THE PSYCHOLOGY OF THINKING: REASONING IN ARITHMETIC
THE ESSENTIALS OF ARITHMETICAL REASONING
We distinguish aimless reverie, as when a child dreams of a vacation
trip, from purposive thinking, as when he tries to work out the answer
to "How many weeks of vacation can a family have for $120 if the cost is
$22 a week for board, $2.25 a week for laundry, and $1.75 a week for
incidental expenses, and if the railroad fares for the round trip are
$12?" We distinguish the process of response to familiar situations,
such as five integral numbers to be added, from the process of response
to novel situations, such as (for a child who has not been trained with
similar problems):--"A man has four pieces of wire. The lengths are 120
yd., 132 meters, 160 feet, and 1/8 mile. How much more does he need to
have 1000 yd. in all?" We distinguish 'thinking things together,' as
when a diagram or problem or proof is understood, from thinking of one
thing after another as when a number of words are spelled or a poem in
an unknown tongue is learned. In proportion as thinking is purposive,
with selection from the ideas that come up, and in proportion as it
deals with novel problems for which no ready-made habitual response is
available, and in proportion as many bonds act together in an organized
way to produce response, we call it reasoning.
When the conclusion is reached as the effect of many particular
experiences, the reasoning is called inductive. When some principle
already established leads to another principle or to a conclusion about
some particular fact, the reasoning is called deductive. In both cases
the process involves the analysis of facts into their elements, the
selection of the elements that are deemed significant for the question
at hand, the attachment of a certain amount of importance or weight to
each of them, and their use in the right relations. Thought may fail
because it has not suitable facts, or does not select from them the
right ones, or does not attach the right amount of weight to each, or
does not put them together properly.
In the world at large, many of our failures in thinking are due to not
having suitable facts. Some of my readers, for example, cannot solve the
problem--"What are the chances that in drawing a card from an ordinary
pack of playing-cards four times in succession, the same card will be
drawn each time?" And it will be probably because they do not know
certain facts about the theory of probabilities. The good thinkers among
such would look the matter up in a suitable book. Similarly, if a person
did not happen to know that there were fifty-two cards in all and that
no two were alike, he could not reason out the answer, no matter what
his mastery of the theory of probabilities. If a competent thinker, he
would first ask about the size and nature of the pack. In the actual
practice of reasoning, that is, we have to survey our facts to see if we
lack any that are necessary. If we do, the first task of reasoning is to
acquire those facts.
This is specially true of the reasoning about arithmetical facts in
life. "Will 3-1/2 yards of this be enough for a dress?" Reason directs
you to learn how wide it is, what style of dress you intend to make of
it, how much material that style normally calls for, whether you are a
careful or a wasteful cutter, and how big the person is for whom the
dress is to be made. "How much cheaper as a diet is bread alone, than
bread with butter added to the extent of 10% of the weight of the
bread?" Reason directs you to learn the cost of bread, the cost of
butter, the nutritive value of bread, and the nutritive value of butter.
In the arithmetic of the school this feature of reasoning appears in
cases where some fact about common measures must be brought to bear, or
some table of prices or discounts must be consulted, or some business
custom must be remembered or looked up.
Thus "How many badges, each 9 inches long, can be made from 2-1/2 yd.
ribbon?" cannot be solved without getting into mind 1 yd. = 36 inches.
"At Jones' prices, which costs more, 3-3/4 lb. butter or 6-1/2 lb. lard?
How much more?" is a problem which directs the thinker to ascertain
Jones' prices.
It may be noted that such problems are, other things being equal,
somewhat better training in thinking than problems where all the data
are given in the problem itself (_e.g._, "Which costs more, 3-3/4 lb.
butter at 48c per lb. or 6-1/2 lb. lard at 27c per lb.? How much
more?"). At least it is unwise to have so many problems of the latter
sort that the pupil may come to think of a problem in applied arithmetic
as a problem where everything is given and he has only to manipulate the
data. Life does not present its problems so.
The process of selecting the right elements and attaching proper weight
to them may be illustrated by the following problem:--"Which of these
offers would you take, supposing that you wish a D.C.K. upright piano,
have $50 saved, can save a little over $20 per month, and can borrow
from your father at 6% interest?"
A
A Reliable Piano. The Famous D.C.K. Upright. You pay $50 cash down
and $21 a month for only a year and a half. _No interest_ to pay.
We ask you to pay only for the piano and allow you plenty of time.
B
We offer the well-known D.C.K. Piano for $390. $50 cash and $20 a
month thereafter. Regular interest at 6%. The interest soon is
reduced to less than $1 a month.
