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 1-1/2 + 1/2 = 2 is taught early in grade 3, there will be more gain from the use of 1-1/2 inches and 1/2 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. (_a_) Very large numbers, such as 1000, 10,000, 100,000, and 1,000,000, need more concrete aids than are commonly given. Guessing contests about the value in dollars of the school building and other buildings, the area of the schoolroom floor and other surfaces in square inches, the number of minutes in a week, and year, and the like, together with proper computations and measurements, are very useful to reënforce the concrete presentations and supply genuine problems in multiplication and subtraction with large numbers. (_b_) Numbers very much smaller than one, such as 1/32, 1/64, .04, and .002, also need some concrete aids. A diagram like that of Fig. 57 is useful.