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 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]
[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 deg., or the mitre. The
equilateral has three angles of 60 deg. each; the divided equilateral or
right-angled scalene has one angle of 90 deg., one of 60 deg., and one
of 30 deg., while the obtuse isosceles has one angle of 120 deg., and
the remaining two each 30 deg. These are the standard angles (90 deg.,
45 deg., 60 deg., and 30 deg.) 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-class 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 assume 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 God, 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.
Dictations.
The dictations should invariably be given so that opposites and their
intermediates may be readily seen. The different triangles may be
studied each in the same way, introducing them one at a time in the
order named, afterwards allowing as free a combination as will produce
symmetrical figures. It is best always to study one of a new kind,
then two, then gradually give larger numbers.
Great possibilities undoubtedly lie in this gift, but it is well to
remember that with young children it must not be made the vehicle of
too abstract instruction. In order to make the dictations simple, the
child must be perfectly familiar with the terms of direction, up,
down, right, left, centre; with the simple names of the planes
(squares, half-squares, equal-sided, blunt and sharp-angled triangles,
etc.); and he must learn to know the longest edge of each triangle,
that he may be able to place it according to direction.
The children should be encouraged to invent, to give the dictation
exercises to one another, and to copy the simpler forms of the lesson
on blackboard or paper. Some duplicate copies in colored papers may be
made from their inventions, and the walls of the schoolroom ornamented
with them. It will be a pleasure to the little ones themselves, and
demonstrate to others how wonderful a gift this is and how charmingly
the children use it.
No exercise should be given without previous study, and in the first
year's teaching it is wiser to draw or make the figures before giving
the dictations. The materials, too, should be prepared beforehand, in
such a form that they can be given out readily and quietly by the
children at the opening of the exercise. To require a class of a dozen
or more pupils to wait while the kindergartner assorts and counts the
various colors and shapes of tablets to be used is positively to
invite loss of interest on the children's part, and to produce in the
teacher a hurry and worry and nervous tension which will infallibly
ruin the play.
Life Forms.
The Life forms are no longer absolute representations, but only more
or less suggestive images of certain objects, and thus show still more
clearly the orderly movement from concrete to abstract.
Hitherto in Life forms the child has produced more or less real
objects,--for instance, he built a miniature house, a fountain, a
chair, or a sofa. They were not absolutely real, and therefore in one
way merely images; but they were bodily images. He could place a
little dish on the table, a tiny cup on the edge of the fountain, a
doll could sit in the chair, and therefore they were all real for
purposes of play, at least.
With the tablets, however, the child can no longer make a chair,
though by a certain arrangement of them he can make an image of it.
The child will notice that many of the forms made with squares are
flat pictures of those made with the third gift, and with the addition
of the right isosceles triangles he can reproduce the facades of many
of the elaborate object forms of the fifth. The various triangles
differ greatly in their capabilities of producing Life forms, the
equilateral and the obtuse isosceles being especially deficient in
this regard and requiring to be combined with the other tablets. The
fact that both the right isosceles and right scalene triangles produce
Life forms in great variety seems to prove that, as Goldammer says,
"the right angle predominates in the products of human activity."
Symmetrical Forms.
The symmetrical forms are more varied and innumerable than those of
any other gift, and with the addition of the brilliant colors of the
pasteboard, or the soft shades of the wooden tablets, make figures
which are undeniably beautiful, and which are mosaic-like in their
effect.
The whirling figures are interesting and new, and the child with
developed eye and growing artistic taste will delight in their oddity,
and yet be able to find opposites and their intermediates and make
them as correctly as in the more methodical figures, where the exact
right and left balanced the upper and lower extremes. Here we note
that the equilateral and obtuse isosceles triangles, so ill fitted
to produce Life forms, lend themselves to forms of symmetry in great
variety. The various sequences of the latter in the third and fifth
gifts may of course be faithfully reproduced in surface-extension
with the tablets, and thus gain an added charm.
The amount of material given to the child is now a matter for the
decision of the kindergartner, and is dependent only on the ability of
the child to use it to advantage. This increase of material presents a
further difficulty, and it is time for us to add still another, that
is, to expect more of the child, and to require that he produce not
only something original, but something which shall, though simple, be
really beautiful.
Inventions in borders are a new and charming feature of this gift, and
the circular and oblong tablets as well as the squares and various
triangles are well adapted to produce them. The various borders laid
horizontally across the tablets may be divided by lines of sticks, and
thus make an effect altogether different from anything we have had
before.