C
The D.C.K. Piano. Special Offer, $375, cash. Compare our prices
with those of any reliable firm.
If you consider chiefly the "only," "No interest to pay," "only," and
"plenty of time" in offer A, attaching much weight to them and little to
the thought, "How much will $50 plus (18 x $21) be?", you will probably
decide wrongly.
The situations of life are often complicated by many elements of little
or even of no relevance to the correct solution. The offerer of A may
belong to your church; your dearest friend may urge you to accept offer
B; you may dislike to talk with the dealer who makes offer C; you may
have a prejudice against owing money to a relative; that prejudice may
be wise or foolish; you may have a suspicion that the B piano is
shopworn; that suspicion may be well-founded or groundless; the salesman
for C says, "You don't want your friends to say that you bought on the
installment plan. Only low-class persons do that," etc. The statement of
arithmetical problems in school usually assists the pupil to the extent
of ruling out all save definitely quantitative elements, and of ruling
out all quantitative elements except those which should be considered.
The first of the two simplifications is very beneficial, on the whole,
since otherwise there might be different correct solutions to a problem
according to the nature and circumstances of the persons involved. The
second simplification is often desirable, since it will often produce
greater improvement in the pupils, per hour of time spent, than would be
produced by the problems requiring more selection. It should not,
however, be a universal custom; for in that case the pupils are tempted
to think that in every problem they must use all the quantities given,
as one must use all the pieces in a puzzle picture.
It is obvious that the elements selected must not only be right but also
be in the right relations to one another. For example, in the problems
below, the 6 must be thought of in relation to a dozen and as being half
of a dozen, and also as being 6 times 1. 1 must be mentally tied to
"each." The 6 as half of a dozen must be related to the $1.00, $1.60,
etc. The 6 as 6 times 1 must be related to the $.09, $.14, etc.
Buying in Quantity
These are a grocer's prices for certain things by the dozen and
for a single one. He sells a half dozen at half the price of a
dozen. Find out how much you save by buying 6 all at one time
instead of buying them one at a time.
Doz. Each
1. Evaporated Milk $1.00 $.09
2. Puffed Rice 1.60 .14
3. Puffed Wheat 1.10 .10
4. Canned Soup 1.90 .17
5. Sardines 1.80 .16
6. Beans (No. 2 cans) 1.50 .13
7. Pork and Beans 1.70 .15
8. Peas (No. 2 cans) 1.40 .12
9. Tomatoes (extra cans) 3.20 .28
10. Ripe olives (qt. cans) 7.20 .65
It is obvious also that in such arithmetical work as we have been
describing, the pupil, to be successful, must 'think things together.'
Many bonds must cooeperate to determine his final response.
As a preface to reasoning about a problem we often have the discovery of
the problem and the classification of just what it is, and as a
postscript we have the critical inspection of the answer obtained to
make sure that it is verified by experiment or is consistent with known
facts. During the process of searching for, selecting, and weighting
facts, there may be similar inspection and validation, item by item.
REASONING AS THE COOePERATION OF ORGANIZED HABITS
The pedagogy of the past made two notable errors in practice based on
two errors about the psychology of reasoning. It considered reasoning as
a somewhat magical power or essence which acted to counteract and
overrule the ordinary laws of habit in man; and it separated too sharply
the 'understanding of principles' by reasoning from the 'mechanical'
work of computation, reading problems, remembering facts and the like,
done by 'mere' habit and memory.
Reasoning or selective, inferential thinking is not at all opposed to,
or independent of, the laws of habit, but really is their necessary
result under the conditions imposed by man's nature and training. A
closer examination of selective thinking will show that no principles
beyond the laws of readiness, exercise, and effect are needed to explain
it; that it is only an extreme case of what goes on in associative
learning as described under the 'piecemeal' activity of situations; and
that attributing certain features of learning to mysterious faculties of
abstraction or reasoning gives no real help toward understanding or
controlling them.
It is true that man's behavior in meeting novel problems goes beyond, or
even against, the habits represented by bonds leading from gross total
situations and customarily abstracted elements thereof. One of the two
reasons therefor, however, is simply that the finer, subtle,
preferential bonds with subtler and less often abstracted elements go
beyond, and at times against, the grosser and more usual bonds. One set
is as much due to exercise and effect as the other. The other reason is
that in meeting novel problems the mental set or attitude is likely to
be one which rejects one after another response as their unfitness to
satisfy a certain desideratum appears. What remains as the apparent
course of thought includes only a few of the many bonds which did
operate, but which, for the most part, were unsatisfying to the ruling
attitude or adjustment.
Successful responses to novel data, associations by similarity and
purposive behavior are in only apparent opposition to the fundamental
laws of associative learning. Really they are beautiful examples of it.
