The repet.i.tion soon becomes monotonous, but the exercises may be most easily changed, taking again the set of long rods, and instead of placing rod number one after nine, place it after ten. In the same way, place two after nine, and three after eight. In this way we make rods of a greater length than ten; lengths which we must learn to name eleven, twelve, thirteen, etc., as far as twenty. The little cubes, too, may be used to fix these higher numbers.
Having learned the operations through ten, we proceed with no difficulty to twenty. The one difficulty lies in the _decimal numbers_ which require certain lessons.
LESSONS ON DECIMALS: ARITHMETICAL CALCULATIONS BEYOND TEN
The necessary didactic material consists of a number of square cards upon which the figure ten is printed in large type, and of other rectangular cards, half the size of the square, and containing the single numbers from one to nine. We place the numbers in a line; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Then, having no more numbers, we must begin over again and take the 1 again. This 1 is like that section in the set of rods which, in rod number 10, extends beyond nine. Counting along _the stair_ as far as nine, there remains this one section which, as there are no more numbers, we again designate as 1; but this is a higher 1 than the first, and to distinguish it from the first we put near it a zero, a sign which means nothing. Here then is 10. Covering the zero with the separate rectangular number cards in the order of their succession we see formed: 11, 12, 13, 14, 15, 16, 17, 18, 19. These numbers are composed by adding to rod number 10, first rod number 1, then 2, then 3, etc., until we finally add rod number 9 to rod number 10, thus obtaining a very long rod, which, when its alternating red and blue sections are counted, gives us nineteen.
The directress may then show to the child the cards, giving the number 16, and he may place rod 6 after rod 10. She then takes away the card bearing 6, and places over the zero the card bearing the figure 8, whereupon the child takes away rod 6 and replaces it with rod 8, thus making 18. Each of these acts may be recorded thus: 10 + 6 = 16; 10 + 8 = 18, etc. We proceed in the same way to subtraction.
When the number itself begins to have a clear meaning to the child, the combinations are made upon one long card, arranging the rectangular cards bearing the nine figures upon the two columns of numbers shown in the figures A and B.
A B +----+ +----+101010201030104010501060107010801090+----+ +----+
Upon the card A we superimpose upon the zero of the second 10, the rectangular card bearing the 1: and under this the one bearing two, etc.
Thus while the one of the ten remains the same the numbers to the right proceed from zero to nine, thus:
In card B the applications are more complex. The cards are superimposed in numerical progression by tens.
+----+1011121314151617181920+----+
Almost all our children count to 100, a number which was given to them in response to the curiosity they showed in regard to learning it.
I do not believe that this phase of the teaching needs further ill.u.s.trations. Each teacher may multiply the practical exercises in the arithmetical operations, using simple objects which the children can readily handle and divide.
CHAPTER XX
SEQUENCE OF EXERCISES
In the practical application of the method it is helpful to know the sequence, or the various series, of exercises which must be presented to the child successively.
In the first edition of my book there was clearly indicated a progression for each exercise; but in the "Children"s Houses" we began contemporaneously with the most varied exercises; and it develops that there exist _grades_ in the presentation of the material in its entirety. These grades have, since the first publication of the book, become clearly defined through experience in the "Children"s Houses."
SEQUENCE AND GRADES IN THE PRESENTATION OF MATERIAL AND IN THE EXERCISES
_First Grade_
As soon as the child comes to the school he may be given the following exercises:
Moving the seats, in silence (practical life).
Lacing, b.u.t.toning, hooking, etc.
The cylinders (sense exercises).
Among these the most useful exercise is that of the cylinders (solid insets). The child here begins to _fix his attention_. He makes his first comparison, his first selection, in which he exercises judgment.
Therefore he exercises his intelligence.
Among these exercises with the solid insets, there exists the following progression from easy to difficult:
(a) The cylinders in which the pieces are of the same height and of decreasing diameter.
(b) The cylinders decreasing in all dimensions.
(c) Those decreasing only in height.
_Second Grade_
_Exercises of Practical Life._ To rise and be seated in silence. To walk on the line.
