2789 sq. mi.
------ ------- ------- ------------ ------------
#Step 1.#
_a_ _b_ _c_ _d_ _e_
$18.64 $25.39 $56.70 819.4 mi. 67.55 mi.
7.40 13.37 45.60 209.2 mi. 36.14 mi.
------ ------ ------ --------- ---------
An Average Ladder [Thorndike, '17, III, 132]
Find the average of the quantities on each line. Begin with #Step 1#.
Climb to the top without making a mistake. Be sure to copy the numbers
correctly. Extend the division to two decimal places if necessary.
#Step 6.#
_a_. 2-2/3, 1-7/8, 2-3/4, 4-1/4, 3-5/8, 3-1/2
_b_. 62-1/2c, 66-2/3c, 40c, 83-1/3c, $1.75, $2.25
_c_. 3-11/16, 3-9/32, 3-3/8, 3-17/32, 3-7/16
_d_. .17, 19, .16-2/3, .15-1/2, .23-1/4, .18
#Step 5.#
_a_. 5 ft. 3-1/2 in., 61-1/4 in., 58-3/4 in., 4 ft. 11 in.
_b_. 6 lb. 9 oz., 6 lb. 11 oz., 7-1/4 lb., 7-3/8 lb.
_c_. 1 hr. 4 min. 40 sec., 58 min. 35 sec., 1-1/4 hr.
_d_. 2.8 miles, 3-1/2 miles, 2.72 miles
#Step 4.#
_a._ .03-1/2, .06, .04-3/4, .05-1/2, .05-1/4
_b._ .043, .045, .049, .047, .046, .045
_c._ 2.20, .87-1/2, 1.18, .93-3/4, 1.2925, .80
_d._ .14-1/2, .12-1/2, .33-1/3, .16-2/3, .15, .17
#Step 3.#
_a._ 5-1/4, 4-1/2, 8-3/8, 7-3/4, 6-5/8, 9-3/8
_b._ 9-5/8, 12, 8-1/2, 8-3/4, 6, 5-1/4, 9
_c._ 9-3/8, 5-3/4, 4-1/8, 7-1/2, 6
_d._ 11, 9-1/2, 10-1/3, 13, 16-2/3, 9-1/2
#Step 2.#
_a._ 13.05, .97, 4.8, 10.625, 3.37
_b._ 1.48, 7.02, .93, 5.307, 4.1, 7, 10.4
_c._ 68, 71.4, 59.8, 112, 96.1, 79.8
_d._ 2.079, 3.908, 4.165, 2.74
#Step 1.#
_a._ 4, 9-1/2, 6, 5, 7-1/2, 8, 10, 9
_b._ 6, 5, 3.9, 7.1, 8
_c._ 1086, 1141, 1059, 1302, 1284
_d._ $100.82, $206.49, $317.25, $244.73
As such tests are widened to cover the whole task of the elementary
school in respect to arithmetic, and accepted by competent authorities
as adequate measures of achievement in computing, they will give, as has
been said, a working definition of the task. The reader will observe,
for example, that work such as the following, though still found in many
textbooks and classrooms, does not, in general, appear in the modern
tests and scales.
Reduce the following improper fractions to mixed numbers:--
19/13 43/21 176/25 198/14
Reduce to integral or mixed numbers:--
61381/37 2134/67 413/413 697/225
Simplify:--
3/4 of 8/9 of 3/5 of 15/22
Reduce to lowest terms:--
357/527 264/312 492/779 418/874 854/1769 30/735 44/242
77/847 18/243 96/224
Find differences:--
6-2/7 8-5/11 8-4/13 5-1/4 7-1/8
3-1/14 5-1/7 3-7/13 2-11/14 2-1/7
------ ------ ------ ------- ------
Square:--
2/3 4/5 5/7 6/9 10/11 12/13 2/7 15/16 19/20 17/18
25/30 41/53
Multiply:--
2/11 x 33 32 x 3/14 39 x 2/13 60 x 11/28 77 x 4/11
63 x 2/27 54 x 8/45 65 x 3/13 344-16/21 432-2/7
MEASUREMENTS OF ABILITY IN APPLIED ARITHMETIC: THE SOLUTION OF PROBLEMS
Stone ['08] measured achievement with the following problems, fifteen
minutes being the time allowed.
