among numbers known as addition, subtraction, multiplication
and division; (4) The fact that the various articles are put to
gether probably suggests the putting together of the numbers
representing their value. That is, it acts as a stimulus to arouse
the association process of addition rather than subtraction or
multiplication or division; (5) tlie occurrence of the successive
37^
ARITHMETIC 375
links of the mechanical associations of the numbers leading to the
sum and the continued association of countinj by units (pennies)
until loo is reached.
In brief the steps in the solution of a representative arithmetical
problem would be as follows:
(i) Number concepts previously acquired.
(.?) Ability to read, write or speak symbols for the numbers or
objects to which they refer. The steps involved in these particular
processes have been analyzed in the preceding chapters and hence
need not be repeated here.
(3) Numerous connections previously established between the
various numbers, which are known as the fundamental operations
with numbers, fractions, and so on.
(4) A clue as to the particular connection to be made at succes
sive points.
(5) The execution of the successive acts with the particular
numbers involved.
(i) The number concept. This is partly acquired by the child
before he enters school. It arises out of, and develops through,
the endless experiences of the child in contact with objects and
repeated occurrences of events. Some investigators of the genesis
of the number idea believe it to arise by means of counting; others
believe it to arise by means of grouping. The probability is that
both factors contribute. What happens is substantially this: The
child deals with various objects and learns through handling them
that they are separate things and that there are several of them.
As he continues to handle them, he established definite meanings
and associations with each one. He furthermore develops the
idea of number or quantity by finding possibly that one or more
may sometimes be missing. He thus acquires a group perception
and gradually widens it to the ability to count, which essentially
consists of associating certain sounds and certain motor speech
processes in speaking the sounds, with successive objects or acts,
probably both, as the successive objects are touched and handled
or at least looked at in turn. These elementary number ideas are
then enlarged rapidly upon entering school by providing more
extensive materials and by associating larger numbers with
them.
(2) This step involves all the detailed and complicated elements
enumerated in the reading and writing processes and as such need
not be considered again as they contain no new elements. All the
376 EDUCATIONAL PSYCHOLOGY
other steps are specifically arithmetical processes and require de
tailed consideration.
(3) The establishment of associations among the numbers.
The processes involved in the four fundamental operations of
adding, subtracting, multiplying and dividing are pure association
processes supplemented by experiences with illustrative material
to show the meaning of the various operations. Ultimately and
fundamentally they are established in the neural and mental
machinery as associative links which, in the trained individual,
become purely automatic. Six plus 2 equals 8, 6 minus 2 equals 4,
6 times 2 equals 12, and 6 divided by 2 equals 3, are pure associa
tion processes so that when the two numbers with a certain symbol
between them appear and are spoken in succession, the appro
priate last link is brought up thereby.
(4) The clue as to which process is to take place. The element
in a given situation or problem that suggests which of the four op
erations at a given point shall take place is the clue which acts as a
stimulus to arouse the appropriate associative reaction. In the
illustration used, the fact of putting the articles together probably
acts as the clue which starts the association of 10 plus 10 plus 45
plus 7 equals 72. Step (4) in this problem is what we ordinarily
call reasoning. But fundamentally, arithmetical reasoning, as well
as socalled reasoning in general, is essentially a matter of the
perception of difTerences and similarities and a matter of selective
association processes which are largely automatic in the perfected
stages of arithmetical skill and largely trial and error in the early
stages. The child associates the putting together of objects with
addition; the taking away of objects from a group with subtraction;
the putting together of groups of equal size with multiplication;
and the taking away of groups of equal size with division. This at
first probably develops through the multiplication of objects in
these various ways. Something in any given situation or problem
suggests one or the other of the four operations. Through in
definitely repeated situations, the correct operation is probably
suggested as a matter of association. In a more complex situation
in which the clue does not operate as automatically, the mind pro
ceeds largely by trial and error by bringing up in turn different
associations until the correct one arises, 6r until one arises that
satisfies the circumstances.
