GEOMETRY
I
PLANE GEOMETRY
The geometric insets used for sensorial exercises in the "Children's
House" made it possible for the child to become familiar with many
figures of plane geometry: the square, rectangle, triangle, polygon,
circle, ellipse, etc. By means of the third series of corresponding
cards, where the figures are merely outlined, he formed the habit of
recognizing a geometric figure represented merely by a line.
Furthermore, he has used a series of iron insets reproducing some of the
geometric figures which he previously had learned through the use of
wooden geometric insets. He used these iron insets to draw the outline
of a figure, which he then filled in with parallel lines by means of
colored pencils (an exercise in handling the instruments of writing).
The geometric material here presented to the elementary classes
supplements that used in the "Children's House." It is similar to the
iron insets; but in this material each frame is fastened to an iron
foundation of exactly the same size as the frame. Since each piece is
complete in itself, no rack is needed to hold them.
The frame of the inset is green, the foundation is white, and the inset
itself--the movable portion--is red. When the inset is in the frame, the
red surface and the green frame are in the same plane.
This material further differs from the other in that each inset is
composed not of a single piece, as in the first material, but of many
pieces which, when put together on the white foundation, exactly
reproduce the geometric figure there designated.
The use to which these modified insets may be put is most varied. The
main purpose is to facilitate the child's auto-education through
exercises in geometry and often through the solution of real problems.
The fact of being able actually to "handle geometric figures," to
arrange them in different ways, and to judge of the relations between
them, commands the child's absorbed attention. The putting together of
the insets, which deal with equivalent figures, reminds one of the
"games of patience"--picture puzzles--which have been invented for
children but which, while amusing them, have no definite educational
aim. Here, however, the child leaves the exercises with "clear concepts"
and not merely with general "notions" of the principles of geometry, a
thing which is very hard to accomplish by the methods common to the
older schools. The difference between like figures, similar figures, and
equivalent figures, the possibility of reducing every regular plane
figure to an equivalent rectangle, and finally the solution of the
theorem of Pythagoras--all these are acquired eagerly and spontaneously
by the child. The same may be said about work in fractions, which is
made most interesting by the exercises with the circular insets. The
real meaning of the word _fraction_, operations in fractions, the
reduction of common fractions to decimal fractions--all of this is
mastered and becomes perfectly clear in the child's mind.
These are formative conquests and at the same time a dynamic part of the
child's intellectual activity. A child who works spontaneously and for a
long period of time with this material not only strengthens his
reasoning powers and his character but acquires higher and clearer
cognitions, which increase his mental capacity. In his succeeding
spontaneous flights into the abstract he will show ability for
surprising progress. While a high school child is still wasting his
mental effort in trying to understand the relation between geometrical
figures, which it seems impossible for him to comprehend, our child in
the primary grades is "finding it out for himself" and is so elated by
his discovery that he immediately begins the search for other
geometrical relations. Our children gallop freely along over a smooth
road, urged on by the inner energy of their growing psychic organism,
while many other children plod on barefooted and in shackles over stony
paths.
Every positive conquest gained through objects with our method of
freedom--allowing the child to exercise himself at the time when he is
most ready for the exercise and permitting him to complete this
exercise--results in spontaneous abstractions. How is it possible to
lead a child to perform abstractions if his mind is not sufficiently
mature and he is without adequate information? These two points of
support are, as it were, the feet of the psychic man who is traveling
toward his highest mental activities. We shall always see the repetition
of this phenomenon. Every ulterior exercise of inner development, every
ulterior cognition, will lead the child to new and ever higher flights
into the realm of the abstract. It is well, however, to emphasize this
principle: that the mind, in order to fly, must leave from some point of
contact, just as the aeroplane starts from its hangar, and that it must
have reached a certain degree of maturity, as is the case with the small
bird when it tries its wings and starts on its first flight from the
nest where it was born and gained his strength. An aeroplane of
perpetual flight without a means of replenishing its supplies, and a
bird with only an "instinct of flight" without the process of
development that takes place from the egg to the first flight, are
things that do not exist.
A machine flying perpetually without need of replenishing the fuel for
its propelling energy, and an instinct without a corresponding organism,
are pure fancies. The same is true of the flight of man's imagination,
which soars through space and creates. Though this is the mind's "manner
of being," its "highest instinct," yet it also needs to find support in
reality, to organize its inner forces from time to time. The longer a
material can claim and hold a child's attention, the greater promise it
gives that an "abstract process," an "imaginative creation" will follow
as the result of a developed potentiality. This creative imagination,
which is ever returning to reality to gain inspiration and to acquire
new energies, will not be a vain, exhaustible, and fickle thing, like
the so-called imagination which our ordinary schools are trying to
develop.
