"There is nothing in the whole present system of education more deserving of serious consideration than the sudden and violent transition from the material to the abstract which our children have to go through on quitting the parental house to enter a school. Froebel therefore made it a point to bridge over this transition by a whole series of play-material, and in this series it is the laying-tablets which occupy the first place." H. GOLDAMMER.
1. The seventh gift consists of variously colored square and triangular tablets made of wood or pasteboard, the sides of the pieces being about one inch in length. Circular and oblong pasteboard tablets have lately been introduced, as well as whole and half circles in polished woods.
2. The first six gifts ill.u.s.trated solids, while the seventh, moving from the concrete towards the abstract, makes the transition to the surface.
The Building Gifts presented to the child divided units, from which he constructed new wholes. Through these he became familiar with the idea of a whole and parts, and was prepared for the seventh gift, which offers him not an object to transform, but independent elements to be combined into varied forms. These divided solids also offered the child a certain fixed amount of material for his use; after the introduction of the seventh gift, the amount to be used is optional with the kindergartner.
3. The child up to this time has seen the surface in connection with solids. He now receives the embodied surface separated from the solid, and gradually abstracts the general idea of "surface," learning to regard it not only as a part, but as an individual whole.
This gift also emphasizes color and the various triangular forms, besides imparting the idea of pictorial representation, or the representation of objects by means of plane surfaces.
4. The gift leads the child from the object itself towards the representation of the object, thus sharpening the observation and preparing the way for drawing.
It is also less definitely suggestive than previous gifts, and demands more creative power for its proper use. It appeals to the sense of form, sense of place, sense of color, and sense of number.
5. The geometrical forms ill.u.s.trated in this gift are:--
Squares.
{ Right isosceles.
{ Obtuse isosceles.
Triangles. { Equilateral.
{ Right-angled scalene.
{ Oblong.
{ Rhombus.
{ Rhomboid.
{ Trapezoid.
In combination. { Trapezium.
{ Pentagon.
{ Hexagon.
{ Heptagon.
{ Octagon.
6. The law of Mediation of Contrasts is shown in the forms of the gift. We have in the triangles, for instance, two lines running in opposite directions, connected by a third, which serves as the mediation. Contrasts and their mediations are also shown in the squares and in the forms made by combination. This gift, representing the plane, is a link between the divided solid and the line.
Step from Solid to Plane.
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 princ.i.p.al advantages of the kindergarten system is that it lays the foundation for a systematic, scientific education which will help the ma.s.ses to become expert and artistic workmen in whatever occupation they may be engaged."[62]
[62] _Pamphlet on the Seventh Gift_. (Milton Bradley Co.)
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.
Edward Wiebe says in regard to the relation of the seventh gift to geometry and general mathematical instruction: "Who can doubt that the contemplation of these figures and the occupations with them must tend to facilitate the understanding of geometrical axioms in the future, and who can doubt that all mathematical instruction by means of Froebel"s system must needs be facilitated and better results obtained? That such instruction will be rendered fruitful in practical life is a fact which will be obvious to all who simply glance at the sequence of figures even without a thorough explanation, for they contain demonstratively the larger number of those axioms in elementary geometry which relate to the conditions of the plane in regular figures."
As the tablets are used in the kindergarten, they are intended only "to increase the sum of general experience in regard to the qualities of things," but they may be made the medium of really advanced instruction in mathematics, such as would be suitable for a connecting-cla.s.s or a primary school. All this training, too, may be given in the concrete, and so lay the foundation for future mathematical work on the rock of practical observation.
The kindergarten child is expected only to know the different kinds of triangles from each other, and to be familiar with their simple names, to recognize the standard angles, and to know practically that all right angles are equally large, obtuse angles greater, and acute less than right angles. All this he will learn by means of play with the tablets, by dictations and inventions, and by constant comparison and use of the various forms.
How and when Tablets should be introduced.
As to the introduction of the tablets, the square is first of all of course given to the child. A small cube of the third gift may be taken and surrounded on all its faces by square tablets, and then each one "peeled off," disclosing, as it were, the hidden solid. We may also mould cubes of clay and have the children slice off one of the square faces, as both processes show conclusively the relation the square plane bears to the cube whose faces are squares. If the first tablets introduced are of pasteboard, as probably will be the case, the new material should be noted and some idea given of the manufacture of paper.
There is a vast difference in opinion concerning the introduction of this seventh gift, and it is used by the child in the various kindergartens at all times, from the beginning of his ball plays up to his laying aside of the fifth gift. It seems very clear, however, that he should not use the square plane until after he has received some impression of the three dimensions as they are shown in solid bodies, and this Mr. Hailmann tells us he has no proper means of gaining, save through the fourth gift.[63]
[63] "The perception of the difference between a surface-extension and an extension in three dimensions begins late and is established slowly."--W. Preyer, _The Mind of the Child_, page 180.
As to the triangular tablets, it is evident enough they should not be dealt with until after the child has seen the triangular plane on the solid forms of the fifth gift. Mr. Hailmann says that a clear idea of the extension of solids in three dimensions can only come from a familiarity with the bricks, and again that the abstractions of the tablet should not be obtruded on the child"s notice until he has that clear idea.
Though the six tablets which surround the cube may be given to the child at the first exercise, it is better to dictate simple positions of one or two squares first, and let him use the six in dictation and many more in invention.
Order of introducing Triangles.
The first triangle given is the right isosceles, showing the angle of forty-five degrees, and formed by bisecting the square with a diagonal line. The child should be given a square of paper and scissors and allowed to discover the new form for himself, letting him experiment until the desired triangle is obtained. He should then study the new form, its edges and angles, and then join his two right-angled triangles into a square, a larger triangle, etc. Then let him observe how many positions these triangles may a.s.sume by moving one round the other. He will find them acting according to the law of opposites already familiar to him, and if not comprehended,[64] yet furnishing him with an infallible criterion for his inventive work.
[64] "With this law I give children a guide for creating, and because it is the law according to which they, as creatures of G.o.d, have themselves been created, they can easily apply it. It is born with them."--_Reminiscences of Froebel_, page 73.
The equilateral is then taken up, is compared with the half-square, and then studied by itself, its three equal sides and angles (each sixty degrees) being noted as well as the obtuse angles made by all possible combinations of the equilateral.
Next, as we have said, comes the right-angled scalene triangle, with its inequality of sides and angles, which must be studied and compared with the equilateral; and last of all, the obtuse isosceles triangle, which is dealt with in the same way.
Here, again, it should be noted that the two last forms should always be discovered by the child in his play with the equilateral, and that he should cut them himself from paper before he is given the regular pasteboard or wooden triangles for study. If presented for the first time in this latter form, they can never mean as much to him as if he had found them out for himself.