Fig. 55.—Clear, simple, and well spaced.
Fig. 56.—Well arranged, though a little wider spacing between the squares would make it even better.
THE USE OF CONCRETE OBJECTS IN ARITHMETIC
We mean by concrete objects actual things, events, and relations presented to sense, in contrast to words and numbers and symbols which mean or stand for these objects or for more abstract qualities and relations. Blocks, tooth-picks, coins, foot rules, squared paper, quart measures, bank books, and checks are such concrete things. A foot rule put successively along the three thirds of a yard rule, a bell rung five times, and a pound weight balancing sixteen ounce weights are such concrete events. A pint beside a quart, an inch beside a foot, an apple shown cut in halves display such concrete relations to a pupil who is attentive to the issue.
Concrete presentations are obviously useful in arithmetic to teach meanings under the general law that a word or number or sign or symbol acquires meaning by being connected with actual things, events, qualities, and relations. We have also noted their usefulness as means to verifying the results of thinking and computing, as when a pupil, having solved, "How many badges each 5 inches long can be made from 31⁄3 yd. of ribbon?" by using 10 × 12⁄5, draws a line 31⁄3 yd. long and divides it into 5-inch lengths.
Concrete experiences are useful whenever the meaning of a number, like 9 or 7⁄8 or .004, or of an operation, like multiplying or dividing or cubing, or of some term, like rectangle or hypothenuse or discount, or some procedure, like voting or insuring property against fire or borrowing money from a bank, is absent or incomplete or faulty. Concrete work thus is by no means confined to the primary grades but may be appropriate at all stages when new facts, relations, and procedures are to be taught.
How much concrete material shall be presented will depend upon the fact or relation or procedure which is to be made intelligible, and the ability and knowledge of the pupil. Thus 'one half' will in general require less concrete illustration than 'five sixths'; and five sixths will require less in the case of a bright child who already knows 2⁄3, 3⁄4, 3⁄8, 5⁄8, 7⁄8, 2⁄5, 3⁄5, and 4⁄5 than in the case of a dull child or one who only knows 2⁄3 and 3⁄4. As a general rule the same topic will require less concrete material the later it appears in the school course. If the meanings of the numbers are taught in grade 2 instead of grade 1, there will be less need of blocks, counters, splints, beans, and the like. If 1½ + ½ = 2 is taught early in grade 3, there will be more gain from the use of 1½ inches and ½ inch on the foot rule than if the same relations were taught in connection with the general addition of like fractions late in grade 4. Sometimes the understanding can be had either by connecting the idea with the reality directly, or by connecting the two indirectly via some other idea. The amount of concrete material to be used will depend on its relative advantage per unit of time spent. Thus it might be more economical to connect 5⁄12, 7⁄12, and 11⁄12 with real meanings indirectly by calling up the resemblance to the 2⁄3, 3⁄4, 3⁄8, 5⁄8, 7⁄8, 2⁄5, 3⁄5, 4⁄5, and 5⁄6 already studied, than by showing 5⁄12 of an apple, 7⁄12 of a yard, 11⁄12 of a foot, and the like.
In general the economical course is to test the understanding of the matter from time to time, using more concrete material if it is needed, but being careful to encourage pupils to proceed to the abstract ideas and general principles as fast as they can. It is wearisome and debauching to pupils' intellects for them to be put through elaborate concrete experiences to get a meaning which they could have got themselves by pure thought. We should also remember that the new idea, say of the meaning of decimal fractions, will be improved and clarified by using it (see page 183 f.), so that the attainment of a perfect conception of decimal fractions before doing anything with them is unnecessary and probably very wasteful.
A few illustrations may make these principles more instructive.