Without positive replenishment in reality there never will be a
spontaneous flight of the mind; this is the unsurmountable difficulty of
the common schools in their attempt to "develop the imagination" and to
"lead to education." The child who without any impelling force from
within is artificially "borne aloft" by the teacher, who forces him into
the "abstract," can at most learn only how to descend slowly like a
parachute. He can never learn to "lift himself energetically to dizzy
heights." This is the difference; hence the necessity for considering
the positive basis which holds the mind of the child to systematic
auto-exercises of preparation. After this it suffices merely to grant
freedom to the child's genius in order that it may take its own flight.
I need not repeat that even in the period of replenishing, freedom is
the guide in finding the "particular moment" and the "necessary time";
for I already have spoken insistently and at length concerning this. It
is well, however, to reaffirm here even more clearly that a material for
development predetermined by experimental research and put into relation
with the child (through lessons) accomplishes so complete a work by the
psychic reactions which it is capable of stimulating that marvelous
phenomena of intellectual development may be obtained. These geometric
insets furnish rich materials for the application of this principle and
respond wonderfully to the "instinct for work" in the child mind.
The exercises with this material not only are exercises of composition
with the pieces of an inset or of the substitution of them into their
relative metal plates; they are also exercises in drawing which, because
of the labor they require, allow the child to take cognizance of every
detail and to meditate upon it.
The designing done with these geometric insets, as will be explained, is
of two kinds: geometric and artistic (mechanical and decorative). And
the union of the two kinds of drawings gives new ways of applying the
material.
The geometric design consists in reproducing the figure outlined by the
corresponding insets. In this way the child learns to use the different
instruments of drawing--the square, the ruler, the compass, and the
protractor. In these exercises he acquires, with the aid of the special
portfolio which comes with the material, actual and real cognitions in
geometry.
Artistic designs are made by combining the small pieces of the various
geometric insets. The resulting figures are then outlined and filled in
with colored pencils or watercolors. Such combinations on the part of
the child are real esthetic creations. The insets are of such reciprocal
proportions that their combination results in an artistic harmony which
facilitates the development of the child's esthetic sense. With our
insets we were able to reproduce some of the classic decorations found
in our masterpieces of art, such as decorations by Giotto.
A combination of geometric design and artistic design is formed by
decorating the different parts of the geometric figure--as the center,
the sides, the angles, the circumference, etc.; or by elaborating with
free-hand details the decorations which have resulted from the
combination of the insets. But a far better concept of all this will be
gained as we pass on to explain our didactic material.
II
THE DIDACTIC MATERIAL USED FOR GEOMETRY
EQUIVALENT, IDENTICAL AND SIMILAR FIGURES
FIRST SERIES OF INSETS: _Squares and Divided Figures._ This is a series
of nine square insets, ten by ten centimeters, each of which has a white
foundation of the same size as the inset.
One inset consists of an entire square; the others are made up in the
following manner:
A square divided into two equal rectangles
" " " " four equal squares
" " " " eight equal rectangles
" " " " sixteen equal squares
" " " " two equal triangles
" " " " four equal triangles
" " " " eight equal triangles
" " " " sixteen equal triangles
The child can take the square divided into two rectangles and the one
divided into two triangles and interchange them: that is, he can build
the first square with triangles and the second with rectangles. The two
triangles can be superimposed by placing them in contact at the under
side where there is no knob, and the same can be done with the
rectangles, thus showing their equivalence by placing one on the other.
But there also is a certain relation between the triangles and the
rectangles; indeed, they are each half of the same square; yet they
differ greatly in form. Inductively the child gains an idea of
equivalent figures. The two triangles are identical; the two rectangles
also are identical; whereas the triangle and the rectangle are
equivalents. The child soon makes comparisons by placing the triangle on
the rectangle, and he notices at once that the small triangle which is
left over on the rectangle equals the small triangle which remains
uncovered on the larger triangle, and therefore that the triangle and
the rectangle, though they do not have the same form, have the same
area.
[Illustration]
This exercise in observation is repeated in a like manner with all the
other insets, which are divided successively into four, eight, and
sixteen parts. The small square which is a fourth of the original
square, resulting from the division of this latter by two medial lines,
is equivalent to the triangle which was formed by dividing this same
original square into four triangles by two diagonal lines. And so on.
By comparing the different figures the child learns the difference
between _equivalent_ figures and _identical_ figures. The two rectangles
are the result of dividing the large square by a medial line and are
identical; the two triangles are formed by dividing the original square
by a diagonal line, etc. _Similar_ figures, on the other hand, are those
which have the same form but differ in dimension. For example, the
rectangle which is half of the original square and the one which is half
of the smaller square--that is, an eighth of the original square--are
neither identical nor equivalent but they are _similar_ figures. The
same may be said of the large square and of the smaller ones which
represent a fourth, a sixteenth, etc.
Through these divisions of the square an idea of fractions is gained
intuitively. However, this is not the material used for the study of
fractions. For this purpose there is another series of insets.
