Construction of the number in the child

The construction of the number in the child is a way of research of the Psychologie of the development. The first scientific research on this subject, was carried out by Jean Piaget.

History

Pythagore then, later, Descartes, places the origin of the Intelligence in the gift of God: “it is of this universal doubt that, like fixed and immutable point, I solved to derive the Connaissance from God, of yourself, and all that the world contains. ” (Descartes, Research of the truth by the natural lights ). Darwin, in his Théorie of the evolution speaks about the “natural evolution of the intelligence”, and thus excludes God from his analysis.

Jean Piaget and Jean-Pierre Changeux will take up this idea with the “mental neuronal Darwinism”. Consequently, this topic of the construction of the number remained in the foreground of psychology développementale, and knew an emergence of many research, which led to many discoveries. Point of considering practice, one can think that these last could be exploited within the framework of the improvement of the training of the Arithmétique for the Enfant.

The theory of Piaget

For Piaget and Szeminska (1941; 1967), the concept of number in the child, occurs only at the time of the Stade of the concrete operations, resting and exceeding former levels of acquisitions. The key test of this Piagétien stage, is that of the conservation of the number (Opération logico-mathematics), which east succeeds by the child around seven or eight years. It is thus only towards this age, that the child reaches the level of “conservation acquired”, where it aligns the same number of tokens as the experimenter, by exceeding the simple perception which induced it in error before. The child then gives answers characteristic of the acquisition of the Conservation according to Piaget.

This vision thus goes in the direction of a performance of the number acquired by the deployment of essential successive capacities, to end up reaching an optimal level, which corresponds to the period when the child becomes “preserving”. “Thus, Piaget conceived the genesis of the concept of number like a primarily endogenous process of coordination of actions becoming gradually reversible”.

But this design was strongly hustled these last years, with the assistance of the evolution of the technical possibilities offered to the contemporary researchers, who can today study more precisely the capacities of the children of age préverbal. This research could show a certain early numerical capacity.

The child and numbers before speaking

According to Fayol, Camos, and Roussel, it is by the entry of charge of the number, that the “numerical conduits” become possible, and thus make reach the child more pointed numerary aptitudes. Thus one can wonder, which are the aptitudes of counting, former to the age where the child with the capacity to charge the numbers, which will evolve/move thereafter.

Several authors were interested in this question, and we here will present two significant.

The first are Strauss and Curtis, which especially stuck to the possibilities of the babies of ten and twelve months to discriminate groups of two, three, four or five objects on photographs. The second independent variable is the fact of presenting groups of objects homogeneous or heterogeneous. The authors used the experimental Paradigme of the Habituation, and thus measured the duration of fixing of the child on the image.

The results showed that even for groups Hétérogène S, the babies are able to discriminate perfectly groups of two compared to three objects, but they cannot do it more but partially when it is a question of doing it with three and four elements. Thus, arrived at four and five, the babies do not manage any more to separate the two groups from objects.

Lastly, Lipton and Spelke wondered about the founded good of the results of other authors who advanced that the newborn could not discriminate four points of six, and thus set up a series of four experiments on a population of older children of six and nine months, in connection with discrimination of long sound sequences.

1st experiment

It was installation in order to study the discrimination of eight different natural sound elements (of the bells, the chirps, the buzzes…) compared to 16 units of the same type. The population is made up of eight boys and eight girls in low age, whose average reaches six months.

The procedure consists in setting up a first phase of familiarisation of the sounds containing eight and twelve sounds (six of the one and six of the other) alternatively in the high speaker of right-hand side then of left. At the time of the phase of test, the researchers measure the time of fixing of the high speaker towards which the child turns the head.

Result 1:

The analysis of variance examining the effects of the familiarisation (8 or 16), indicated a principal effect. Indeed, twelve of the sixteen infants longer turned the head towards the new sequences, providing a clear result in favor of successful discrimination.

2nd experiment :

It was installation in order to study the Discrimination of eight sound elements compared to twelve units of the same type. All being equal in addition compared to the preceding one, this experiment makes it possible to check the smoothness of discrimination of the six month old children.

Result 2:

The children did not show any significant answer of orientation at the time of the presentation of the sequences which varied of number. Thus, this result shows that to the six months age, the children are assigned by the report/ratio in the face between the units to discriminate, and that they could not make a success of this test, because this report/ratio is too small in comparison with experiment 1.