In ordinary terminology, we call this process reasoning. It may
seem as though we were reducing reasoning to mechanical associa
ARITHMETIC 377
tion which naively we do not consider to be reasoning. Perhaps
in a sense that is really what it amounts to. There probably is no
such thing as reasoning in the sense of forcing one's thought proc
esses into a given desired direction in a straight and direct line by
sheer force of will. Reasoning, even that of the most original
and inventive type, probably consists fundamentally in starting
with a certain idea, desire, or problem, in short, with a stimulus,
and in waiting for associations to arise and then in following out
in turn by trial and error, one link after another, and in waiting for
each one to bring up its links until a chain of successful links arises
which satisfies the desire or which meets no opposition and which
is then selected. Probably all that voluntary effort will do is to
stimulate possibly a more rapid arousal of associative links, and to
stir up by virtue of stimulating greater neural activity, such addi
tional associations as ordinarily do not come up quite so easily.
This is apparently what happens even in the inventive and most
original type of thinking or reasoning. The statements of scientists
and inventors bear witness to the error of the usual belief that
original thinking goes straight to the goal with unerring step. Thus
Jevons remarked:
"In all probability the errors of the great mind exceed in number those
of the less vigorous one. Fertility of imagination and abundance of
guesses at truth are among the first requisites of discovery."
And Faraday said :
"The world little knows how many of the thoughts and theories which
have passed through the mind of a scientific investigator have been
crushed in silence and secrecy by his own severe criticism and adverse
examination; that in the most successful instances not a tenth of the
suggestions, the hopes, the wishes, the preliminary conclusions have
been realized." (Quoted from Lindley.)
Edison has described the invention of the electric lamp as follows:
"During all those years of experimentation and research, I never
once made a discovery. All my work was deductive, and the results I
achieved were those of invention, pure and simple. I would construct
a theory and work on its lines until I found it was untenable. Then
it would be discarded at once and another theory evolved. This was
the only possible way for me to work out the problem. ... I speak
without exaggeration when I say that I have constructed 3,000 different
theories in connection with the electric light, each one of them reasonable
378 EDUCATIONAL PSYCHOLOGY
and apparently likely to be true. Yet only in two cases did my experi
ments prove the truth of my theory. My chief difficulty was in con
structing the carbon filament. ... Every quarter of the globe was
ransacked by my agents, and all sorts of the queerest materials used,
until finally the shred of bamboo, now utilized by us, was settled upon."
(G. C. Lathrop, " Talks with Edison," Harpers, Vol. 80, p. 425.)
However, it must not be supposed that our analysis has more
than sketched in its broadest outlines one of the most complex
and difficult mental activities which human beings ordinarily per
form. Indeed arithmetic is psychologically not a single activity but
a group of fairly distinct activities, some of which are as unrelated
to each other as arithmetic as a whole is to English. Thorndike
found that the average arithmetic grades of the children in a cer
tain school for a period of two and onehalf years gave a correlation
(Pearson) of .39 with the grades in Enghsh, and of .36 with those
in geography. On the other hand, Stone tested 500 children on the
four fundamental processes and arithmetical reasoning and ob
tained the following correlations:
Addition with subtraction 50
Addition with multiplication 65
Addition with division 56
Siibtriiction with multiplication 89
Subtraction with division 95
Multiplication with division 95
Arithmetical reasoning with addition 28
Arithmetical reasoning with subtraction 32
Arithmetical reasoning with multiplicati::i 34
Arithmetical reasoning with division 36
These results are especially striking in their low correlation of
the four fundamental processes with arithmetical reasoning. It
is significant also that subtraction, which might be thought to be
closely related to addition, turns out to be much more closely re
lated to division, the correlations being .50 and .95 respectively.
These results have added significance when it is remembered that
all elements of subtraction or multiplication, for example, which
are normally present in div^ision fundamentals were carefully
eliminated from the scores.