SECOND SERIES OF INSETS: _Fractions._ There are ten metal plates, each
of which has a circular opening ten centimeters in diameter. One inset
is a complete circle; the other circular insets are divided respectively
into 2, 3, 4, 5, 6, 7, 8, 9, and 10 equal parts.
[Illustration]
The children learn to measure the angles of each piece, and so to count
the degrees. For this work there is a circular piece of white
card-board, on which is drawn in black a semicircle with a radius of the
same length as that of the circular insets. This semicircle is divided
into 18 sectors by radii which extend beyond the circumference on to the
background; and these radii are numbered by tens from 0° to 180°. Each
sector is then subdivided into ten parts or degrees.
[Illustration]
The diameter from 0° to 180° is outlined heavily and extends beyond the
circumference, in order to facilitate the adjustment of the angle to be
measured and to give a strict exactness of position. This is done also
with the radius which marks 90°. The child places a piece of an inset in
such a way that the vertex of the angle touches the middle of the
diameter and one of its sides rests on the radius marked 0°. At the
other end of the arc of the inset he can read the degrees of the angle.
After these exercises, the children are able to measure any angle with a
common protractor. Furthermore, they learn that a circle measures 360°,
half a circle 180°, and a right angle 90°. Once having learned that a
circumference measures 360° they can find the number of degrees in any
angle; for example, in the angle of an inset representing the seventh of
the circle, they know that 360° ÷ 7 = (approximately) 51°. This they can
easily verify with their instruments by placing the sector on the
graduated circle.
These calculations and measurements are repeated with all the different
sectors of this series of insets where the circle is divided into from
two to ten parts. The protractor shows approximately that:
1/3 circle = 120° and 360° ÷ 3 = 120°
1/4 " = 90° " 360° ÷ 4 = 90°
1/5 " = 72° " 360° ÷ 5 = 72°
1/6 " = 60° " 360° ÷ 6 = 60°
1/7 " = 51° " 360° ÷ 7 = 51°
1/8 " = 45° " 360° ÷ 8 = 45°
1/9 " = 40° " 360° ÷ 9 = 40°
1/10 " = 36° " 360° ÷ 10 = 36°
In this way the child learns to write fractions:
1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10
He has concrete impressions of them as well as an intuition of their
arithmetical relationships.
The material lends itself to an infinite number of combinations, all of
which are real arithmetical exercises in fractions. For example, the
child can take from the circle the two half circles and replace them by
four sectors of 90°, filling the same circular opening with entirely
different pieces. From this he can draw the following conclusion:
1/2 + 1/2 = 1/4 + 1/4 + 1/4 + 1/4.
He also may say that two halves are equal to four fourths, and write
accordingly:
2/2 = 4/4.
[Illustration]
This is merely the expression of the same thing. Seeing the pieces, he
has done an example mentally and then has written it out. Let us write
it according to the first form, which is, in reality, an analysis of
this example:
1/2 + 1/2 = 1/4 + 1/4 + 1/4 + 1/4.
When the denominator is the same, the sum of the fractions is found by
adding the numerators:
1/2 + 1/2 = 2/2; 1/4 + 1/4 + 1/4 + 1/4 = 4/4.
The two halves make an entire circle, as do the four fourths.
Now let us fill a circle with different pieces: for example, with a half
circle and two quarter circles. The result is 1 = 1/2 + 2/4. And in the
inset itself it is shown that 1/2 = 2/4. If we should wish to fill the
circle with the largest piece (1/2) combined with the fewest number of
pieces possible, it would be necessary to withdraw the two quarter
sectors and replace them by another half circle; result:
1 = 1/2 + 1/2 = 2/2 = 1.
Let us fill a circle with three 1/5 sectors and four 1/10 sectors:
1 = 3/5 + 4/10.
If the larger pieces are left in and the circle is then filled with the
fewest number of pieces possible, it would necessitate replacing the
four tenths by two fifths. Result:
1 = 3/5 + 2/5 = 5/5 = 1.
Let us fill the circle thus: 5/10 + 1/4 + 2/8 = 1.
Now try to put in the largest pieces possible by substituting for
several small pieces a large piece which is equal to them. In the space
occupied by the five tenths may be placed one half, and in that occupied
by the two eighths, one fourth; then the circle is filled thus:
1 = 1/2 + 1/4 + 1/4 = 1/2 + 2/4.
We can continue to do the same thing, that is to replace the smaller
pieces by as large a sector as possible, and the two fourths can be
replaced by another half circle. Result:
1 = 1/2 + 1/2 = 2/2 = 1.
All these substitutions may be expressed in figures thus:
5/10 + 1/4 + 2/8 = 1/2 + 1/4 + 1/4 =
1/2 + 2/4 = 1/2 + 1/2 = 2/2 = 1.
This is one means of initiating a child intuitively into the operations
used for the reduction of fractions to their lowest terms.
