Example of writing done with pen, by a child five years. One-fourth reduction.Example of writing done with pen, by a child five years. One-fourth reduction.Translation: "We would like to wish a joyous Easter to the civil engineer Edoardo Talamo and the Princess Maria. We will ask them to bring their pretty children here. Leave it to me: I will write for all. April 7, 1909."
Translation: "We would like to wish a joyous Easter to the civil engineer Edoardo Talamo and the Princess Maria. We will ask them to bring their pretty children here. Leave it to me: I will write for all. April 7, 1909."
Graphic language, comprising dictation and reading, contains articulate language in its complete mechanism (auditory channels, central channels, motor channels), and, in the manner of development called forth by my method, is based essentially on articulate language.
Graphic language, therefore, may be considered from two points of view:
(a) That of the conquest of a new language of eminent social importance which adds itself to the articulate language of natural man; and this is the cultural significance which is commonly given to graphic language, which is therefore taught in the schools without any consideration of its relation to spoken language, but solely with the intention of offering to the social being a necessary instrument in his relations with his fellows.
(b) That of the relation between graphic and articulate language and, in this relation, of an eventual possibility of utilising the written language to perfect the spoken: a new consideration upon which I wish to insist and which gives to graphic language aphysiological importance.
Moreover, as spoken language is at the same time anatural functionof man and an instrument which he utilises for social ends, so written language may be considered in itself, in itsformation, as an organicensembleof new mechanisms which are established in the nervous system, and as an instrument which may be utilised for social ends.
In short, it is a question of giving to written language not only a physiological importance, but also aperiod of developmentindependent of the high functions which it is destined to perform later.
It seems to me that graphic language bristles with difficulties in its beginning, not only because it has heretofore been taught by irrational methods, but because we have tried to make it perform, as soon as it has been acquired, the high function of teachingthe written languagewhich has been fixed by centuries of perfecting in a civilised people.
Think how irrational have been the methods we have used! We have analysed the graphic signs rather than the physiological acts necessary to produce the alphabetical signs; and this without considering thatany graphic signis difficult to achieve, because the visual representation of the signs have no hereditary connection with the motor representations necessary for producing them; as, for example, the auditory representations of the word have with the motor mechanism of the articulate language. It is, therefore, always a difficult thing to provoke a stimulative motor action unless we have already established the movement before the visual representation of the sign is made. It is a difficult thing to arouse an activity that shall produce a motion unless that motion shall have been previously established by practice and by the power of habit.
Thus, for example, the analysis of writing intolittle straight lines and curveshas brought us to present to the child a sign without significance, which therefore doesnot interest him, and whose representation is incapable of determining a spontaneous motor impulse. The artificial act constituted, therefore, aneffortof the will which resulted for the child in rapid exhaustion exhibited in the form of boredom and suffering. To this effort was added the effort of constitutingsynchronouslythe muscular associations co-ordinating the movements necessary to the holding and manipulating the instrument of writing.
All sorts ofdepressingfeelings accompanied such efforts and conduced to the production of imperfect and erroneous signs which the teachers had to correct, discouraging the child still more with the constant criticism of the error and of the imperfection of the signs traced. Thus, while the child was urged to make an effort, the teacher depressed rather than revived his psychical forces.
Although such a mistaken course was followed, the graphic language, so painfully learned, was nevertheless to beimmediatelyutilised for social ends; and, still imperfect and immature, was made to do service in thesyntactical construction of the language, and in the ideal expression of the superior psychic centres. One must remember that in nature the spoken language is formed gradually; and it is already established inwordswhen the superior psychic centres use these words in what Kussmaul callsdictorium, in the syntactical grammatical formation of language which is necessary to the expression of complex ideas; that is, in the language of thelogical mind.
In short the mechanism of language is a necessary antecedent of the higher psychic activities which are toutilise it.
There are, therefore, two periods in the development of language: a lower one which prepares the nervouschannel and the central mechanisms which are to put the sensory channels in relation with the motor channels; and a higher one determined by the higher psychic activities which areexteriorizedby means of the preformed mechanisms of language.