Our numerous previous observations of the specific nature of
psychological activities, however, so far from making us surprised
at Stone's results would rather lead us to expect still greater dif
ferentiation of the arithmetical processes. Howell found with re
ARITHMETIC 379
gard to the combination 12 â€” 8=4 and 12 â€” 4 = 8 that knowing
tlie one by no means assured knowing the other. Table 112 shows
the total number of errors of 300 children in grades three to eight
on eight pairs of complementary subtraction combinations:
TABLE 112
Total Errors Total Errors
124 T4 127 17
1 28 o 1 25 o
13S 8 158 17
138 IS 157 o
1 16 10 1 19 15
115 o 112 o
1 17 22 I7~9 '2
114 s 178 I
Here there is evidently a strong tendency for the smaller of two
amounts subtracted to yield a surprisingly greater degree of accu
racy than its complement.
Now each of these numerous specialized arithmetical activities
has a psychology more or less of its own. Most of these processes
are as yet very imperfectly explored. The psychology of comple
mentary processes just considered offers an excellent example.
Other important ones are concerned with borrowing, carrying,
and decimal points. Perhaps one of the most significant of all is a
peculiar type of attention, or memory span, such as is involved in
accurately keeping in mind and adding one to the tens at appro
priate intervals while at the same time adding the units of a long
column of digits; or, in subtracting, in which it is necessary to re
member when borrowing has taken place so as to make the appro
priate compensation. Among the specialized aspects of the psy
chology of arithmetic is the phenomenon of number preferences.
The United States census reports show that when people are not
certain of their ages they tend strongly to give even numbers and
to avoid odd numbers. Phillips obtained introspections showing
that people like the even numbers but feel uncomfortable at the
thought of odd numbers. Jastrow in an unpublished investigation
had university students estimate the number of steps of two short
flights of stairs which all climbed nearly every day. The avoidance
of the odd numbers and particularly of the prime numbers is very
38o
EDUCATIONAL PSYCHOLOGY
Strikingly shown by Figure 75 where the results of both sets of judg
ments are combined. The avoidance of a given odd number is
shown by comparing its frequency with the frequency of the num
bers at each side of it. Eight is an especial favorite.
28r
cm
r>n n
3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20
Estimates
Fig. 75. â€” The number of persons whose estimates of the number of steps
fall on various numbers showing a tendency to avoid odd and prime numbers.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17i
Totals in Addition
Fig. 76. â€” Showing the relative numbers of addition combinations amounting
to the various totals. Adapted from Table 128,
ARITHMETIC 38 1
These number preferences are chiefly significant to education in
their genesis. Some indications as to their origin are incidentally
furnished by two studies to be described presently, the results of
which are represented graphically by Figures 76 and 80. Figure
76 shows very clearly that there is a strong tendency for additions
of digits which result in totals of odd numbers and especially of
prime numbers, to be more difficult and thus to cause the learner
more trouble than evennumbered totals. From quite a different
source, namely, the difficulty of mastering number pictures (Fig
ure 80), there comes evidence tending toward the same conclusion.
Indeed it is possible that early experiences of a disagreeable nature
involving these odd and prime numbers may have caused the
unpleasant feelings reported by Phillips' subjects to be associated
with them, and that this unpleasant feelingtone may have in
hibited to a certain extent the choice of odd numbers in Jastrow's
experiment according to the wellknown tendency for the un
pleasant to produce avoiding reactions. Moreover, the special
difficulty encountered with prime numbers suggests that the diffi
culty of the various totals, apart from the factor of size, is closely
related in its turn to indirect assistance derived from multiplication.