Improper fractions also interest them very much. They come to these by
adding a number of sectors which fill two, three, or four circles. To
find the whole numbers which exist under the guise of fractions is a
little like putting away in their proper places the circular insets
which have been all mixed up. The children manifest a desire to learn
the real operations of fractions. With improper fractions they originate
most unusual sums, like the following:
[8 + (7/7 + 18/9 + 24/2) + 1] =
------------------------------
8
[8 + (1 + 2 + 12) + 1]
---------------------- =
8
8 + 15 + 1
---------- = 24/8 = 3.
8
We have a series of commands which may be used as a guide for the
child's work. Here are some examples:
--Take 1/5 of 25 beads
--Take 1/4 " 36 counters
--Take 1/6 " 24 beans
--Take 1/3 " 27 beans
--Take 1/10 " 40 beans
--Take 2/5 " 60 counters
In this last there are two operations:
60 ÷ 5 = 12; 12 X 2 = 24; or 2 X 60 = 120; 120 ÷ 5 = 24, etc.
REDUCTION OF COMMON FRACTIONS TO DECIMAL FRACTIONS: The material for
this purpose is similar to that of the circular insets, except that the
frame is white and is marked into ten equal parts, and each part is then
subdivided into ten. In these subdivisions the little line which marks
the five is distinguished from the others by its greater length. Each of
the larger divisions is marked respectively with the numbers, 10, 20,
30, 40, 50, 60, 70, 80, 90, and 0. The 0 is at the top and there is a
raised radius against which are placed the sectors to be measured.
[Illustration]
To reduce a common fraction to a decimal fraction the sector is placed
carefully against the raised radius, with the arc touching the
circumference of the inset. Where the arc ends there is a number which
represents _the hundredths_ corresponding to the sector. For example, if
the 1/4 sector is used its arc ends at 25; hence 1/4 equals 0.25.
[Illustration]
[Illustration]
Page 275 shows in detail the practical method of using our material to
reduce common fractions to decimal fractions. In the upper figure the
segments correspond to 1/3, 1/4, and 1/8 of a circle are placed within
the circle divided into hundredths. Result:
1/3 + 1/4 + 1/8 = 0.70.
The lower figure shows how the 1/3 sector is placed: 1/3 = 0.33.
If instead we use the 1/5 sector we have: 1/5 = 0.20, etc.
Numerous sectors may be placed within the circle; for example:
1/4 + 1/7 + 1/9 + 1/10.
In order to find the sum of the fraction reduced to decimals, it is
necessary to read only the number at the outer edge of the last sector.
[Illustration]
Using this as a basis, it is very easy to develop an arithmetical idea.
Instead of 1, which represents the whole circle, let us write 100, which
represents its subdivisions when used for decimals, and let us divide
the 100 into as many parts of a circle as there are sectors in the
circle, and the reduction is made. All the parts which result are so
many hundredths. Hence:
1/4 = 100 ÷ 4 = 25 hundredths: that is, 25/100 or 0.25.
The division is performed by dividing the numerator by the demoninator:
1 ÷ 4 = 0.25.
THIRD SERIES OF INSETS: _Equivalent Figures._ Two concepts were given by
the squares divided into rectangles and triangles: that of fractions and
that of equivalent figures.
There is a special material for the concept of fractions which, besides
developing the intuitive notion of fractions, has permitted the solution
of examples in fractions and of reducing fractions to decimals; and it
has furthermore brought cognizance of other things, such as the
measuring of angles in terms of degrees.
For the concept of equivalent figures there is still another material.
This will lead to finding the area of different geometric forms and also
to an intuition of some theorems which heretofore have been foreign to
elementary schools, being considered beyond the understanding of a
child.
MATERIAL: Showing that a triangle is equal to a rectangle which has one
side equal to the base of the triangle, the other side equal to half of
the altitude of the triangle.
In a large rectangular metal frame there are two white openings: the
triangle and the equivalent rectangle. The pieces which compose the
rectangle are such that they may fit into the openings of either the
rectangle or the triangle. This demonstrates that the rectangle and the
triangle are equivalent. The triangular space is filled by two pieces
formed by a horizontal line drawn through the triangle parallel to the
base and crossing at half the altitude. Taking the two pieces out and
putting them one on top of the other the identity of the height may be
verified.
[Illustration]
Already the work with the beads and the squaring of numbers has led to
finding the area of a square by multiplying one side by the other; and
in like manner the area of a rectangle is found by multiplying the base
by half other. Since a triangle may be reduced to a rectangle, it is
easy to find its area by multiplying the base by half the height.
MATERIAL: Showing that a rhombus is equal to a rectangle which has one
side equal to one side of the rhombus and the other equal to the height
of the rhombus.
The frame contains a rhombus divided by a diagonal line into two
triangles and a rectangle filled with pieces which can be put into the
rhombus when the triangles have been removed, and will fill it
completely. In the material there are also an entire rhombus and an
entire rectangle. If they are placed one on top of the other they will
be found to have the same height. As the equivalence of the two figures
is demonstrated by these pieces of the rectangle which may be used to
fill in the two figures, it is easily seen that the area of a rhombus is
found by multiplying the side or base by the height.
