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 3½ 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 2½ yd. ribbon?" cannot be solved without getting into mind 1 yd. = 36 inches. "At Jones' prices, which costs more, 3¾ lb. butter or 6½ 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, 3¾ lb. butter at 48¢ per lb. or 6½ lb. lard at 27¢ 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.