Thus for example in the scheme which Kussmaul gives on the mechanism of articulate language we must first of all distinguish a sort of cerebral diastaltic arc (representing the pure mechanism of the word), which is established in the first formation of the spoken language. Let E be the ear, and T the motor organs of speech, taken as a whole and here represented by the tongue, A the auditory centre of speech, and M the motor centre. The channels EA and MT are peripheral channels, the former centripetal and the latter centrifugal, and the channel AM is the inter-central channel of association.
The centre A in which reside the auditive images of words may be again subdivided into three, as in the following scheme, viz.: Sound (So), syllables (Sy), and words (W).
That partial centres for sounds and syllables can really be formed, the pathology of language seems to establish, for in some forms of centro-sensory dysphasia, the patients can pronounce only sounds, or at most sounds and syllables.
Small children, too, are, at the beginning, particularly sensitive to simple sounds of language, with which indeed, and especially withs, their mothers caress them and attract their attention; while later the child is sensitive tosyllables, with which also the mother caresses him, saying: "ba, ba, punf, tuf!"
Finally it is the simple word, dissyllabic in most cases, which attracts the child's attention.
But for the motor centres also the same thing may be repeated; the child utters at the beginning simple or double sounds, as for examplebl,gl,ch, an expression which the mother greets with joy; then distinctly syllabic sounds begin to manifest themselves in the child:ga,ba; and, finally, the dissyllabic word, usually labial:mama.
We say that the spoken language begins with the child when the word pronounced by him signifies anidea; when for example, seeing his mother and recognising her he says "mamma;" and seeing a dog says, "tettè;" and wishing to eat says: "pappa."
Thus we considerlanguage begunwhen it is established in relation to perception; while the language itself is still, in its psycho-motor mechanism, perfectly rudimentary.
That is, when above the diastaltic arc where the mechanical formation of the language is still unconscious, the recognition of the word takes place, that is, the word is perceived and associated with the object which it represents, language is considered to have begun.
On this level,later, language continues the process of perfecting in proportion as the hearing perceives better the component sounds of the words and the psycho-motor channels become more permeable to articulation.
This is the first stage of spoken language, which has its own beginning and its own development, leading, through the perceptions, to theperfectingof the primordial mechanism of the language itself; and at this stage precisely is established what we callarticulate language, which will later be the means which the adult will have at his disposal to express his own thoughts, and which the adult will have great difficulty in perfecting or correcting when it has once been established: in fact a high stage of culture sometimes accompanies an imperfect articulate language which prevents the æsthetic expression of one's thought.
The development of articulate language takes place in the period between the age of two and the age of seven: the age ofperceptionsin which the attention of the child is spontaneously turned towards external objects, and the memory is particularly tenacious. It is the age also ofmotilityin which all the psycho-motor channels are becoming permeable and the muscular mechanisms establish themselves. In this period of life by the mysterious bond between the auditory channel and the motor channel of the spoken language it would seem that the auditory perceptions have the direct power ofprovokingthe complicated movements of articulate speech which develop instinctively after such stimuli as if awaking from the slumber of heredity. It is well known that it is only at this age that it is possible to acquire all the characteristic modulations of a language which it would be vain to attempt to establish later. The mother tongue alone is well pronounced because it was established in the period of childhood; and the adult who learns to speak a new language must bring to it the imperfections characteristic of the foreigner's speech: only children who under theage of seven years learn several languages at the same time can receive and reproduce all the characteristic mannerisms of accent and pronunciation.
Thus also thedefectsacquired in childhood such as dialectic defects or those established by bad habits, become indelible in the adult.
What develops later, thesuperiorlanguage, thedictorium, no longer has its origin in the mechanism of language but in the intellectual development which makes use of the mechanical language. As the articulate language develops by the exercise of its mechanism and is enriched by perception, thedictoriumdevelops with syntax and is enriched byintellectual culture. Going back to the scheme of language we see that above the arc which defines the lower language, is established thedictorium,D,—from which now come the motor impulses of speech—which is established asspoken languagefit to manifest the ideation of the intelligent man; this language will be enriched little by little by intellectual culture and perfected by the grammatical study of syntax.
Hitherto, as a result of a preconception, it has been believed that written language should enter only into the development of thedictorium, as the suitable means for the acquisition of culture and of permitting grammatical analysis and construction of the language. Since "spoken words have wings" it has been admitted that intellectual culture could only proceed by the aid of a language which was stable, objective, and capable of being analysed, such as the graphic language.