[Illustration]
[Illustration]
* * * * *
MATERIAL: To show the equivalence of a trapezoid and a rectangle having
one side equal to the sum of the two bases and the other equal to half
the height.
The child himself can make the other comparison: that is, a trapezoid
equals a rectangle having one side equal to the height and the other
equal to one-half the sum of the bases. For the latter it is only
necessary to cut the long rectangle in half and superimpose the two
halves.
The large rectangular frame contains three openings: two equal
trapezoids and the equivalent rectangle having one side equal to the sum
of the two bases and the other side equal to half the height. One
trapezoid is made of two pieces, being cut in half horizontally at the
height of half its altitude; the identity in height may be proved by
placing one piece on top of the other. The second trapezoid is composed
of pieces which can be placed in the rectangle, filling it completely.
Thus the equivalence is proved and also the fact that the area of a
trapezoid is found by multiplying the sum of the bases by half the
height, or half the sum of the bases by the height.
[Illustration]
With a ruler the children themselves actually calculate the area of the
geometrical figures, and later calculate the area of their little
tables, etc.
MATERIAL: To show the equivalence between a regular polygon and a
rectangle having one side equal to the perimeter and the other equal to
half of the hypotenuse.
[Illustration: The analysis of the decagon.]
In the material there are two decagon insets, one consisting of a
whole decagon and the other of a decagon divided into ten triangles.
Page 281 shows a table taken from our geometry portfolio, representing
the equivalence of a decagon to a rectangle having one side equal to the
perimeter and the other equal to half the hypotenuse.
[Illustration: The bead number cubes built into a tower.]
The photograph shows the pieces of the insets--the decagon and the
equivalent rectangle--and beneath each one there are the small equal
triangles into which it can be subdivided. Here it is demonstrated that
a rectangle equivalent to a decagon may have one side equal to the whole
hypotenuse and the other equal to half of the perimeter.
Another inset shows the equivalence of the decagon and a rectangle which
has one side equal to the perimeter of the decagon and the other equal
to half of the altitude of each triangle composing the decagon. Small
triangles divided horizontally in half can be fitted into this figure,
with one of the upper triangles divided in half lengthwise.
Thus we demonstrate that the surface of a regular polygon may be found
by multiplying the perimeter by half the hypotenuse.
SOME THEOREMS BASED ON EQUIVALENT FIGURES
_A._ All triangles having the same base and altitude are equal.
This is easily understood from the fact that the area of a triangle is
found by multiplying the base by half the altitude; therefore triangles
having the same base and the same altitude must be equal.
For the inductive demonstration of this theorem we have the following
material: The rhombus and the equivalent rectangle are each divided
into two triangles. The triangles of the rhombus are different, for they
are divided by opposite diagonal lines. The three different triangles
resulting from these divisions have the same base (this can be actually
verified by measuring the bases of the different pieces) and fit into
the same long rectangle which is found below the first three figures.
Therefore, it is demonstrated that the three triangles have the same
altitude. They are equivalent because each one is the half of an
equivalent figure.
[Illustration: The decagon and the rectangle can be composed of the same
triangular insets.]
[Illustration: The triangular insets fitted into their metal plates.]
[Illustration]
_B._ THE THEOREM OF PYTHAGORAS: In a right-angled triangle the square of
the hypotenuse is equal to the sum of the squares of the two sides.
MATERIAL: The material illustrates three different cases:
First case: In which the two sides of the triangle are
equal.
Second case: In which the two sides are in the
proportion of 3:4.
Third case: General.
_First case:_ The demonstration of this first case affords an impressive
induction.
In the frame for this, shown below, the squares of the two sides are
divided in half by a diagonal line so as to form two triangles and the
square of the hypotenuse is divided by two diagonal lines into four
triangles. The eight resulting triangles are all identical; hence the
triangles of the squares of the two sides will fill the square of the
hypotenuse; and, vice versa, the four triangles of the square of the
hypotenuse may be used to fill the two squares of the sides. The
substitution of these different pieces is very interesting, and all the
more because the triangles of the squares of the sides are all of the
same color, whereas the triangles formed in the square of the hypotenuse
are of a different color.
[Illustration]
_Second case:_ Where the sides are as the proportion of 3:4.
In this figure the three squares are filled with small squares of three
different colors, arranged as follows: in the square on the shorter
side, 3^{2} = 9; in that on the larger side, 4^{2} = 16; in that on the
hypotenuse, 5^{2} = 25.
[Illustration: Second Case]
The substitution game suggests itself. The two squares formed on the
sides can be entirely filled by the small squares composing the square
on the hypotenuse, so that they are both of the same color; while the
square formed on the hypotenuse can be filled with varied designs by
various combinations of the small squares of the sides which are in two
different colors.
