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 materialthere 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.
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 ofthe 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.
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.
The analysis of the decagon.
In the material there are two decagon insets, one consistingof a whole decagon and the other of a decagon divided into ten triangles.
Page 281shows 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.
photographThe 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.
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 equivalentrectangle 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.
photographThe decagon and the rectangle can be composed of the same triangular insets.
photographThe triangular insets fitted into their metal plates.
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.
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, 32= 9; in that on the larger side, 42= 16; in that on the hypotenuse, 52= 25.
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 thesemovable 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 thenA 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.
photographShowing 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.
. . . . . . .
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.
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 thetriangles 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 centimeterssquare 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).
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 onemust 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 aboveA quadrangular pyramid (10 X 10 X 20 cm.)A triangular prism (10 X 20 cm.)A triangular prism equal to 1/3 of aboveThe corresponding pyramid (10 X 20 cm.)A cylinder (10 cm. diameter, 20 altitude)A cylinder equal to 1/3 of aboveA cone (10 cm. diameter, 20 altitude)A sphere (10 cm. diameter)An ovoid (maximum diameter 10 cm.)An ellipsoid (maximum diameter 10 cm.)Regular PolyhedronsTetrahedronHexahedron (cube)OctahedronDodecahedronIcosahedron
(The faces of these polyhedrons are in different colors.)
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 = 22.On top of these is placed the double prism, making a cube with 4 cm. on a side = 23.One of the seven cubes is put beside this = 24.
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 = 22.
On top of these is placed the double prism, making a cube with 4 cm. on a side = 23.
One of the seven cubes is put beside this = 24.
In front are placed two more of the seven cubes = 25.
On top are put the remaining four equal cubes = 26.
In this way we have made a cube measuring 8 cm. on a side. From this we see that:
23, 26have the form of a cube.22, 25have the form of a square.2, 24have a linear form.
The Cube of a Binomial:(a + b)3= a3+ b3+ 3a2b + 3ba.
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.
photographsHollow 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.
. . . . . . .
photographsDesigns 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 innercauseshave been adequately stimulated.
DRAWING
LINEAR GEOMETRIC DESIGN DECORATION
I already have mentioned the fact that the material of the geometric insets may be applied also to design.
It is through design that the child may be led to ponder on the geometric figures which he has handled, taken out, combined in numerous ways, and replaced. In doing this he completes an exercise necessitating much use of the reasoning faculties. Indeed, he reproduces all of the figures by linear design, learning to handle many instruments—the centimeter ruler, the double decimeter, the square, the protractor, the compass, and the steel pen used for line ruling. For this work we have included in the geometric material a large portfolio where, together with the pages reproducing the figures, there are also some illustrative sheets with brief explanations of the figures and containing the relative nomenclature. Aside from copying designs the child may copy also the explanatory notes and thus reproduce the whole geometry portfolio. These explanatory notes are very simple. Here, for example, is the one which refers to the square:
"Square:The side or base is divided into 10 cm. All the other sides are equal, hence each measures 10 cm. The square has four equal sides and four equal angles which are always right angles. The number 4 and the identity of the sides and angles are the distinguishing characteristics of the square."
The children measure paper and construct the figurewith attention and application that are truly remarkable. They love to handle the compasses and are very proud of possessing a pair.
One child asked her mother for a Christmas gift of "onelastdoll and a box of compasses," as if she were ending one epoch of her life and beginning another. One little boy begged his mother to let him accompany her when she went to buy the compass for him. When they were in the store the salesman was surprised to find that so young a child was to use the compass and gave them a box of the simplest kind. "Not those," protested the little fellow; "I want an engineer's compass;" and he picked out one of the most complicated ones. This was the very reason why he was so anxious to go with his mother.
As the children draw, they learn many particulars concerning the geometric figures: the sides, angles, bases, centers, median lines, radii, diameters, sectors, segments, diagonals, hypotenuses, circumferences, perimeters, etc. They do not, however, learn all this as so much dry information nor do they limit themselves to reproducing the designs in the geometry portfolio. Each child adds to his own portfolio other designs which he chooses and sometimes originates. The designs reproduced in the portfolio are drawn on plain white drawing paper with China inks, but the children's special designs are drawn on colored paper with different colored inks and with gildings (silver, gold). The children reproduce the geometric figures and then they fill them in with decorations made either with pen or water-colors. These decorations serve especially to emphasize, in a geometric analysis, the various parts of the figure, such as center, angles, circumference, medians, diagonals, etc.