But why, when we acknowledge the graphic language as a precious, nay indispensable, instrument of intellectual education, for the reason that itfixes the ideasof men and permits of their analysis and of their assimilation in books, where they remain indelibly written as an ineffaceable memory of words which are therefore always present and by which we can analyse the syntactical structure of the language, why shall we not acknowledge that it isusefulin the more humble task offixingthewordswhich represent perception and of analysing their component sounds?
Compelled by a pedagogical prejudice we are unable to separate the idea of a graphic language from that of a function which heretofore we have made it exclusively perform; and it seems to us that by teaching such a language to children still in the age of simple perceptions and of motility we are committing a serious psychological and pedagogical error.
But let us rid ourselves of this prejudice and consider the graphic language in itself, reconstructing its psycho-physiological mechanism. It is far more simple than the psycho-physiological mechanism of the articulate language, and is far more directly accessible to education.
Writingespecially is surprisingly simple. For let us considerdictatedwriting: we have a perfect parallel with spoken language since amotoraction must correspond withheardspeech. Here there does not exist, to be sure, the mysterious hereditary relations between the heard speech and the articulate speech; but the movements of writing are far simpler than those necessary to the spoken word, and are performed by large muscles, all external,upon which we can directly act, rendering the motor channels permeable, and establishing psycho-muscular mechanisms.
This indeed is what is done by my method, whichprepares the movements directly; so that the psycho-motor impulse of the heard speechfinds the motor channels already established, and is manifested in the act of writing, like an explosion.
The real difficulty is in theinterpretation of the graphic signs; but we must remember that we are in theage of perceptions, where the sensations and the memory as well as the primitive associations are involved precisely in the characteristic progress of natural development. Moreover our children are already prepared by various exercises of the senses, and by methodical construction of ideas and mental associations to perceive the graphic signs; something like a patrimony of perceptive ideas offers material to the language in the process of development. The child who recognises a triangle and calls it a triangle can recognise a lettersand denominate it by the sounds. This is obvious.
Let us not talk of premature teaching; ridding ourselves of prejudices, let us appeal to experience which shows that in reality children proceed without effort, nay rather with evident manifestations of pleasure to the recognition of graphic signs presented as objects.
And with this premise let us consider the relations between the mechanisms of the two languages.
The child of three or four has already long begun his articulate language according to our scheme. But he finds himself in the period in which themechanism ofarticulate language is being perfected; a period contemporary with that in which he is acquiring a content of language along with the patrimony of perception.
The child has perhaps not heard perfectly in all their component parts the words which he pronounces, and, if he has heard them perfectly, they may have been pronounced badly, and consequently have left an erroneous auditory perception. It would be well that the child, by exercising the motor channels of articulate language should establish exactly the movements necessary to a perfect articulation,beforethe age of easy motor adaptations is passed, and, by the fixation of erroneous mechanisms, the defects become incorrigible.
To this end theanalysis of speechis necessary. As when we wish to perfect the language we first start children at composition and then pass to grammatical study; and when we wish to perfect the style we first teach to write grammatically and then come to the analysis of style—so when we wish to perfect thespeechit is first necessary that the speechexist, and then it is proper to proceed to its analysis. When, therefore, the childspeaks, but before the completion of the development of speech which renders it fixed in mechanisms already established, the speech should be analysed with a view to perfecting it.
Now, as grammar and rhetoric are not possible with the spoken language but demand recourse to the written language which keeps ever before the eye the discourse to be analysed, so it is with speech.
The analysis of the transient is impossible.
The language must be materialised and made stable. Hence the necessity of the written word or the word represented by graphic signs.
In the third stage of my method for writing, that is, composition of speech, is included theanalysis of the wordnot only into signs, but into the component sounds; the signs representing its translation. The child, that is,dividesthe heard word which he perceives integrally as aword, knowing also its meanings, into sounds and syllables.
Let me call attention to the following diagram which represents the interrelation of the two mechanisms for writing and for articulate speech.