_Third case:_ This is the general case.
The large frame is somewhat complicated and difficult to describe. It
develops a considerable intellectual exercise. The entire frame measures
44 × 24 cm. and may be likened to a chess-board, where the movable
pieces are susceptible of various combinations. The principles already
proved or inductively suggested which lead to the demonstration of the
theorem are:
(1) That two quadrilaterals having an equal base and equal altitude are
equivalent.
(2) That two figures equivalent to a third figure are equivalent to each
other.
In this figure the square formed on the hypotenuse is divided into two
rectangles. The additional side is determined by the division made in
the hypotenuse by dropping a perpendicular line from the apex of the
triangle to the hypotenuse. There are also two rhomboids in this frame,
each of which has one side equal respectively to the large and to the
small square of the sides of the triangle and the other side equal to
the hypotenuse.
The shorter altitude of the two rhomboids, as may be seen in the figure
itself, corresponds to the respective altitudes, or shorter sides, of
the rectangles. But the longer side corresponds respectively to the side
of the larger and of the smaller squares of the sides of the triangle.
It is not necessary that these corresponding dimensions be known by the
child. He sees red and yellow pieces of an inset and simply moves them
about, placing them in the indentures of the frame. It is the fact that
these movable pieces actually fit into this white background which
gives the child the opportunity for reasoning out the theorem, and not
the abstract idea of the corresponding relations between the dimensions
of the sides and the different heights of the figures. Reduced to these
terms the exercise is easily performed and proves very interesting.
This material may be used for other demonstrations:
DEMONSTRATION A: _The substitution of the pieces._ Let us start with the
frame as it should be filled originally. First take out the two
rectangles formed on the hypotenuse; place them in the two lateral
grooves, and lower the triangle. Fill the remaining empty space with the
two rhomboids.
The same space is filled in both cases with:
A triangle plus two rectangles, and then
A triangle plus two rhomboids.
Hence the sum of the two rectangles (which form the square of the
hypotenuse) is equal to the sum of the two rhomboids.
In a later substitution we consider the rhomboids instead of the
rectangles in order to demonstrate their respective equivalence to the
two squares formed on the sides of the triangle. Beginning, for example
with the larger square, we start with the insets in the original
position and consider the space occupied by the triangle and the larger
square. To analyze this space the pieces are all taken out and then it
is filled successively by:
The triangle and the large square in their original positions.
The triangle and the large rhomboid.
[Illustration: Showing that the two rhomboids are equal to the two
rectangles.]
DEMONSTRATION B: _Based on Equivalence_. In this second demonstration
the relative equivalence of the rhomboid, the rectangles, and the
squares is shown outside the figure by means of the parallel indentures
which are on both sides of the frame. These indentures, when the pieces
are placed in them, show that the pieces have the same altitude.
This is the manner of procedure: Starting again with the original
position, take out the two rectangles and place them in the parallel
indentures to the left, the larger in the wider indenture and the
smaller in the narrower indenture. The different figures in the same
indenture have the same altitude; therefore the pieces need only to be
placed together at the base to prove that they are equal--hence the
figures are equal in pairs: the smaller rectangle equals the smaller
rhomboid and the larger rectangle equals the larger rhomboid.
Starting again from the original position you proceed analogously with
the squares. In the parallel indentures to the right the large square
may be placed in the same indenture with the large rhomboid, which,
however, must be turned in the opposite direction (in the direction of
its greatest length); and the smaller square and the smaller rhomboid
fit into the narrower indenture. They have the same altitude; and that
the bases are equal is easily verified by putting them together;
therefore here is proof that the squares and the rhomboids are
respectively equivalent.
Rectangles and squares which are equivalent to the same rhomboids are
equivalent to each other. Hence the theorem is proved.
. . . . . . . .
[Illustration: Showing that the two rhomboids are equal to the two
squares.]
This series of geometric material is used for other purposes, but they
are of minor importance.
FOURTH SERIES OF INSETS: _Division of a Triangle_. This material made up
of four frames of equal size, each containing an equilateral triangle
measuring ten centimeters to a side. The different pieces should fill
the triangular spaces exactly.
One is filled by an entire equilateral triangle.
One is filled by two rectangular scalene triangles, each equal to half
of the original equilateral triangle, which is bisected by dropping a
line perpendicularly to the base.
[Illustration]
The third is filled by three obtuse isosceles triangles, formed by lines
bisecting the three angles of the original triangle.
The fourth is divided into four equilateral triangles which are similar
in shape to the original triangle.
With these triangles a child can make a more exact analytical study than
he made when he was observing the triangles of the plane insets used in
the "Children's House." He measures the degrees of the angles and learns
to distinguish a right angle (90°) from an acute angle (<90°) and from
an obtuse angle (>90°).
