We have now left the solid and are approaching abstraction when we begin the study of planes. All mental development has ever begun and must begin with the concrete, and progress by successive stages toward the abstract, and it was Froebel's idea that his play-material might be used to form a series of steps up which the child might climb in his journey toward the abstract.
Beginning with the ball, a perfect type of wholeness and unity, we are led through diversity, as shown in the three solids of the second gift, toward divisibility in the Building Gifts, and approximation to surface in the sixth gift. The next move in advance is the partial abstraction of surface, shown in the tablets of the seventh gift.
The tablets show two dimensions, length and breadth, the thickness being so trifling relatively that it need not be considered, as it does not mar the child's perception and idea of the plane. They are intended to represent surfaces, and should be made as thin as is consistent with durability.
Systematic Relation between the Tablets.
The various tablets as first introduced in Germany and in this country were commonly quite different in size and degrees of angles in the different kindergartens, as they were either cut out hastily by the teachers themselves, or made by manufacturers who knew very little of the subject. The former practice of dividing an oblong from corner to corner to produce the right-angled scalene triangle was much to be condemned, as it entirely set aside the law of systematic relation between the tablets and rendered it impossible to produce the standard angles, which are so valuable a feature of the gift.
"One of the principal advantages of the kindergarten system is that it lays the foundation for a systematic, scientific education which will help the masses to become expert and artistic workmen in whatever occupation they may be engaged."[62]
In this direction the seventh gift has doubtless immense capabilities, but much of its force and value has been lost, much of the work thrown away which it has accomplished, for want of proper and systematic relation between the tablets. The order in which these are now derived and introduced is as follows:—
The square tablet is, of course, the type of quadrilaterals, and when it is divided from corner to corner a three-sided figure is seen,—the half square or right isosceles triangle; but one which is not the type of three-sided figures. The typical and simplest triangle, the equilateral, is next presented, and if this be divided by a line bisecting one angle, the result will be two triangles of still different shape, the right-angled scalene. If these two are placed with shortest sides together, we have another form, the obtuse-angled triangle, and this gives us all the five forms of the seventh gift.
The square educates the eye to judge correctly of a right angle, and the division of the square gives the angle of 45°, or the mitre. The equilateral has three angles of 60° each; the divided equilateral or right-angled scalene has one angle of 90°, one of 60°, and one of 30°, while the obtuse isosceles has one angle of 120°, and the remaining two each 30°. These are the standard angles (90°, 45°, 60°, and 30°) used by carpenter, joiner, cabinet-maker, blacksmith,—in fact, in all the trades and many of the professions, and the child's eye should become as familiar with them as with the size of the squares on his table.
Possibilities of the Gift in Mathematical Instruction.