The decorated motif is selected or else invented by the child himself. He is allowed the same freedom of choice in his backgrounds as he enjoys for his inks or water-colors. The observation of nature (flowers and their different parts—pollen, leaves, a section of some part observed under the microscope, plant seeds, shells, etc.) serves to nourish the child's æsthetic imagination. The children also have access to artistic designs, collections of photographs reproducing the great masterpieces, and Haeckel's famous work,Nature's Artistic Forms, all of which equipment is so interesting and delightful to a child.
The children work many, many hours on drawing. This is the time we seize for reading to them (see above p. 197) and almost all their history is learned during this quiet period of copy and simple decoration which is so conducive to concentration of thought.
Copying some design, or drawing a decoration which has been directly inspired by something seen; the choice of colors to fill in a geometric figure or to bring out, by small and simple designs, the center or side of the figure; the mechanical act of mixing a color, of dissolving the gildings, or of choosing one kind of ink from a series of different colors; sharpening a pencil, or getting one's paper in the proper position; determining through tentative means the required extension of the compass—all this is a complex operation requiring patience and exactitude. But it does not require great intellectual concentration. It is, therefore, a work of application rather than of inspiration; and the observation of each detail, in order to reproduce it exactly, clarifies and rests the mind instead of rousing it to the intense activity demanded by the labor of association and creation. Thechild is busy with his hands rather than with his mind; but yet his mind is sufficiently stimulated by this work as not easily to wander away into the world of dreams.
These are quiet hours of work in which the children use only a part of their energies, while the other part is reaching out after something else; just as a family sits quietly by the fireside in long winter evenings engaged in light manual labors requiring little intelligence, watching the flames with a sense of enjoyment, willing to pass in this way many peaceful hours, yet feeling that a certain side of their needs is not satisfied. This is the time chosen for story telling or for light reading. Similarly this is the best time for our little children to listen to reading of all kinds.
During these hours they listened to the reading of books likeThe Betrothed(of Manzoni), psychological books like Itard'sEducation of the Savage of Aveyron, or historical narratives. The children took a deep interest in the reading. Each child may be occupied with his own design as well as with the facts which he is hearing described. It seems as though the one occupation furnishes the energy necessary for perfection in the other. The mechanical attention which the child gives to his design frees his mind from idle dreaming and renders it more capable of completely absorbing the reading that is going on; and the pleasure gained from the reading which, little by little, penetrates his whole being seems to give new energy to both hand and eye. His lines become most exact and the colors more delicate.
When the reading has reached some point of climax we hear remarks, exclamations, applause or discussions, which animate and lighten the work without interrupting it. But there are times when, with one accord, our childrenabandon their drawing so as to act out some humorous selection or to represent an historical fact which has touched them deeply; or, indeed, as happened during the reading of theSavage of Aveyron, their hands remained almost unconsciously raised in the intensity of their emotion, while on their faces was an expression of ecstasy, as if they were witnessing wonderful unheard-of things. Their actions seemed to interpret the well-known sentiment: "Never have I seen woman like unto this."
Artistic Composition with the Insets:Our geometric insets, which are all definitely related to one another in dimensions and include a series of figures which can be contained one within the other, lend themselves to very beautiful combinations. With these the children make real creations and often follow out their artistic ideas for days and even weeks. By moving the small pieces or by combining them in different ways on the white background, these very insets produce various decorations. The ease with which the child may form designs by arranging the little pieces of iron on a sheet of paper and then outlining them, and the harmony which is thus so easily obtained, affords endless delight. Really wonderful pieces of work are often produced in this way.
During these periods of creative design, as indeed during the periods of drawing from life, the child is deeply and wholly concentrated. His entire intellect is at work and no kind of instructive reading would be at all fitting while he is engaged in drawing or designing of this nature.
With the insets, as we have said, we have reproduced some of the classic decorations so greatly admired in theItalian masterpieces; for instance, those of Giotto in Florentine Art. When the children try with the insets to reproduce these classic decorations from photographs they are led to make most minute observations, which may be considered a real study of art. They judge the relative proportions of the various figures in such a way that their eye learns to appreciate the harmony of the work. And thus, even in childhood, a fine æsthetic enjoyment begins to engage their minds on the higher and more noble planes.