The peripheric channels are indicated by heavy lines; the central channels of association by dotted lines; and those referring to association in relation to the development of the heard speech by light lines.The peripheric channels are indicated by heavy lines; the central channels of association by dotted lines; and those referring to association in relation to the development of the heard speech by light lines.Eear;Soauditory centre of sounds;Syauditory centre of syllables;Wauditory centre of word;Mmotor centre of the articulate speech;Texternal organs of articulate speech (tongue);Hexternal organs of writing (hand);MCmotor centre of writing;VCvisual centre of graphic signs;Vorgan of vision.
Eear;Soauditory centre of sounds;Syauditory centre of syllables;Wauditory centre of word;Mmotor centre of the articulate speech;Texternal organs of articulate speech (tongue);Hexternal organs of writing (hand);MCmotor centre of writing;VCvisual centre of graphic signs;Vorgan of vision.
Whereas in the development of spoken language the sound composing the word might be imperfectly perceived, here in the teaching of the graphic sign corresponding to the sound (which teaching consists in presenting to the child a sandpaper letter, naming itdistinctlyand makingthe childseeit andtouchit), not only is the perception of the heard soundclearlyfixed—separately and clearly—but this perception is associated with two others: the centro-motor perception and the centro-visual perception of the written sign.
The triangleVC,MC,Sorepresents the association of three sensations in relation with the analysis of speech.
When the letter is presented to the child and he is made to touch and see it, while it is being named, the centripetal channelsESo;H,MC,So;V,VC,Soare acting and when the child is made to name the letter, alone or accompanied by a vowel, the external stimulus acts inVand passes through the channelsV,VC,So,M,T; andV,CV,So,Sy,M,T.
When these channels of association have been established by presenting visual stimuli in the graphic sign, the corresponding movements of articulate language can be provoked and studied one by one in their defects; while, by maintaining the visual stimulus of the graphic sign which provokes articulation and accompanying it by the auditory stimulus of the correspondingsounduttered by the teacher, their articulation can be perfected; this articulation is by innate conditions connected with theheard speech; that is, in the course of the pronunciation provoked by the visual stimulus, and during the repetition of the relative movements of the organs of language, the auditory stimulus which is introduced into the exercise contributes to the perfecting of the pronunciation of the isolated or syllabic sounds composing the spoken word.
When later the child writes under dictation, translating into signs the sounds of speech, he analyses the heard speech into its sounds, translating them into graphic movements through channels already rendered permeable by the corresponding muscular sensations.
Defects and imperfections of language are in part due to organic causes, consisting in malformations or in pathological alterations of the nervous system; but in part they are connected with functional defects acquired in the period of the formation of language and consist in an erratic pronunciation of the component sounds of the spoken word. Such errors are acquired by the child who hears words imperfectly pronounced, orhears bad speech, The dialectic accent enters into this category; but there also enter vicious habits which make the natural defects of the articulate language of childhood persist in the child, or which provoke in him by imitation the defects of language peculiar to the persons who surrounded him in his childhood.
The normal defects of child language are due to the fact that the complicated muscular agencies of the organs of articulate language do not yet function well and are consequently incapable of reproducing thesoundwhich was the sensory stimulus of a certain innate movement. The association of the movements necessary to the articulation of the spoken words is established little by little. The result is a language made of words with sounds which are imperfect and often lacking (whence incomplete words). Such defects are grouped under the nameblæsitasand are especially due to the fact that the child is not yet capable of directing the movements of his tongue. They comprise chiefly:sigmatismor imperfect pronunciation ofs;rhotacismor imperfect pronunciation ofr;lambdacismor imperfect pronunciation ofl;gammacismor imperfect pronunciation, ofg;iotacism, defective pronunciation of the gutturals;mogilalia, imperfect pronunciation of the labials, and according to some authors, as Preyer, mogilalia is made to include also the suppression of the first sound of a word.
Some defects of pronunciation which concern the utterance of the vowel sound as well as that of the consonant are due to the fact that the childreproduces perfectlysounds imperfectly heard.
In the first case, then, it is a matter of functional insufficiencies of the peripheral motor organ and hence of the nervous channels, and the cause lies in the individual; whereas in the second case the error is caused by the auditory stimulus and the cause lies outside.
These defects often persist, however attenuated, in the boy and the adult: and produce finally an erroneous language to which will later be added in writing orthographical errors, such for example as dialectic orthographical errors.