Man's successful responses to novel data--as when he argues that the
diagonal on a right triangle of 796.278 mm. base and 137.294 mm.
altitude will be 808.022 mm., or that Mary Jones, born this morning,
will sometime die--are due to habits, notably the habits of response to
certain elements or features, under the laws of piecemeal activity and
assimilation.
Nothing is less like the mysterious operations of a faculty of reasoning
transcending the laws of connection-forming, than the behavior of men in
response to novel situations. Let children who have hitherto confronted
only such arithmetical tasks, in addition and subtraction with one- and
two-place numbers and multiplication with one-place numbers, as those
exemplified in the first line below, be told to do the examples shown in
the second line.
ADD ADD ADD SUBT. SUBT. MULTIPLY MULTIPLY MULTIPLY
8 37 35 8 37 8 9 6
5 24 68 5 24 5 7 3
-- -- 23 -- -- -- -- --
19
--
MULTIPLY MULTIPLY MULTIPLY
32 43 34
23 22 26
-- -- --
They will add the numbers, or subtract the lower from the upper number,
or multiply 3 x 2 and 2 x 3, etc., getting 66, 86, and 624, or respond
to the element of 'Multiply' attached to the two-place numbers by "I
can't" or "I don't know what to do," or the like; or, if one is a child
of great ability, he may consider the 'Multiply' element and the bigness
of the numbers, be reminded by these two aspects of the situation of the
fact that
'9
9 multiply'
--
gave only 81, and that
'10
10 multiply'
--
gave only 100, or the like; and so may report an intelligent and
justified "I can't," or reject the plan of 3 x 2 and 2 x 3, with 66, 86,
and 624 for answers, as unsatisfactory. What the children will do will,
in every case, be a product of the elements in the situation that are
potent with them, the responses which these evoke, and the further
associates which these responses in turn evoke. If the child were one of
sufficient genius, he might infer the procedure to be followed as a
result of his knowledge of the principles of decimal notation and the
meaning of 'Multiply,' responding correctly to the 'place-value' element
of each digit and adding his 6 tens and 9 tens, 20 twos and 3 thirties;
but if he did thus invent the shorthand addition of a collection of
twenty-three collections, each of 32 units, he would still do it by the
operation of bonds, subtle but real.
Association by similarity is, as James showed long ago, simply the
tendency of an element to provoke the responses which have been bound to
it. _abcde_ leads to _vwxyz_ because _a_ has been bound to _vwxyz_ by
original nature, exercise, or effect.
Purposive behavior is the most important case of the influence of the
attitude or set or adjustment of an organism in determining (1) which
bonds shall act, and (2) which results shall satisfy. James early
described the former fact, showing that the mechanism of habit can give
the directedness or purposefulness in thought's products, provided that
mechanism includes something paralleling the problem, the aim, or need,
in question.
The second fact, that the set or attitude of the man helps to determine
which bonds shall satisfy, and which shall annoy, has commonly been
somewhat obscured by vague assertions that the selection and retention
is of what is "in point," or is "the right one," or is "appropriate," or
the like. It is thus asserted, or at least hinted, that "the will," "the
voluntary attention," "the consciousness of the problem," and other such
entities are endowed with magic power to decide what is the "right" or
"useful" bond and to kill off the others. The facts are that in
purposive thinking and action, as everywhere else, bonds are selected
and retained by the satisfyingness, and are killed off by the
discomfort, which they produce; and that the potency of the man's set or
attitude to make this satisfy and that annoy--to put certain
conduction-units in readiness to act and others in unreadiness--is in
every way as important as its potency to set certain conduction-units in
actual operation.
Reasoning is not a radically different sort of force operating against
habit but the organization and cooeperation of many habits, thinking
facts together. Reasoning is not the negation of ordinary bonds, but
the action of many of them, especially of bonds with subtle elements of
the situation. Some outside power does not enter to select and
criticize; the pupil's own total repertory of bonds relevant to the
problem is what selects and rejects. An unsuitable idea is not killed
off by some _actus purus_ of intellect, but by the ideas which it itself
calls up, in connection with the total set of mind of the pupil, and
which show it to be inadequate.
Almost nothing in arithmetic need be taught as a matter of mere
unreasoning habit or memory, nor need anything, first taught as a
principle, ever become a matter of mere habit or memory. 5 x 4 = 20
should not be learned as an isolated fact, nor remembered as we remember
that Jones' telephone number is 648 J 2. Almost everything in arithmetic
should be taught as a habit that has connections with habits already
acquired and will work in an organization with other habits to come. The
use of this organized hierarchy of habits to solve novel problems is
reasoning.