Furthermore he finds in measuring the angles of any triangle that their
sum is always equal to 180° or to two right angles.
He can observe that in equilateral triangles all the angles are equal
(60°); that in the isosceles triangle the two angles at the opposite
ends of the unequal side are equal; while in the scalene triangle no two
angles are alike. In the right-angled triangle the sum of the two acute
angles is equal to a right angle. A general definition is that those
triangles are similar in which the corresponding angles are equal.
MATERIAL FOR INSCRIBED AND CONCENTRIC FIGURES: In this material, which
for the most part is made up of that already described, and which is
therefore merely an application of it, inscribed or concentric figures
may be placed in the white background of the different inset frames. For
example, on the white background of the large equilateral triangle the
small red equilateral triangle, which is a fourth of it, may be placed
in such a way that each vertex is tangent to the middle of each side of
the larger triangle.
There are also two squares, one of 7 centimeters on a side and the other
3.5. They have their respective frames with white backgrounds. The 7
centimeters square may be placed on the background of the 10 centimeters
square in such a way that each corner touches the middle of each side of
the frame. In like manner the 5 centimeters square, which is a fourth of
the large square, may be put in the 7 centimeters square; the 3.5
centimeters square in the 5 centimeters square; and finally the tiny
square, which is 1/16 part of the large square, in the 3.5 centimeters
square.
There is also a circle which is tangent to the edges of the large
equilateral triangle. This circle may be placed on the background of the
10 centimeters circle, and in that case a white circular strip remains
all the way round (concentric circles). Within this circle the smaller
equilateral triangle (1/4 of the large triangle) is perfectly inscribed.
Then there is a small circle which is tangent to the smallest
equilateral triangle.
Besides these circles which are used with the triangles there are two
others tangent to the squares: one to the 7 centimeters square and the
other to the 3.5 centimeters square. The large circle, 10 centimeters in
diameter, fits exactly into the 10 centimeters square; and the other
circles are concentric to it.
These corresponding relations make the figures easily adaptable to our
artistic composition of decorative design (see following chapter).
Finally, together with the other material, there are two stars which are
also used for decorative design. The two stars, or "flowers," are based
on the 3.5 centimeters square. In one the circle rests on the side as a
semi-circle (simple flower); and in the other the same circle goes
around the vertex and beyond the semi-circle until it meets the
reciprocal of four circles (flower and foliage).
III
SOLID GEOMETRY
Since the children already know how to find the area of ordinary
geometric forms it is very easy, with the knowledge of the arithmetic
they have acquired through work with the beads (the square and cube of
numbers), to initiate them into the manner of finding the volume of
solids. After having studied the cube of numbers by the aid of the cube
of beads it is easy to recognize the fact that the volume of a prism is
found by multiplying the area by the altitude.
In our didactic material we have three objects for solid geometry: a
prism, a pyramid having the same base and altitude, and a prism with the
same base but with only one-third the altitude. They are all empty. The
two prisms have a cover and are really boxes; the uncovered pyramid can
be filled with different substances and then emptied, serving as a sort
of scoop.
These solids may be filled with wheat or sand. Thus we put into practise
the same technique as is used to calculate capacity, as in anthropology,
for instance, when we wish to measure the capacity of a cranium.
It is difficult to fill a receptacle completely in such a way that the
measured result does not vary; so we usually put in a scarce measure,
which therefore does not correspond to the exact volume but to a smaller
volume.
One must know how to fill a receptacle, just as one must know how to do
up a bundle, so that the various objects may take up the least possible
space. The children like this exercise of shaking the receptacle and
getting in as great a quantity as possible; and they like to level it
off when it is entirely filled.
The receptacles may be filled also with liquids. In this case the child
must be careful to pour out the contents without losing a single drop.
This technical drill serves as a preparation for using metric measures.
By these experiments the child finds that the pyramid has the same
volume as the small prism (which is one-third of the large prism); hence
the volume of the pyramid is found by multiplying the area of the base
by one-third the altitude. The small prism may be filled with clay and
the same piece of clay will be found to fill the pyramid. The two solids
of equal volume may be made of clay. All three solids can be made by
taking five times as much clay as is needed to fill the same prism.
. . . . . . . .
Having mastered these fundamental ideas, it is easy to study the rest,
and few explanations will be needed. In many cases the incentive to do
original problems may be developed by giving the children definite
examples: as, how can the area of a circle be found? the volume of a
cylinder? of a cone? Problems on the total area of some solids also may
be suggested. Many times the children will risk spontaneous inductions
and often of their own accord proceed to measure the total surface area
of all the solids at their disposal, even going back to the materials
used in the "Children's House."
The material includes a series of wooden solids with a base measurement
of 10 cm.:
A quadrangular parallelopiped (10 X 10 X 20 cm.)
A quadrangular parallelopiped equal to 1/3 of above
A quadrangular pyramid (10 X 10 X 20 cm.)
A triangular prism (10 X 20 cm.)
