“The hints that are here given suffice to show that the knowledge forms are adapted to children of three and four years of age, and that they incite plays which are both spontaneous and nourishing to heart and intellect.… These few indications for the use of these forms must suffice; they already show sufficiently clearly that the observation and comprehension of them are perfectly suited to the active, intellectual and emotional sides of children three and four years of age, and to actual free play which strengthens intellect and feeling.”--_P., p. 185._ Now the “hints” refer to making clear to the child, always in justice, be it remembered, in the concrete, “as perceptible facts only,” such points as “similarity of size with dissimilarity of shape and position, in such words as: “Twice as long and half as wide, Half as long and twice as wide, The same size are we two.” Certainly children differ very much, and some have a special aptitude for mathematical relations, but to most children under five these words would convey nothing. _Half_ may have a meaning, though at that age and for some time after we hear of “a fair half” and “quarter” is generally used as a name for any fraction recognized as not a half, even if it should be greater. Such words as _fourth_ and _eighth_ can have no meaning for a child who shows no consciousness of difference when shown six, seven or eight objects. At the age of three, an average child recognizes three objects, but when a fourth is added, he proceeds to count one by one, he does not recognize three plus one. Again, we must repeat that Froebel never intended any mathematical ideas to be forced upon unwilling children. He constantly tells the mother not to force, and he frequently speaks of the child’s “accidental productions which will become a point of departure for his self-development,” through the explanatory rhymes, to be sung by the mother in order to call the child’s attention to the results of his own action. It is true, too, that it is in connection with this kind of work, or play, that Froebel writes of “the knowledge-acquiring side of the game, which is the quickly tiring side.”