The Psychology of ArithmeticThorndike, Edward L. (Edward Lee)
Science
The Psychology of Arithmetic
Thorndike, Edward L. (Edward Lee)
Arithmetic -- Study and teaching
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.
Public-domain text, read in full here on John Shaqi.
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