Mathematical Forms.
The work with forms of knowledge, as has been fully shown, will be in
geometry than in arithmetic, to which indeed the gift is not
especially well adapted. In addition to the study and comparison of
the various forms, their lines and angles, we have a great variety of
figures to be produced by combination. We can make the nine regular
forms already mentioned in the introduction in a variety of ways, and
thus give new charm to the old truths. We must allow the child to
experiment by himself very frequently, and interpret to him his
discoveries when he makes them.
The Seventh Gift in Weaving.
The square tablets afford a valuable aid to the occupation of weaving,
as all the simple patterns can be formed with them, the child laying
them upon his table until he has mastered the numerical principle upon
which they are constructed. We can easily see how these same patterns
may be further utilized as designs for inlaid tiles, or parquetry
floors. Thus the seventh gift may introduce children to subsequent
practical life, and serve as a useful preparation for various branches
of art-work.
Seventh Gift Parquetry.
It is easy to see when we begin the practical use of the tablets that
the essential characteristics of the gifts in their progress from
solid to point are now becoming less marked, and that they begin to
merge into the occupations, which develop from point to solid. The
meeting-place of the two series is close at hand, and, like drops of
water fallen near each other, they tremble with impatience to rush
into one.
The inventions which the child makes with tablets he now very commonly
expresses a desire to give away, or to take home with him,--a thought
which he seldom had with the gifts, wishing rather to show them in
their place upon the tables. As this is a natural and legitimate
desire, a supplement to the seventh gift has been devised, consisting
of paper substitutes for the various forms, of the same size and
appropriate coloring, and to be had either plain or gummed on the
back. After the inventions have been made, they are easily transferred
to paper with parquetry, and so can be bestowed according to the will
of the inventor.
Group Work.
The parquetry of the seventh gift lends an added grace to cooperative
work, for the children can now combine all their material in one form
to decorate the room, or perhaps to send as a gift to an absent
playmate. They may make an inlaid floor for the doll's house, a
brightly colored windowpane for the sun to stream through, and with
larger forms may even design an effective border for the wainscoting
of the schoolroom.[65]
[65] "The utility of this united action is not to be
overlooked. The children all proceed according to one and the
same law, they all work to produce one and the same result,
the same purpose unites them all; in short, we see here in
the children's play all that forms the base of every human
society, all that renders it possible for men to act together
in organized communities, such as are the family, the state,
and the church. And to prepare for the future, to be mindful
even amidst play of that which a child will afterwards
require in order worthily to fill his place in the world,
ought surely not to be among the least important ends of an
education claiming to be in conformity with nature and
reason."--H. Goldammer, _The Kindergarten_, page 135.
The group work at the square tables is also carried on very fully with
the tablets, the symmetrical figures when the colors are well combined
being quite dazzling in beauty.
Color with Seventh Gift.
In this connection, a danger may be noted in the treatment of the
gifts, both by kindergartner and children. Color appears again here in
almost bewildering profusion after its long absence in the series, and
is another straw to prove that the wind is blowing strongly toward the
occupations. Many of the pasteboard tablets are of different colors on
the opposite sides, and though this is of great use in Beauty forms,
when properly treated, it is quite often unfortunate in forms of life,
unless careful attention is given to arranging the material
beforehand. The effect of a barn, for instance, with its front view
checkered with violet, red, and yellow squares, may be imagined, or
of a pigeon-house with a parti-colored green and blue roof, an orange
standard, and red supports. Yet these are no fancy pictures I have
painted, and if the child places the tablets in this fashion, they
are often allowed so to remain without criticism from the purblind
kindergartner. She even sometimes dictates, herself, extravagant and
vulgar combinations of color, such as a violet centre-piece with
green corners and an orange border.
There needs no reasoning to prove that such a person is radically
unfit to handle the subject of color-teaching, and is sure to corrupt
the children under her charge; for in general, if ordinarily well
trained, they should now be far beyond the stage in which they would
be satisfied with such crudity of combination. They have had their
season of "playing with brightness," as Mr. Hailmann calls it, and
should now begin to have really good ideas as to harmonious
arrangement of hues. If they have not, if they really seem to prefer
the pigeon-house or barn above mentioned, then they are viciously
ill-taught, or altogether deficient in color sense.
It has been noted that the older children often choose the light and
dark wooden tablets, for invention, rather than the gay pasteboard
forms; but this may be on account of the high polish of the wood, and
its novelty in this guise, rather than because, as has been suggested,
they have been surfeited with brightness.