If one considers the charm of human speech one is bound to acknowledge the inferiority of one who does not possess a correct spoken language; and an æsthetic conception in education cannot be imagined unless special care be devoted to perfecting articulate language. Although the Greeks had transmitted to Rome the art of educating in language, this practice was not resumed by Humanism which cared more for the æsthetics of the environment and the revival of artistic works than for the perfecting of the man.
To-day we are just beginning to introduce the practice of correcting by pedagogical methods the serious defects of language, such as stammering; but the idea oflinguistic gymnasticstending to its perfection has not yet penetratedinto our schools as auniversal method, and as a detail of the great work of the æsthetic perfecting of man.
Some teachers of deaf mutes and intelligent devotees of orthophony are trying nowadays with small practical success to introduce into the elementary schools the correction of the various forms ofblæsitas, as a result of statistical studies which have demonstrated the wide diffusion of such defects among the pupils. The exercises consist essentially insilencecures which procure calm and repose for the organs of language, and in patientrepetitionof theseparatevowel and consonantsounds; to these exercises is added also respiratory gymnastics. This is not the place to describe in detail the methods of these exercises which are long and patient and quite out of harmony with the teachings of the school. But in my methods are to be found all exercises for the corrections of language:
(a)Exercises of Silence, which prepare the nervous channels of language to receive new stimuli perfectly;
(b)Lessonswhich consist first of the distinct pronunciation by the teacher offew words(especially ofnounswhich must be associated with a concrete idea); by this means clear and perfectauditory stimuliof language are started, stimuli which arerepeatedby the teacher when the child has conceived the idea of the object represented by the word (recognition of the object); finally of the provocation of articulate language on the part of the child who must repeatthat word alonealoud, pronouncing its separate sounds;
(c)Exercises in Graphic Language, which analyse the sounds of speech and cause them to be repeated separately in several ways: that is, when the child learns the separate letters of the alphabet and when he composes orwrites words, repeating their sounds which he translates separately into composed or written speech;
(d)Gymnastic Exercises, which comprise, as we have seen, bothrespiratory exercisesand those ofarticulation.
I believe that in the schools of the future the conception will disappear which is beginning to-day of "correcting in the elementary schools" the defects of language; and will be replaced by the more rational one ofavoiding them by caring for the development of languagein the "Children's Houses"; that is, in the very age in which language is being established in the child.
Children of three years already know how to count as far as two or three when they enter our schools. They thereforevery easilylearn numeration, which consistsin counting objects. A dozen different ways may serve toward this end, and daily life presents many opportunities; when the mother says, for instance, "There are two buttons missing from your apron," or "We need three more plates at table."
One of the first means used by me, is that of counting with money. I obtainnewmoney, and if it were possible I should have good reproductions made in cardboard. I have seen such money used in a school for deficients in London.
Themaking of changeis a form of numeration so attractive as to hold the attention of the child. I present the one, two, and four centime pieces and the children, in this way learn to count toten.
No form of instruction is morepracticalthan that tending to make children familiar with the coins in common use, and no exercise is more useful than that of making change. It is so closely related to daily life that it interests all children intensely.
Having taught numeration in this empiric mode, I pass to more methodical exercises, having as didactic materialone of the sets of blocks already used in the education of the senses; namely, the series of ten rods heretofore used for the teaching of length. The shortest of these rods corresponds to a decimetre, the longest to a metre, while the intervening rods are divided into sections a decimetre in length. The sections are painted alternately red and blue.
Some day, when a child has arranged the rods, placing them in order of length, we have him count the red and blue signs, beginning with the smallest piece; that is,one; one, two; one, two, three, etc., always going back to one in the counting of each rod, and starting from the side A. We then have him name the single rods from the shortest to the longest, according to the total number of the sections which each contains, touching the rods at the sidesB, on which side the stair ascends. This results in the same numeration as when we counted the longest rod—1, 2, 3, 4, 5, 6, 7, 8, 9, 10. Wishing to know the number of rods, we count them from the side A and the same numeration results; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. This correspondence of the three sides of the triangle causes the child to verify his knowledge and as the exercise interests him he repeats it many times.