"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?
7. If 3-1/2 tons of coal cost $21, what will 5-1/2 tons cost?
8. A news dealer bought some magazines for $1. He sold them for
$1.20, gaining 5 cents on each magazine. How many magazines were
there?
9. A girl spent 1/8 of her money for car fare, and three times as
much for clothes. Half of what she had left was 80 cents. How much
money did she have at first?
10. Two girls receive $2.10 for making buttonholes. One makes 42,
the other 28. How shall they divide the money?
11. Mr. Brown paid one third of the cost of a building; Mr. Johnson
paid 1/2 the cost. Mr. Johnson received $500 more annual rent than
Mr. Brown. How much did each receive?
12. A freight train left Albany for New York at 6 o'clock. An
express left on the same track at 8 o'clock. It went at the rate of
40 miles an hour. At what time of day will it overtake the freight
train if the freight train stops after it has gone 56 miles?"
The criteria he had in mind in selecting the problems were as follows:--
"The main purpose of the reasoning test is the determination of the
ability of VI A children to reason in arithmetic. To this end, the
problems, as selected and arranged, are meant to embody the following
conditions:--
1. Situations equally concrete to all VI A children.
2. Graduated difficulties.
_a._ As to arithmetical thinking.
_b._ As to familiarity with the situation presented.
3. The omission of
_a._ Large numbers.
_b._ Particular memory requirements.
_c._ Catch problems.
_d._ All subject matter except whole numbers, fractions, and
United States money.
The test is purposely so long that only very rarely did any pupil fully
complete it in the fifteen minute limit."
Credits were given of 1, for each of the first five problems, 1.4, 1.2,
and 1.6 respectively for problems 6, 7, and 8, and of 2 for each of the
others.
Courtis sought to improve the Stone test of problem-solving, replacing
it by the two tests reproduced below.
ARITHMETIC--Test No. 6. Speed Test--Reasoning
#Do not work# the following examples. Read each example through, make
up your mind what operation you would use if you were going to work it,
then write the name of the operation selected in the blank space after
the example. Use the following abbreviations:--"Add." for addition,
"Sub." for subtraction, "Mul." for multiplication, and "Div." for
division.
+-----------+----+
| OPERATION | |
|-----------+----|
1. A girl brought a collection of 37 colored postal | | |
cards to school one day, and gave away 19 cards to | | |
her friends. How many cards did she have left to | | |
take home? | | |
|-----------+----|
2. Five boys played marbles. When the game was | | |
over, each boy had the same number of marbles. If | | |
there were 45 marbles altogether, how many did each | | |
boy have? | | |
|-----------+----|
3. A girl, watching from a window, saw 27 | | |
automobiles pass the school the first hour, and | | |
33 the second. How many autos passed by the | | |
school in the two hours? | | |
|-----------+----|
4. In a certain school there were eight rooms and | | |
each room had seats for 50 children. When all the | | |
places were taken, how many children were there in | | |
the school? | | |
|-----------+----|
5. A club of boys sent their treasurer to buy | | |
baseballs. They gave him $3.15 to spend. How many | | |
balls did they expect him to buy, if the balls cost | | |
45c. apiece? | | |
|-----------+----|
6. A teacher weighed all the girls in a certain | | |
grade. If one girl weighed 79 pounds and another | | |
110 pounds, how many pounds heavier was one girl | | |
than the other? | | |
|-----------+----|
7. A girl wanted to buy a 5-pound box of candy to | | |
give as a present to a friend. She decided to get | | |
the kind worth 35c. a pound. What did she pay for | | |
the present? | | |
|-----------+----|
8. One day in vacation a boy went on a fishing trip | | |
and caught 12 fish in the morning, and 7 in the | | |
afternoon. How many fish did he catch altogether? | | |
|-----------+----|
9. A boy lived 15 blocks east of a school; his chum | | |
lived on the same street, but 11 blocks west of the | | |
school. How many blocks apart were the two boys' | | |
houses? | | |
|-----------+----|
10. A girl was 5 times as strong as her small | | |
sister. If the little girl could lift a weight of | | |
20 pounds, how large a weight could the older girl | | |
lift? | | |
|-----------+----|
11. The children of a school gave a sleigh-ride | | |
party. There were 270 children to go on the ride | | |
and each sleigh held 30 children. How many sleighs | | |