3rd experiment :

Being given the preceding result, the authors wondered whether this report/ratio decreased quickly or not according to the age of the subjects. They thus set up a group of nine month old children, and theirs made pass a test strictly identical that to the precedents.

Result 3:

These children, indeed, directed longer their head towards the Haut-parleur S which diffused unknown sequences (phase of familiarisation) that at the time of the phase of test. Thus, these results show that, only three months after, the children increased their discriminatory capacity.

4th experiment :

The purpose of this last experimentation is to complicate the test by decreasing the report/ratio of number of the sounds to be discriminated, in order to see up to which point the children progress. Thus the researchers reproduce an experimental procedure identical to the preceding one, but this time, the nine months subjects will have to discriminate between eight and ten sounds.

Result 4:

The babies showed a tendency to fix the new numerosities, but this tendency is not really significant, F (1,15) = 2,72 (p>.05).

Conclusion

This experiment is astonishing, and that under several aspects. First of all, it shows that the young children (six months) have the capacity to discriminate groups of relatively large “objects”. But also that there is a significant progress three months afterwards, i.e. at the nine months age. That means that there is a progress of the numerical capacities well, and that before to acquire the language well.

Calculation in the young child

In 1992, a researcher publishes an article on the capacity of calculation at the human baby, who will start many reactions.

Reaction to the impossible events

Wynn, in 1992, has establishes an experimental procedure, in order to study at babies of four and five months, their capacity to make simple calculations such as the Addition and the Soustraction.

Thus it uses small a Puppet theater S, with attracting characters the attention of the children (Mickey Mouse), and it introduces event impossible in order to measure the time of fixing of the child. This time will have to determine if the child “considers” the event possible, or transgressing a physical law.

In the situation of addition, the children react to the impossible event (1+1=1), by fixing it longer. In the same way, in the situation of subtraction, the author notes that it is the same for the event (2 -1=2).

Thus Wynn from of concluded that the children from four and five months, have precise capacities of the number, and not only one dichotomy between single and several. Moreover, one can note that to make a success of the test, the babies were to have acquired the Permanence of the object.

Criticisms made with Wynn and their answer by Wynn and Houdé

In the letter of the LPEQ , one can read a summary of a conference of Jacqueline Bideaud, which says that “the results of Wynn would be more related on the permanence of the object and the capacity of the child to represent two discrete objects (on the basis of space-time information in particular) that with preliminary precise arithmetic competences. This construction of discrete would be the root of exact counting”.

It is true that the question which one can put following the experiment of Wynn, it is one can be on the child treated the objects numerically well, and not in a total way, i.e. by “thinking” that a mickey more a mickey was going to make more than one. That would thus come from a physical and nonmathematical treatment.

Wynn thus, in answer, remade its experiment, but by integrating the impossible event 1+1=3. Thus, if the baby reacts to this event, one will have to exclude the fact that the child reasons in term of “more than one”. Made checking, the experiment confirmed the preceding one.

If one wished to compare the results of Piaget with the latter, it was necessary to set up an experiment with great numbers, and either that with small numbers, and it was necessary to test the child at the linguistic level. Thus, Olivier Houdé, carries out an experiment also based on the reactions to the impossible events, but in children of two and three years (age of the Langage articulated). The question that was posed the author was “the young child will be the cognitive, intellectual heir to the qualified baby who he was, one will find this synchronism of the reactions to the impossible events, 1+1=1 and 1+1=3, which was the criterion of a precise calculation? ”

Consequently, it will use the same of the same number object (Babar S), to test the numerical capacity of the children, in the situation of Wynn and Piaget.

In the child of nursery school, there is a shift between the first and the second experiment. Indeed, in the procedure of Piaget, one does not note a use of the number, whereas in the situation of Wynn, one notices it. Moreover, the two year old children react well vis-a-vis 1+1=1, but they do not react any more to the event 1+1=3. Thus, one does not find this “synchronism of the reactions to the impossible events” in the two year old child, which shows that it is able than the child of four or five months.

On the other hand, at three years, the children make a success of the observation again that 1+1=3 is impossible, but always fail in the situation of Piaget. Consequently, one can wonder that it is the reason.