We now unite to the exercises innumerationthe earlier, sensory exercises in which the child recognised the long and short rods. Having mixed the rods upon a carpet, the directress selects one, and showing it to the child, has him count the sections; for example, 5. She then asks him to give her the one next in length. He selects itby his eye, and the directress has himverifyhis choice byplacing the two pieces side by side and by counting their sections. Such exercises may be repeated in great variety and through them the child learns to assign aparticular name to each one of the pieces in the long stair. We may now call them piece number one; piece number two, etc., and finally, for brevity, may speak of them in the lessons as one, two, three, etc.
At this point, if the child already knows how to write, we may present the figures cut in sandpaper and mounted upon cards. In presenting these, the method is the same used in teaching the letters. "This is one." "This is two." "Give me one." "Give me two." "Whatnumberis this?" The child traces the number with his finger as he did the letters.
Exercises with Numbers.Association of the graphic sign with the quantity.
I have designed two trays each divided into five little compartments. At the back of each compartment may be placed a card bearing a figure. The figures in the first tray should be 0, 1, 2, 3, 4, and in the second, 5, 6, 7, 8, 9.
The exercise is obvious; it consists in placing within the compartments a number of objects corresponding to the figure indicated upon the card at the back of the compartment. We give the children various objects in order to vary the lesson, but chiefly make use of large wooden pegs so shaped that they will not roll off the desk. We place a number of these before the child whose part is to arrange them in their places, one peg corresponding to the card marked one, etc. When he has finished he takes his tray to the directress that she may verify his work.
The Lesson on Zero.We wait until the child, pointing to the compartment containing the card marked zero, asks, "And what must I put in here?" We then reply, "Nothing; zero is nothing." But often this is not enough. It is necessary to make the childfeelwhat we mean bynothing. To this end we make use of little games which vastly entertain the children. I stand among them, and turning to one of them who has already used this material, I say, "Come, dear, come to mezerotimes." The child almost always comes to me, and then runs back to his place. "But, my boy, you cameonetime, and I told you to comezerotimes." Then he begins to wonder. "But what must I do, then?" "Nothing; zero is nothing." "But how shall I do nothing?" "Don't do anything. You must sit still. You must not come at all, not any times. Zero times. No times at all." I repeat these exercises until the children understand, and they are then, immensely amused at remaining quiet when I call to them to come to me zero times, or to throw me zero kisses.They themselves often cry out, "Zero is nothing! Zero is nothing!"
When the children recognise the written figure, and when this figure signifies to them the numerical value, I give them the following exercise:
I cut the figures from old calendars and mount them upon slips of paper which are then folded and dropped into a box. The children draw out the slips, carry them still folded, to their seats, where they look at them and refold them,conserving the secret. Then, one by one, or in groups, these children (who are naturally the oldest ones in the class) go to the large table of the directress where groups of various small objects have been placed. Each one selectsthe quantityof objects corresponding to the number he has drawn. The number, meanwhile, has been leftat the child's place, a slip of paper mysteriously folded. The child, therefore, mustrememberhis number not only during the movements which he makes in coming and going, but while he collects his pieces, counting them one by one. The directress may here make interesting individual observations upon the number memory.
When the child has gathered up his objects he arranges them upon his own table, in columns of two, and if the number is uneven, he places the odd piece at the bottom and between the last two objects. The arrangement of the pieces is therefore as follows:—
The crosses represent the objects, while the circle stands for the folded slip containing the figure. Having arranged his objects, the child awaits the verification. The directress comes, opens the slip, reads the number, and counts the pieces.
When we first played this game it often happened that the children tookmore objectsthan were called for upon the card, and this was not always because they did not remember the number, but arose from a mania for the having the greatest number of objects. A little of that instinctive greediness, which is common to primitive and uncultured man. The directress seeks to explain to the children that it is useless to have all those things upon the desk, and that the point of the game lies in taking the exact number of objects called for.
Little by little they enter into this idea, but not so easily as one might suppose. It is a real effort of self-denial which holds the child within the set limit, and makes him take, for example, only two of the objects placed at his disposal, while he sees others taking more. I therefore consider this game more an exercise of will power than of numeration. The child who has thezero, should not move from his place when he sees all his companions rising and taking freely of the objects which are inaccessible to him. Many times zero falls to the lot of a child who knows how to count perfectly, and who would experience great pleasure in accumulating and arranging a fine group of objects in the proper order upon his table, and in awaiting with security the teacher's verification.