were needed? | | |
|-----------+----|
12. In September there were 43 children in the | | |
eighth grade of a certain school; by June there | | |
were 59. How many children entered the grade | | |
during the year? | | |
|-----------+----|
13. A girl who lived 17 blocks away walked to | | |
school and back twice a day. What was the total | | |
number of blocks the girl walked each day in | | |
going to and from school? | | |
|-----------+----|
14. A boy who made 67c. a day carrying papers, was | | |
hired to run on a long errand for which he received | | |
50c. What was the total amount the boy earned that | | |
day? | | |
|-----------+----|
Total Right | | |
+-----------+----+
(Two more similar problems follow.)
Test 6 and Test 8 are from the Courtis Standard Test. Used by permission
of S. A. Courtis.
ARITHMETIC--Test No. 8. Reasoning
In the blank space below, work as many of the following examples as
possible in the time allowed. Work them in order as numbered, entering
each answer in the "answer" column before commencing a new example. Do
not work on any other paper.
+--------+-+
| ANSWER | |
|--------+-|
1. The children in a certain school gave a Christmas | | |
party. One of the presents was a box of candy. In filling | | |
the boxes, one grade used 16 pounds of candy, another 17 | | |
pounds, a third 12 pounds, and a fourth 13 pounds. What | | |
did the candy cost at 26c. a pound? | | |
|--------+-|
2. A school in a certain city used 2516 pieces of chalk | | |
in 37 school days. Three new rooms were opened, each | | |
room holding 50 children, and the school was then found | | |
to use 84 sticks of chalk per day. How many more sticks | | |
of chalk were used per day than at first? | | |
|--------+-|
3. Several boys went on a bicycle trip of 1500 miles. | | |
The first week they rode 374 miles, the second week 264 | | |
miles, the third 423 miles, the fourth 401 miles. They | | |
finished the trip the next week. How many miles did they | | |
ride the last week? | | |
|--------+-|
4. Forty-five boys were hired to pick apples from 15 | | |
trees in an apple orchard. In 50 minutes each boy had | | |
picked 48 choice apples. If all the apples picked were | | |
packed away carefully in 8 boxes of equal size, how many | | |
apples were put in each box? | | |
|--------+-|
5. In a certain school 216 children gave a sleigh-ride | | |
party. They rented 7 sleighs at a cost of $30.00 and paid | | |
$24.00 for the refreshments. The party travelled 15 miles | | |
in 2-1/2 hours and had a very pleasant time. What was | | |
each child's share of the expense? | | |
|--------+-|
6. A girl found, by careful counting, that there were | | |
2400 letters on one page of her history, and only 2295 | | |
letters on a page of her reader. How many more letters | | |
had she read in one book than in the other if she had | | |
read 47 pages in each of the books? | | |
|--------+-|
7. Each of 59 rooms in the schools of a certain city | | |
contributed 25 presents to a Christmas entertainment for | | |
poor children. The stores of the city gave 1986 other | | |
articles for presents. What was the total number of | | |
presents given away at the entertainment? | | |
|--------+-|
8. Forty-eight children from a certain school paid 10c. | | |
apiece to ride 7 miles on the cars to a woods. There in a | | |
few hours they gathered 2765 nuts. 605 of these were bad, | | |
but the rest were shared equally among the children. How | | |
many good nuts did each one get? | | |
|--------+-|
Total | | |
+--------+-+
These proposed measures of ability to apply arithmetic illustrate very
nicely the differences of opinion concerning what applied arithmetic and
arithmetical reasoning should be. The thinker who emphasizes the fact
that in life out of school the situation demanding quantitative
treatment is usually real rather than described, will condemn a test all
of whose constituents are _described_ problems. Unless we are
excessively hopeful concerning the transfer of ideas of method and
procedure from one mental function to another we shall protest against
the artificiality of No. 3 of the Stone series, and of the entire
Courtis Test 8 except No. 4. The Courtis speed-reasoning test (No. 6) is
a striking example of the mixture of ability to understand quantitative
relations with the ability to understand words. Consider these five, for
example, in comparison with the revised versions attached.[3]
[3] The form of Test 6 quoted here is that given by Courtis ['11-'12,
p. 20]. This differs a little from the other series of Test 6,
shown on pages 43 and 44.