_Sense Exercises._ Material dealing with dimensions. The Long Stair. The prisms, or Big Stair. The cubes. Here the child makes exercises in the recognition of dimensions as he did in the cylinders but under a very different aspect. The objects are much larger. The differences much more evident than they were in the preceding exercises, but here, _only the eye of the child_ recognises the differences and controls the errors. In the preceding exercises, the errors were mechanically revealed to the child by the didactic material itself. The impossibility of placing the objects in order in the block in any other than their respective s.p.a.ces gives this control. Finally, while in the preceding exercises the child makes much more simple movements (being seated he places little objects in order with his hands), in these new exercises he accomplishes movements which are decidedly more complex and difficult and makes small muscular efforts. He does this by moving from the table to the carpet, rises, kneels, carries heavy objects.
We notice that the child continues to be confused between the two last pieces in the growing scale, being for a long time unconscious of such an error after he has learned to put the other pieces in correct order.
Indeed the difference between these pieces being throughout the varying dimensions the same for all, the relative difference diminishes with the increasing size of the pieces themselves. For example, the little cube which has a base of 2 centimetres is double the size, as to base, of the smallest cube which has a base of 1 centimetre, while the largest cube having a base of 10 centimetres, differs by barely 1/10 from the base of the cube next it in the series (the one of 9 centimetres base).
Thus it would seem that, theoretically, in such exercises we should begin with the smallest piece. We can, indeed, do this with the material through which size and length are taught. But we cannot do so with the cubes, which must be arranged as a little "tower." This column of blocks must always have as its base the largest cube.
The children, attracted above all by the tower, begin very early to play with it. Thus we often see very little children playing with the tower, happy in believing that they have constructed it, when they have inadvertently used the next to the largest cube as the base. But when the child, repeating the exercise, _corrects himself of his own accord_, in a permanent fashion, we may be certain that _his eye_ has become trained to perceive even the slightest differences between the pieces.
In the three systems of blocks through which dimensions are taught that of length has pieces differing from each other by 10 centimetres, while in the other two sets, the pieces differ only 1 centimetre.
Theoretically it would seem that the long rods _should be the first to attract the attention_ and to exclude errors. This, however, is not the case. The children are attracted by this set of blocks, but they commit the greatest number of errors in using it, and only after they have for a long time eliminated every error in constructing the other two sets, do they succeed in arranging the Long Stair perfectly. This may then be considered as the most difficult among the series through which dimensions are taught.
Arrived at this point in his education, the child is capable of fixing his attention, with interest, upon the thermic and tactile stimuli.
The progression in the sense development is not, therefore, in actual practice identical with the theoretical progression which psychometry indicates in the study of its subjects. Nor does it follow the progression which physiology and anatomy indicate in the description of the relations of the sense organs.
In fact, the tactile sense is the _primitive_ sense; the organ of touch is the most _simple_ and the most widely diffused. But it is easy to explain how the most simple sensations, the least complex organs, are not the first through which to attract the _attention_ in a didactic presentation of sense stimuli.
Therefore, when the _education of the attention has been begun_, we may present to the child the rough and smooth surfaces (following certain thermic exercises described elsewhere in the book).
These exercises, if presented at the proper time, _interest_ the children _immensely_. It is to be remembered that these games are of the _greatest importance_ in the method, because upon them, in union with the exercises for the movement of the hand, which we introduce later, we base the acquisition of writing.
Together with the two series of sense exercises described above, we may begin what we call the "pairing of the colours," that is, the recognition of the ident.i.ty of two colours. This is the first exercise of the chromatic sense.
Here, also, it is only the _eye_ of the child that intervenes in the judgment, as it was with the exercises in dimension. This first colour exercise is easy, but the child must already have acquired a certain grade of education of the attention through preceding exercises, if he is to repeat this one with interest.
Meanwhile, the child has heard music; has walked on the line, while the directress played a rhythmic march. Little by little he has learned to accompany the music spontaneously with certain movements. This of course necessitates the repet.i.tion of the same music. (To acquire the sense of rhythm _the repet.i.tion of the same exercise is necessary_, as in all forms of education dealing with spontaneous activity.)
The exercises in silence are also repeated.