A triangular prism equal to 1/3 of above
The corresponding pyramid (10 X 20 cm.)
A cylinder (10 cm. diameter, 20 altitude)
A cylinder equal to 1/3 of above
A cone (10 cm. diameter, 20 altitude)
A sphere (10 cm. diameter)
An ovoid (maximum diameter 10 cm.)
An ellipsoid (maximum diameter 10 cm.)
Regular Polyhedrons
Tetrahedron
Hexahedron (cube)
Octahedron
Dodecahedron
Icosahedron
(The faces of these polyhedrons are in different colors.)
APPLICATIONS: _The Powers of Numbers_.
MATERIAL: Two equal cubes of 2 cm. on a side; a prism twice the size of
the cubes; a prism double this preceding prism; seven cubes 4 cm. on a
side.
The following combinations are made:
The two smaller cubes are placed side by side = 2.
In front of these is placed the prism which is twice
as large as the cube = 2^{2}.
On top of these is placed the double prism, making a
cube with 4 cm. on a side = 2^{3}.
One of the seven cubes is put beside this = 2^{4}.
In front are placed two more of the seven cubes = 2^{5}.
On top are put the remaining four equal cubes = 2^{6}.
In this way we have made a cube measuring 8 cm. on a side. From this we
see that:
2^{3}, 2^{6} have the form of a cube.
2^{2}, 2^{5} have the form of a square.
2, 2^{4} have a linear form.
_The Cube of a Binomial:_ (a + b)^{3} = a^{3} + b^{3} + 3a^{2}b +
3b^{a}.
MATERIAL: A cube with a 6 cm. edge, a cube with a 4 cm. edge; three
prisms with a square base of 4 cm. on a side and 6 cm. high; three
prisms with a square base of 6 cm. to a side and 4 cm. high. The 10 cm.
cube can be made with these.
These two combinations are in special cube-shaped boxes into which the
10 cm. cube fits exactly.
. . . . . . . .
_Weights and Measures:_ All that refers to weights and measures is
merely an application of similar operations and reasonings.
The children have at their disposal and learn to handle many of the
objects which are used for measuring both in commerce and in every-day
life. In the "Children's House" days they had the long stair rods which
contain the meter and its decimeter subdivisions. Here they have a
tape-measure with which they measure floors, etc., and find the area.
They have the meter in many forms: in the anthropometer, in the ruler.
Then, too, they use the metal tape, the dressmaker's tape measure, and
the meterstick used by merchants.
[Illustration: Hollow geometric solids, used for determining equivalence
by measuring sand, sugar, etc.]
The twenty centimeter ruler divided into millimeters they use constantly
in design; and they love to calculate the area of the geometric figures
they have designed or of the metal insets. Often they calculate the
surface of the white background of an inset and that of the different
pieces which exactly fit this opening, so as to verify the former. As
they already have some preparation in decimals it is no task for them to
recognize and to remember that the measures increase by tens and take on
new names each time. The exercises in grammar have greatly facilitated
the increase in their vocabulary.
They calculate the reciprocal relations between length, surface, and
volume by going back to the three sets which first represented "long,"
"thick," and "large."
The objects which differ in length vary by 10's; those differing in
areas vary by 100's; and those which differ in volume vary by 1000's.
The comparison between the bead material and the cubes of the pink tower
(one of the first things they built) encourages a more profound study of
the sensory objects which were once the subject of assiduous
application.
By the aid of the double decimeter the children make the calculations
for finding the volume of all the different objects graded by tens, such
as the rods, the prisms of the broad stair, the cubes of the pink tower.
By taking the extremes in each case they learn the relations between
objects which differ in one dimension, in two dimensions, and in three
dimensions. Besides, they already know that the square of 10 is 100, and
the cube of 10 is 1000.
. . . . . . . .
[Illustration: Designs formed by arranging sections of the insets within
the frames.]
The children make use of various scientific instruments: thermometers,
distillers, scales, and, as previously stated, the principal measures
commonly used.
By filling an empty metal cubical decimeter, which like the geometric
solids is used for the calculation of volume, they have a liter measure
of water, which may be poured into a glass liter bottle. All the decimal
multiples and subdivisions of the liter are easily understood. Our
children spent much time pouring liquids into all the small measures
used in commerce for measuring wine and oil.
They distil water with the distiller. They use the thermometer to
measure the temperature of water in ebullition and the temperature of
the freezing mixture. They take the water which is used to determine the
weight of the kilogram, keeping it at the temperature of 4°C.
The objects which serve to measure capacity also are at the disposal of
the children.
There is no need to go into more details upon the multitudinous
consequences resulting from both a methodical preparation of the
intellect and the possibility of actually being in contact with real
objects.
A great number of problems given by us, as well as problems originated
by the children themselves, bear witness to the ease with which external
effects may he spontaneously produced when once the inner _causes_ have
been adequately stimulated.