1. The children of a school gave a sleigh-ride party. There were 9
sleighs, and each sleigh held 30 children. How many children were
there in the party?
REVISION. _If one sleigh holds 30 children, 9 sleighs hold ....
children._
2. Two school-girls played a number-game. The score of the girl
that lost was 57 points and she was beaten by 16 points. What was
the score of the girl that won?
REVISION. _Mary and Nell played a game. Mary had a score of 57.
Nell beat Mary by 16. Nell had a score of ...._
3. A girl counted the automobiles that passed a school. The total
was 60 in two hours. If the girl saw 27 pass the first hour how
many did she see the second?
REVISION. _In two hours a girl saw 60 automobiles. She saw 27 the
first hour. She saw .... the second hour._
4. On a playground there were five equal groups of children each
playing a different game. If there were 75 children all together,
how many were there in each group?
REVISION. _75 pounds of salt just filled five boxes. The boxes were
exactly alike. There were .... pounds in a box._
5. A teacher weighed all the children in a certain grade. One girl
weighed 70 pounds. Her older sister was 49 pounds heavier. How many
pounds did the sister weigh?
REVISION. _Mary weighs 70 lb. Jane weighs 49 pounds more than Mary.
Jane weighs .... pounds._
The distinction between a problem described as clearly and simply as
possible and the same problem put awkwardly or in ill-known words or
willfully obscured should be regarded; and as a rule measurements of
ability to apply arithmetic should eschew all needless obscurity or
purely linguistic difficulty. For example,
_A boy bought a two-cent stamp. He gave the man in the store 10
cents. The right change was .... cents._
is better as a test than
_If a boy, purchasing a two-cent stamp, gave a ten-cent stamp in
payment, what change should he be expected to receive in return?_
The distinction between the description of a _bona fide_ problem that a
human being might be called on to solve out of school and the
description of imaginary possibilities or puzzles should also be
considered. Nos. 3 and 9 of Stone are bad because to frame the problems
one must first know the answers, so that in reality there could never be
any point in solving them. It is probably safe to say that nobody in the
world ever did or ever will or ever should find the number of apples in
a box by the task of No. 4 of the Courtis Test 8.
This attaches no blame to Dr. Stone or to Mr. Courtis. Until very
recently we were all so used to the artificial problems of the
traditional sort that we did not expect anything better; and so blind to
the language demands of described problems that we did not see their
very great influence. Courtis himself has been active in reform and has
pointed out ('13, p. 4 f.) the defects in his Tests 6 and 8.
"Tests Nos. 6 and 8, the so-called reasoning tests, have proved the
least satisfactory of the series. The judgments of various teachers and
superintendents as to the inequalities of the units in any one test, and
of the differences between the different editions of the same test, have
proved the need of investigating these questions. Tests of adults in
many lines of commercial work have yielded in many cases lower scores
than those of the average eighth grade children. At the same time the
scores of certain individuals of marked ability have been high, and
there appears to be a general relation between ability in these tests
and accuracy in the abstract work. The most significant facts, however,
have been the difficulties experienced by teachers in attempting to
remedy the defects in reasoning. It is certain that the tests measure
abilities of value but the abilities are probably not what they seem to
be. In an attempt to measure the value of different units, for instance,
as many problems as possible were constructed based upon a single
situation. Twenty-one varieties were secured by varying the relative
form of the question and the relative position of the different phrases.