Houdé explains to it not success of the two year old children, by the fact that the language could interfere on the number. Indeed, it is the period when the child learns, inter alia, the difference between Singulier and Pluriel, which can disturb it. It would be the strategy used for the experiment of Piaget by the three year old child (strategy length equalizes number), who would pose problem, and which would explain why the child fails the Piagétienne experiment, and not with that of Wynn. Indeed, this strategy is very often used by all, and it is often effective. However, in this precise case, it fails.

Houdé puts forth the assumption that the difference between the children of three years and those in seven years, is the capacity to inhibit this strategy current, in order to use the numerical strategy which is more adequate. The child of this age would be thus able to handle the number, but only under certain conditions.

In order to check this assumption, Houdé and Guichart set up an experiment basing itself on a computerized version of the test of Piaget. Thus the authors can use the mental Chronométrie in order to check if there is inhibition of strategy well to make a success of it.

Thus the researchers install some eight year old children in front of a screen of computer where one finds two lines of “digitized tokens”. The subjects must say if there is the same number of tokens on the line of bottom, that on that the top, and their answer must be given as soon as possible. However, two types of situations are presented. Sometimes the strategy “length equalizes number” is effective, whereas other times it are not it. The screen which will follow one or the other will be a screen where the strategy “length equalizes number” is effective.

If the first screen (Amorce) is that where the child must inhibit the strategy “length equalizes number”, then it spends more time to answer the second screen (Cible). On the other hand, if the starter coincides with the strategy “length equalizes number”, then the child answers more quickly.

At the time of the presentation of the first screen (first situation), the child inhibits the strategy “length equalizes number” to succeed, and thus spends more time to make a success of the second screen, because a time is necessary to allow the lifting of inhibition. That reveals a competition between two strategies.

Calculations with great numerosities

McCrink and Wynn carried out in 2003 an experiment, to answer the criticism of its preceding study, which says that the infant would not have real numerical capacity, but would be based on “specialized processes of advance which apply only to small numbers”.

Thus, twenty six six month old older children took part in the study and were divided into two Groupe S (additions Vs subtractions). Each group had roughly equal numbers of Garçon S and girl S. The subjects are placed in front of a screen of Ordinateur, and are exposed either with the procedure of addition, or with that of subtraction.

At the time of the first, five objects go down to the bottom of the screen, and are hidden shortly after by an ocular mask which entirely recovers them. At this point in time five additional objects emergent and disappear in their turn behind the ocular mask.

At the time of the second procedure (subtraction), they are not any more five, but ten objects which are dissimulated behind the ocular mask. Shortly after five objects move apart from the mask and leave the screen. In both cases, the mask is dropped, to let appear, either five, or ten objects, and thus the child discovers, or an impossible event, or a possible event.

Thus, the infants who saw an operation of addition looked at longer when five objects appeared (10,28 S) that when there were ten of them (7,35 S). While the infants who saw an operation of subtraction longer looked at the final screens with ten objects (9.13 S) that those with five (8.00 S).

Conclusion

Many a Recherche S installation in the last few years advanced much knowledge on the subject, but they still reveal many questions. Thus, one is brought to question itself on the operation of the Apprentissage of verbal counting around two three years. Simple calculations, such as the additions and subtractions are treated by the children of this age, by the installation of strategies, on which it is undoubtedly necessary to be based to set up an adequate training. But it is still necessary to increase our knowledge on the Genèse of the number in the child, in order to take the right directions.

All this so that “these algorithmic procedures leave then the place to a strategy of direct recovery of the result in memory. However, the recovery of these numerical facts seems more frequent for the addition than for the subtraction which would remain mainly solved by procedures of counting”. This last remark can also raise question, since we saw as the baby knows early capacities, as well of addition as of subtraction. Thus, we can think that other explanations are to be come on construction from the number in the child.

See too

Internal bonds

External bonds

Documents HTML:

Documents extracted the course of Content A., Universit3e libre de Bruxelles;
Memory of Amélie Lubin on the number;
Documents extracted the course of GUERINI C., Paris VIII;
Knowledge protonumeric in the primate and the young child Centers PsyCLÉ of the University of Provence;
Conference of Houdé O., of January 25th, 2000.

Note and references

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