It is most interesting to study the expressions upon the faces of those who possess zero. The individual differences which result are almost a revelation of the "character" of each one. Some remain impassive, assuming abold front in order to hide the pain of the disappointment; others show this disappointment by involuntary gestures. Still others cannot hide the smile which is called forth by the singular situation in which they find themselves, and which will make their friends curious. There are little ones who follow every movement of their companions with a look of desire, almost of envy, while others show instant acceptance of the situation. No less interesting are the expressions with which they confess to the holding of the zero, when asked during the verification, "and you, you haven't taken anything?" "I have zero." "It is zero." These are the usual words, but the expressive face, the tone of the voice, show widely varying sentiments. Rare, indeed, are those who seem to give with pleasure the explanation of an extraordinary fact. The greater number either look unhappy or merely resigned.
We therefore give lessons upon the meaning of the game, saying, "It is hard to keep the zero secret. Fold the paper tightly and don't let it slip away. It is the most difficult of all." Indeed, after awhile, the very difficulty of remaining quiet appeals to the children, and when they open the slip marked zero it can be seen that they are content to keep the secret.
The didactic material which we use for the teaching of the first arithmetical operations is the same already used for numeration; that is, the rods graduated as to length which, arranged on the scale of the metre, contain the first idea of the decimal system.
The rods, as I have said, have come to be called by the numbers which they represent; one, two, three, etc. Theyare arranged in order of length, which is also in order of numeration.
The first exercise consists in trying to put the shorter pieces together in such a way as to form tens. The most simple way of doing this is to take successively the shortest rods, from one up, and place them at the end of the corresponding long rods from nine down. This may be accompanied by the commands, "Take one and add it to nine; take two and add it to eight; take three and add it to seven; take four and add it to six." In this way we make four rods equal to ten. There remains the five, but, turning this upon its head (in the long sense), it passes from one end of the ten to the other, and thus makes clear the fact that two times five makes ten.
These exercises are repeated and little by little the child is taught the more technical language; nine plus one equals ten, eight plus two equals ten, seven plus three equals ten, six plus four equals ten, and for the five, which remains, two times five equals ten. At last, if he can write, we teach the signsplusandequalsandtimes. Then this is what we see in the neat note-books of our little ones:
9 + 1 = 108 + 2 = 107 + 3 = 106 + 4 = 105 × 2 = 10
When all this is well learned and has been put upon the paper with great pleasure by the children, we call their attention to the work which is done when the pieces grouped together to form tens are taken apart, and put back in their original positions. From the ten last formed we take away four and six remains; from the next we take away three and seven remains; from the next, two and eight remains; from the last, we take away one and nineremains. Speaking of this properly we say, ten less four equals six; ten less three equals seven; ten less two equals eight; ten less one equals nine.
In regard to the remaining five, it is the half of ten, and by cutting the long rod in two, that is dividing ten by two, we would have five; ten divided by two equals five. The written record of all this reads:
10 − 4 = 610 − 3 = 710 − 2 = 810 − 1 = 910 ÷ 2 = 5
Once the children have mastered this exercise they multiply it spontaneously. Can we make three in two ways? We place the one after the two and then write, in order that we may remember what we have done, 2 + 1 = 3. Can we make two rods equal to number four? 3 + 1 = 4, and 4 - 3 = 1; 4 - 1 = 3. Rod number two in its relation to rod number four is treated as was five in relation to ten; that is, we turn it over and show that it is contained in four exactly two times: 4 ÷ 2 = 2; 2 × 2 = 4. Another problem: let us see with how many rods we can play this same game. We can do it with three and six; and with four and eight; that is,
2 × 2 = 43 × 2 = 64 × 2 = 85 × 2 = 1010 ÷ 2 = 58 ÷ 2 = 46 ÷ 2 = 34 ÷ 2 = 2
At this point we find that the cubes with which we played the number memory games are of help:
From this arrangement, one sees at once which are the numbers which can be divided by two—all those which have not an odd cube at the bottom. These are the even numbers, because they can be arranged in pairs, two by two; and the division by two is easy, all that is necessary being to separate the two lines of twos that stand one under the other. Counting the cubes of each file we have the quotient. To recompose the primitive number we need only reassemble the two files thus 2 × 3 = 6. All this is not difficult for children of five years.
The repetition 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 thedecimal numberswhich require certain lessons.