The Measurement of Efficiency in Arithmetical Operations
According to our analysis, the measurement of efficiency in arith
metical operations consists fundamentally in a measurement of
the quickness and correctness with which the various operations
are suggested and carried out under different conditions. Two
types of measuring devices have been worked out. The first type
consists of the wellknown Courtis ('10) tests. Courtis prepared at
the outset a series of eight tests now widely used, known as Scries A,
to measure the following operations: Addition, subtraction, mul
tiplication, division, copying figures, reasoning without performance
of the operations, fundamentals consisting of various combinations
of the four fundamental operations, and reasoning, including the
performance of the operations. The general principle of the tests
consists in the selection of units of the material in each test of
approximately the same type and difficulty throughout the test
and of measuring efficiency by the number of operations attempted
or done correctly in a given period of time. For example, the
addition test is composed of a large number of combinations of
single digits, such as 6 plus 3, 7 plus i, etc. The measure of
382 EDUCATIONAL PSYCHOLOGY
efficiency is the number of such additions made in one minute.
Each of the other tests is constructed on the same general princi
ple. The reasoning tests consist of separate problems of approxi
mately equal length and difficulty, and the measure of efficiency
is the number of problems attempted or solved correctly in a
given numl:)er of minutes.
More recently Courtis prepared another set, Series B, which is
confined to the four fundamental operations and contains conse
quently one test for addition, subtraction, multiplication and
division. The chief difference between the tests in Series B and
the corresponding ones in Series A is that the problems in the
former contain larger numbers arranged in larger columns so as to
introduce more complex elements.
The second general type of tests is constructed on the principle
of a scale of problems of increasing difficulty. The author ('15)
prepared a test for measuring ability in solving concrete prob
lems, designated as Arithmetical Scale A. It is composed of a
series of problems increasing in difficulty by determined steps of
difficulty ranging from o to 15. Ability is measured not in terms
of speed, but in terms of the highest point on the scale at which a
pupil can solve problems correctly. The difficulty of the problem
and the distances between the steps were determined by extensive
experiments with pupils.
,. â€” \ Stone ('08) prepared a set of problems whose difficulty was deter
mined experimentally. The problems are, however, not arranged
in scale form.
Woody ('16) prepared a series of tests on the scale plan for
measuring ability chiefly in the fundamental operations. Each
scale is composed of a series of problems arranged in the order of
increasing difficulty.
Judd and Counts ('16) prepared for the survey of the Cleveland
schools, fifteen sets of arithmetic tests, four for addition, two for
subtraction, three for multiplication, four for division, and two
for operations with fractions. The various sets for any one type
of operation represented successively more and more complex
stages of the operation.
Standards of attainment have been prepared by the authors of
the various tests so that measurements of the abilities of pupils
and schools can be compared directly with the respective norms for
the various grades. These tests have been very useful in getting at
some real facts concerning progress and attainment in arithmetic.
ARITHMETIC
3^3
Figure 19, Chapter III, shows the range of individual abihties
and the overlapping of abilities in the various grades in a certain
school as measured by the author's arithmetical scale. The facts
shown therein, although indicating enormous ranges of differences
and overlappings, are substantially the same as those found in
other subjects and hence are no longer surprising. Figure 20 ex
hibits the same facts for addition as measured by the Courtis tests.
id
11

/
10

9

^ /
/
Pa
/
Si 8
/ y
y'
7
~
^^py^
6

/
/
y^
5
^

"
^/
4
1
1
4
5
Grade
7 8
Fig. 77. â€” DiffcreiiLe between boys and girls in solving arithmetical problems
as measured by the author s scale ( '15).
Figure 77 shows the difference between the sexes and indicates
that in arithmetical reasoning the bcjys surpass the girls. This is
one of the few school abilities in which the boys are in the lead.
Economical Methods of Learning Arithmetical Operations
(i) The Acquisition of the Number Concept. The fundamental
psychological materials which the child must acquire are ideas of
numbers and quantities. He must learn what i, 2, 3, 4, etc., mean.
The practical question is, How may the pupil acquire these number
concepts most economically? A considerable amount of experi
mental work has been done which bears either directly or indirectly
on the processes by which the child develops definite notions of
the significance of number and quantity. The investigations on
the span of attention or apprehension as measured in terms of the
number of objects that can be grasped simultaneously have chiefly
384 EDUCATIONAL PSYCHOLOGY
an indirect bearing. Such researches have been made by Cattell
('86), Messenger ('03), Burnett ('06), Dietze ('85), Warren ('97),
Nanu ('04), Freeman, and others. The general result of these
researches has been to show that the average person has a span of
four, five or six objects or stimuli presented either simultaneously
or in rapid succession to the sense of vision, hearing or touch.
