We have now for the first time the slanting line, the mediation of the two opposites, vertical and horizontal, and by this three of the small cubes are divided into halves and three into quarters. It is advisable, when building the cube, to place nine whole cubes in each of the two lower layers, keeping all the divided cubes in the upper or third layer, halves in the middle row, quarters at the back. Then we may slide the box gently over the cube as in the third and fourth gifts, which enables us to have the blocks separated properly when taken out again, and forms the only expedient way of handling the pieces.[47]
[47] "This procedure is by no means intended merely to make the withdrawal of the box easy for the child, but, on the contrary, brings to him much inner profit. It is well for him to receive his playthings in an orderly manner--not to have them tossed to him as fodder is tossed to animals. It is good for the child to begin his play with the perception of a whole, a simple self-contained unit, and from this unity to develop his representations. Finally, it is essential that the playing child should receive his material so arranged that its various elements are discernible, and that by seeing them his mind may unconsciously form plans for using them.
Receiving his material thus arranged, the child will use it with ever-recurrent and increasing satisfaction, and his play will produce far more abiding results than the play of one whose material lies before him like a heap of cobblestones."--Froebel"s _Pedagogics_, page 205.
The exercises with this gift are like those which have preceded it.
Exercises of the Gift
1. Informal questions by the kindergartner and answers by the children, on its introduction, that it may be well understood. This should be made entirely conversational, familiar, and playful, but a logical plan of development should be kept in mind. A consideration of the various pieces of the gift may occupy a part of each building or number lesson.
2. Dictation, building by suggestion, and cooperative plays in the various forms. With all except advanced children the Life forms are most useful and desirable.[48]
[48] "The child, in a word, follows the same path as the man, and advances from use to beauty and from beauty to truth."--Froebel"s _Pedagogics_, page 219.
3. Free invention with each lesson.
4. Number and form lessons. In number there will of course be some repet.i.tion of what has been done before, but a sufficient amount of new presentation to awaken interest. It is only by constant review and repet.i.tion that we can a.s.sist children to remember these things and to receive them among their natural experiences, and fortunately the habit of repet.i.tion in childhood is a natural one, and therefore seldom irksome.
Errors in Form Teaching.
As to the form lessons, we must remember that our method has nothing to do with scientific geometry, but is based entirely on inspection and practice. It lays the foundation of instruction in drawing, and forms an admirable preparation for different trades, as carpentry, cabinet-making, masonry, lock-smithing, pattern-making, etc. Even in the primary schools, and how much more in the kindergarten, the form or geometrical work should be essentially practical and given by inspection. Even there all scientific demonstration should be prohibited, and the teacher should be sparing in definitions.
It is enough if the children recognize the forms by their special characteristics and by perceiving their relations, and can reproduce the solids in modeling, and the planes and outlines in tablets, sticks, rings, slats, drawing, and sewing.[49]
[49] "The Conference recommends that the child"s geometrical education should begin as early as possible; in the kindergarten, if he attends a kindergarten, or if not, in the primary school. He should at first gain familiarity through the senses with simple geometrical figures and forms, plane and solid; should handle, draw, measure, and model them; and should gradually learn some of their simpler properties and relations."--_Report of Committee of Ten_, page 110.
LIFE FORMS.
We can now be quite methodical and workman-like in our building, and can learn to use all the parts economically and according to principle. We can discuss ground plans, cellars, foundations, bas.e.m.e.nts, roofs, eaves, chimneys, entrances, and windows, and thus can make almost habitable dwellings and miniature models of larger objects.[50]
[50] "The child"s life moves from the house and its living-rooms, through kitchen and cellar, through yard and garden, to the wider s.p.a.ce and activity of street and market, and this expansion of life is clearly reflected in the order and development of his productions."--Froebel"s _Pedagogics_, page 221.
The child is a real carpenter now, and innocently happy in his labor.
Who can doubt that in these cheerful daily avocations he becomes in love with industry and perseverance, and as character is nothing but crystallized habit, he gets a decided bias in these directions which affects him for many a year afterward.[51]
[51] "In some German kindergartens large building-logs are supplied in one corner of the play garden. These logs are a foot or more in length, three inches wide, and one inch thick. Several hundred of these are kept neatly piled against the fence, and the children are expected to leave them in good order. This bit of voluntary discipline has its good uses on the playground, and the free building allowed with this larger material gives rise to individual effort, and tests the power of the children in a way which makes the later, more organized work at the tables far more full of meaning."--_Kindergarten Magazine_, November, 1894.
Objects which he meets in his daily walks are to be constructed, and also objects with which he is not so familiar,[52] so that by pleasant conversation the realm of his knowledge may be extended, and the sphere of his affections and fancies enlarged; for these exercises when properly conducted address equally head, heart, and hand.
[52] "As these building gifts afford a means of clearing the perceptions of the child, they give occasion for extending these perceptions, and for representing in their essential parts objects of which the child has only heard."--Froebel"s _Pedagogics_, page 222.
Froebel says of all this building, "It is essential to proceed from the cube as a whole. In this way the conception of the whole, of uniting, stamps itself upon the child"s mind, and the evolution of the particular, partial, and manifold from unity is ill.u.s.trated."
Group Work.
Our opportunities for group work, or united building, are greatly extended, and none of them should be neglected, as it is essential to inculcate thus early the value of cooperation. We have material enough to call into being many different things on the children"s tables; the house where they live, the church they see on Sunday, the factory where their fathers or brothers work, the schoolhouse, the City Hall, the public fountain, the stable, and the shops. Thus we may create an entire village with united effort, and systematic, harmonious action.
Each object may be brought into intimate relation with the others by telling a story in which every form is introduced. This always increases the interest of the cla.s.s, and the story itself seems to be more distinctly remembered by the child when brought into connection with what he has himself constructed.