FREE-HAND DRAWING—STUDIES FROM LIFE
All the preceding exercises are "formative" for the art of drawing. They develop in the child the manual ability to execute a geometric design and prepare his eye to appreciate the harmony of proportions between geometric figures. The countless observations of drawings, the habit of minute examination of natural objects, constitute so many preparatory drills. We can, however, say that the whole method, educating the eye and the hand at the same time and training the child to observe and execute drawings with intense application, prepares the mechanical means for design, while the mind, left free to take its flight and to create, is ready to produce.
It is by developing the individual that he is prepared for that wonderful manifestation of the human intelligence, which drawing constitutes. The abilityto see realityin form, in color, in proportion, to be master of the movements of one's own hand—that is what is necessary. Inspiration is an individual thing, and when a child possesses these formative elements he can give expression to all he happens to have.
There can be no "graduated exercises in drawing" leading up to an artistic creation. That goal can be attained only through the development of mechanical technique and through the freedom of the spirit. Thatis our reason for not teaching drawing directly to the child. We prepare him indirectly, leaving him free to the mysterious and divine labor of reproducing things according to his own feelings. Thus drawing comes to satisfy a need for expression, as does language; and almost every idea may seek expression in drawing. The effort to perfect such expression is very similar to that which the child makes when he is spurred on to perfect his language in order to see his thoughts translated into reality. This effort is spontaneous; and the real drawing teacher is in the inner life, which of itself develops, attains refinement, and seeks irresistibly to be born into external existence in some empirical form. Even the smallest children try spontaneously to draw outlines of the objects which they see; but the hideous drawings which are exhibited in the common schools, as "free drawings" "characteristic" of childhood, are not found among our children. These horrible daubs so carefully collected, observed, and catalogued by modern psychologists as "documents of the infant mind" are nothing but monstrous expressions of intellectual lawlessness; they show only that the eye of their child is uneducated, the hand inert, the mind insensible alike to the beautiful and to the ugly, blind to the true as well as to the false. Like most documents collected by psychologists who study the children of our schools, they reveal not the soul but the errors of the soul; and these drawings, with their monstrous deformities, show simply what the uneducated human being is like.
Such things are not "free drawings" by children.Free drawingsare possible only when we have afree childwho has been left free to grow and perfect himself in the assimilation of his surroundings and in mechanicalreproduction; and who when left free to create and express himself actually does create and express himself.
The sensory and manual preparation for drawing is nothing more than an alphabet; but without it the child is an illiterate and cannot express himself. And just as it is impossible to study the writing of people who cannot write, so there can be no psychological study of the drawings of children who have been abandoned to spiritual and muscular chaos. All psychic expressions acquire value when the inner personality has acquired value by the development of its formative processes. Until this fundamental principle has become an absolute acquisition we can have no idea of the psychology of a child as regards his creative powers.
Thus, unless we know how a child should develop in order to unfold his natural energies, we shall not know how drawing as a natural expression is developed. The universal development of the wondrous language of the hand will come not from a "school of design" but from a "school of the new man" which will cause this language to spring forth spontaneously like water from an inexhaustible spring. To confer the gift of drawing we must create an eye that sees, a hand that obeys, a soul that feels; and in this task the whole life must cooperate. In this sense life itself is the only preparation for drawing. Once we have lived, the inner spark of vision does the rest.
Designs formed by the use of the geometry squares, circles, and equilateral triangle, modified by free-hand drawing. In the design on the right the "flower" within the cross is made with compasses: the decorative detail in the arms of the cross and the circle in the center are free-hand. The design on the left is similar to a decoration in the Cathedral at Florence, in the windows round the apse.
Leave to man then this sublime gesture which transfers to the canvas the marks of creative divinity. Leave it free to develop from the very time when the tiny child takes a piece of chalk and reproduces a simple outline on the blackboard, when he sees a leaf and makes his first reproduction of it on the white page. Such a child is insearch of every possible means of expression, because no one language is rich enough to give expression to the gushing life within him. He speaks, he writes, he draws, he sings like a nightingale warbling in the springtime.
Let us consider, then, the "elements" which our children have acquired in their development with reference to drawing: they are observers of reality, knowing how to distinguish theformsandcolorsthey see there.