Investigations bearing more directly upon the development of
number concepts have been carried out by Lay ('98 and '07),
Walsemann, Knilling, and others.
Pestalozzi was probably the first to attack the problem by
devising for the purpose of school exercises his Strichlabellen (stroke
' I II III iin nil! iiiiii iiiiiii iiiiiKi iiiiiiiii iiimiip''
2 o 00 000 0000 00000 000000 0000000 00000000 000000000 0000000000
3 o 00 00 000 000 0000 oooo 00000
Â° Â° o 00 000000 00000000
4oo 0000 000000
Â»o 0000 00000 00
00000
500 0000 000 000 000 000 000
o o 00 00 000 000 000 000
o 00 000
o Â° o o o o o^ o o o [Beetz]
o oooooooooooooo
Fig. 78. â€” Number pictures or arrangements. After Howell (p. 154).
tables) which were the forerunners of more recent number pic
tures. These have been devised in two general forms: (a) those
in which the dots are arranged in two parallel horizontal rows
(prepared by Born, Busse, and Behme) and (b) those in which the
dots are arranged vertically (prepared by Hentschel, Beetz, So
belewsky, and Kaselitz). See Figure 78.
Extensive experiments have been made upon school children by
Lay to determine the relative merits of the various devices and
arrangements of the materials used for teaching apprehension of
the number of objects. Lay investigated such problems as whether
it is better for teaching numbers to present groups of stimuU or
objects simultaneously or successively; whether it is better to
present them in single rows or in double rows, in continuous rows
or in quadrats (groups of four); how many objects children can
ARITHMETIC
385
apprehend at one time; what the effect is of distance between the
objects; and so on.
The chief results have been summarized by Howell in the follow
ing Table 113, and graph, Figure 79.
y
f!
\
\
X = Grade
y = % Mistakes
\
\
1
\
\
\
]
S
N
 ,
s
hs
â– â€”
â€” .
1
A.
IB
2A
2B
:
_3
A.
3B
4A
4B
1
1
1
^
'^
sk
X
Fig. 79. â€” Curve of error for the number pictures 5 to 12 as a whole. After
Ilowell ('14, p. 213).
TABLE 113. After Howell
Comparison of the Born, Beetz, and Lay pictures with one another and with
the Russian Machine in the ap[)rehension of the numbers 5 to 10.
P = chances for mistakes. M = number of mistakes.
NUMIIKKS
Lay
Quadratic
Pictures
Firstyear
Pupils
Born
Firstyear
Pupils
Wendling
Arp. (Born)
1irstyear
Pupils
Beetz
Firstyear
Pupils
Russian
Machine
Firstyear
Pupils
5
P
87
87
87
87
87
87
M
22
16
35
23
58
50
0/
/o
25
18
40
26
67
57
P
106
106
95
95
95
82
M
25
iS
38
23
58
39
%
24
17
40
24
61
48
P
106
53
53
53
106
53
M
12
5
II
13
37
25
%
II
10
20
25
33
49
P
58
58
58
58
58
58
M
29
12
42
49
46
50
or
/o
50
20
72
84
80
86
P
106
53
53
53
106
53
M
26
IS
24
31
28
17
/O
25
30
45
40
36
32
6
7
8
9
10
Total
522
12,422
204
173
40
7
579
1,669
201
13.'
35
8
424
r,oii
103
59
24
6
348
228
424
141
Total Training
School
Combined
2,944
377
13
2,248
336
IS
3S6
EDUCATIONAL PSYCHOLOGY
The outcome of these experiments is that the quadratic pictures
are superior to other known pictures, particularly to the rows, and
that the Lay and Born pictures are practically equal in merit.
Howell continued experimentation with quadratic number