The third gift may be used with the fifth if we wish to increase the number of blocks for cooperative work, and is particularly adapted to the laying of foundations for large buildings in the sand-table. A large fifth gift, constructed on the scale of a foot instead of an inch, is very useful for united building. One child or the kindergartner may be the architect of the monument or other large form which is to be erected in the centre of the circle. The various children then bring the whole cubes, the halves, and quarters, and lay them in their appropriate places, and the erection when complete is the work of every member of the community.
SYMMETRICAL FORMS.
These are in number and variety almost endless, as we have thirty-nine pieces of different characters. Edward Wiebe says: "He who is not a stranger in mathematics knows that the number of combinations and permutations of thirty-nine different bodies cannot be counted by hundreds nor expressed by thousands, but that millions hardly suffice to exhaust all possible combinations."
These forms naturally separate themselves, Froebel says, into two distinct series, i. e., the series of squares and the series of triangles, and move from these to the circle as the conclusion of the whole series of representations. "From these forms approximating to the circle there is an easy transition to the representation of the different kinds of cog-wheels, and hence to a crude preliminary idea of mechanics."
If the movements begin with the exterior part of the figure instead of the interior, we should make all the changes we wish in that direction before touching the centre, and _vice versa_.
Each definite beginning conditions a certain process of its own, and however much liberty in regard to changes may be allowed, they are always to be introduced within certain limits.[53]
We should leave ample room for the child"s own powers of creation, but never disregard Froebel"s principle of connection of opposites; this alone will furnish him with the "inward guide" which he needs.[54] It is only by becoming accustomed to a logical mode of action that the child can use this amount of material to good advantage.
[53] "With these forms of beauty it is above all important that they be developed one from another. Each form in the series should be a modification or transformation of its predecessor. No form should be entirely destroyed. It is also essential that the series should be developed so that each step should show either an evolution into greater manifoldness and variety, or a return to greater simplicity."--Froebel"s _Pedagogics_, page 225.
[54] "This free activity ... is only possible when the law of free creativeness is known and applied; for that a free creativeness only can be a lawful one, we are taught by the smallest blade of gra.s.s, whose development takes place only according to immutable laws."--_Reminiscences of Froebel_, page 133.
Dangers of Dictation.
The dictations should be made with great care and simplicity. The child"s mind must never be forced if it shows weariness, nor the more difficult lessons given in too noisy a room, as the nervous strain is very great under such circ.u.mstances. We should remember that great concentration is needed for a young child to follow these dictations, and we must be exceedingly careful in enforcing that strict attention for too long a time. A well-known specialist says that such exercises should not be allowed at first to take up more than a minute or two at a time; then, that their duration should gradually extend to five and ten minutes. The length of time which children closely and voluntarily attend to an exercise is as follows: Children from five to seven years, about fifteen minutes; from seven to ten years, twenty minutes; from twelve to eighteen years, thirty minutes. A magnetic teacher can obtain attention somewhat longer, but it will always be at the expense of the succeeding lesson. "By teachers of high pretensions, lessons are often carried on greatly and grievously in excess of the proper limits; but when the results are examined they show that after a certain time has been exceeded, everything forced upon the brain only tends to drive out or to confuse what has been previously stored in it."
We find, of course, that the mind can sustain more labor for a longer time when all the faculties are employed than when a single faculty is exerted, but the ambitious teacher needs to remind herself every day that no error is more fatal than to overwork the brain of a young child. Other errors may perhaps be corrected, but the effects of this end only with life. To force upon him knowledge which is too advanced for his present comprehension, or to demand from him greater concentration, and for a longer period than he is physically fitted to give, is to produce arrested development.[55]
[55] "Whoever sacrifices health to wisdom has generally sacrificed wisdom, too." (Jean Paul.)
MATHEMATICAL FORMS.
We must beware of abstractions in these forms of knowledge, and let the child see and build for himself, then lead him to express in numbers what he has seen and built. He will not call it Arithmetic, nor be troubled with any visions of mathematics as an abstract science.[56]
[56] "Perceptions and recognitions which are with difficulty gained from _words_ are easily gained from facts and deeds.
Through actual experience the child gains in a trice a total concept, whereas the same concept expressed in words would be only grasped in a partial manner. The rare merit, the vivifying influence of this play-material is that, through the representations it makes possible, concepts are recognized at once in their wholeness and unity, whereas such an idea of a whole can only very gradually be gained from its verbal expression. It must, however, be added that later, through words, the concept can be brought into higher and clearer consciousness."--Froebel"s _Pedagogics_, page 206.
The cube may be divided into thirds, ninths, and twenty-sevenths, and the fact thus practically shown that whether the thirds are in one form or another, in long lines or squares, upright or flat, the contents remain the same. We may also ill.u.s.trate by building, that like forms may be produced which shall have different contents, or different forms having the same contents.
Halves and quarters may be discussed and fully ill.u.s.trated, and addition, subtraction, multiplication, and division may be continued as fully as the comprehension of the child will allow.
During the practice with the forms of knowledge we should frequently ill.u.s.trate the lawful evolution of one form from another, as in the series moving from the parallelopiped to the hexagonal prism.
It should not be forgotten that whenever the cube is separated and divided, recombination should follow, and that the gift plays should always close with synthetic processes.
Some of the mathematical truths shown in the fifth gift were also seen in the third, but "repeated experiences," as Froebel says, "are of great profit to the child."[57]
We should allow no memorizing in any of these exercises or meaningless and sing-song repet.i.tions of words. We must always talk enough to make the lesson a living one, but not too much, lest the child be deprived of the use of his own thoughts and abilities.