One of these proved nineteen times as hard as another as measured by the
number of mistakes made by the children; yet the cause of the difference
was merely the changes in the phrasing. This and other facts of the same
kind seem to show that Tests 6 and 8 measure mainly the ability to
read."
The scientific measurement of the abilities and achievements concerned
with applied arithmetic or problem-solving is thus a matter for the
future. In the case of described problems a beginning has been made in
the series which form a part of the National Intelligence Tests ['20],
one of which is shown on page 49 f. In the case of problems with real
situations, nothing in systematic form is yet available.
Systematic tests and scales, besides defining the abilities we are to
establish and improve, are of very great service in measuring the status
and improvement of individuals and of classes, and the effects of
various methods of instruction and of study. They are thus helpful to
pupils, teachers, supervisors, and scientific investigators; and are
being more and more widely used every year. Information concerning the
merits of the different tests, the procedure to follow in giving and
scoring them, the age and grade standards to be used in interpreting
results, and the like, is available in the manuals of Educational
Measurement, such as Courtis, _Manual of Instructions for Giving and
Scoring the Courtis Standard Tests in the Three R's_ ['14]; Starch,
_Educational Measurements_ ['16]; Chapman and Rush, _Scientific
Measurement of Classroom Products_ ['17]; Monroe, DeVoss, and Kelly,
_Educational Tests and Measurements_ ['17]; Wilson and Hoke, _How to
Measure_ ['20]; and McCall, _How to Measure in Education_ ['21].
TEST 1
National Intelligence Tests.
Scale A. Form 1, Edition 1
Find the answers as quickly as you can.
Write the answers on the dotted lines.
Use the side of the page to figure on.
#Begin here#
1 Five cents make 1 nickel. How many nickels make a
dime? _Answer_ .....
2 John paid 5 dollars for a watch and 3 dollars for a chain.
How many dollars did he pay for the watch and chain? _Answer_ .....
3 Nell is 13 years old. Mary is 9 years old. How much
younger is Mary than Nell? _Answer_ .....
4 One quart of ice cream is enough for 5 persons. How
many quarts of ice cream are needed for 25 persons? _Answer_ .....
5 John's grandmother is 86 years old. If she lives, in
how many years will she be 100 years old? _Answer_ .....
6 If a man gets $2.50 a day, what will he be paid for six
days' work? _Answer_ .....
7 How many inches are there in a foot and a half? _Answer_ .....
8 What is the cost of 12 cakes at 6 for 5 cents? _Answer_ .....
9 The uniforms for a baseball team of nine boys cost $2.50
each. The shoes cost $2 a pair. What was the total
cost of uniforms and shoes for the nine? _Answer_ .....
10 A train that usually arrives at half-past ten was 17
minutes late. When did it arrive? _Answer_ .....
11 At 10c a yard, what is the cost of a piece 10-1/2 ft. long?
_Answer_ .....
12 A man earns $6 a day half the time, $4.50 a day one
fourth of the time, and nothing on the remaining days
for a total period of 40 days. What did he earn in all
in the 40 days? _Answer_ .....
13 What per cent of $800 is 4% of $1000? _Answer_ .....
14 If 60 men need 1500 lb. flour per month, what is the
requirement per man per day counting a month as 30
days? _Answer_ .....
15 A car goes at the rate of a mile a minute. A truck goes
20 miles an hour. How many times as far will the car
go as the truck in 10 seconds? _Answer_ .....
16 The area of the base (inside measure) of a cylindrical
tank is 90 square feet. How tall must it be to hold
100 cubic yards? _Answer_ .....
From National Intelligence Tests by National Research Council.
Copyright, 1920, by The World Book Company, Yonkers-on-Hudson,
New York.
Used by permission of the publishers.