1.2. The Concept of Number and Its Formation in Children

It is known that the concept of number in children is formed through a complex path — at first elementary notions of "multiplicity", denoted by number, later — of the quantity of concrete objects standing behind a number, and then the essential attribute of number is gradually singled out and this attribute is abstracted and generalized. The Russian scholar D.D. Galanin said that defining number as an aggregate of units is one-sided and incorrect. "I think that the concept of number is contained rather in a relation, for which the aggregate of counting units is a particular case". This characteristic of number is manifested most distinctly in complex operations with abstract number. Thus, in Galanin's view, the concept "greater (less) by several times" contains the concept of relation, which in no way reflects the notion of number as an aggregate of counting units. This complex nature of counting must be taken into account in forming the concepts of number and counting in children. As Galanin wrote, in order to form in a child a notion of number, it is not enough to teach him to enumerate objects, since in doing so the child at best acquires a notion of the singularity of objects and their aggregate, but no notion of quantity arises, since number as a definite quantity is not contained in the enumerated objects; the formation of this concept is possible only together with the formation of logical thinking.

Researchers, while disagreeing on many questions concerning the problem of number and counting, agreed on one point — that the formation of number in its genesis is based on the multiple and varied connections into which number enters, and that the discernment and evaluation of these connections becomes possible only with the involvement of higher forms of analysis, requiring a generalized and abstract perception of number and the ability to operate with number itself, rather than with its quantitative essence.

Thus, J. Piaget, investigating the development of which abilities in children is connected with number, established that these abilities concern the essential and fundamental properties of the number system, the initial assumptions about the nature and behavior of numbers that the average adult uses silently in the course of everyday arithmetic operations, because they are so deeply rooted in him that they seem self-evident. Number, says Piaget, makes it possible to group objects into a class, while at the same time establishing certain asymmetric relations among them (order — ordination). Therefore, mastering the concept of number presupposes understanding ordination, cardination (quantity), and their interrelation. Number organizes attention and perception and thus makes it possible to establish the conservation of quantity. But the ability to use the corresponding numerical characteristic does not yet guarantee an understanding of the quantitative aspect of number. For this, one must master not merely the counting of the elements of a set, but also the ordering of these elements on the basis of the place that each of them occupies in the series in relation to the rest.

Piaget found that in a child ordination is not in the required coordination with cardination, and therefore the child does not yet possess the concept of number. The comparison that Piaget draws between the logical operation of grouping and the numerical operation of grouping is of interest. The latter has a more precise composition, i.e., here the relation of the parts to the whole and the relation of the parts to one another are determined by the fact that any of the elements can be taken as a unit, through which all the others can be determined. Logical classes do not possess this property. Mastery of number contributes to the development of logical categorical thinking.

N.A. Menchinskaya, in her study of the formation of number as the result of the development of a complex system of connections between perceived sets and the denoting numerical series, found that the structure of the concept of number at first includes the multiple connections of number, presupposing the varied substitution of one number for others. Subsequently, these connections of numbers, formed in action with objects that are, moreover, situated within a visual spatial field, are used by the subject as automatically actualized numerical tables.

P.Ya. Galperin and his students invest the concept of number with a different content. In their studies, they proceed from an understanding of number as a reflection of quantitative relations between a set and an adopted measure, the magnitude of which is not constant but depends on the scale of the things with which the measuring operations are carried out. It is thanks to the measure that number acquires its quantitative value. Here number is not merely a conventional sign; it is the object of an action. In accordance with this understanding of number, Galperin and his students propose a different path for the formation of number and counting in children, which was subjected to careful theoretical and experimental analysis in the works of V.V. Davydov, N.I. Nepomnyashchaya, L.S. Georgiev, and others.

Galperin and his students showed that the formation of all mental actions passes through a series of complex stages — from their visual-active form to an abstract one, unfolding "in the mind", by way of a transition from a form expanded in terms of the composition of its operations to a contracted one, from one that unfolds voluntarily to one that is automatized, etc. This applies fully to the function of counting as well. At first, number and the counting operation constitute a visual-active form, later — a verbal one, and at the highest stage of formation this function is carried out on the ideal plane. The formation and development of the function of counting is closely connected with speech, which acts, on the one hand, as a means of expressing this complex system of knowledge, and, on the other — as the organizer of the activity of counting. These studies also note the role of the spatial factor in the formation of the concept of number and the function of counting in children.

This factor, i.e., the perception of the optic-spatial relations between objects subject to quantitative measurement, is assigned an unequal role by different authors in the formation of the concept of number and of operations with it. Some authors hold that the spatial factor is merely an external condition, a "spatial field" for operations with objects, while others consider it to be not only a condition but a constituent part of the structure of number, of its essence, and connected with number itself, forming a complex system of interaction between quantity and order.

Our studies have shown the crucial role of this factor in the formation of the concept of number and the function of counting in children. The perception and awareness of the spatial relations in which the objects being measured and the measure stand appear in their most distinct form at the first stage of the formation of the concept of number (the movement of the measure along the object being measured, the separation of one part from another, etc.), but it also remains present at later stages of the formation of number and counting.

Different authors also treat the psychological content of the concept of number differently. Thus, for some researchers number is a set of connections presupposing the substitution of one number for others (Menchinskaya, 1957); others represent number as a relation between a set and an adopted measure (Galperin, 1966); for still others, number is a sign occupying a special place in a temporal ordinal system and characterizing quantitative relations through a concrete set of units — it is an "abstract object" with which operations can be performed, and it is characterized not only through the "measure" (according to Galperin), but also through its movement along the object being measured (Davydov, 1962); for a fourth group, number is the coordination between ordination and cardination, in connection with which mastering the concept of number presupposes understanding the relationship between order and quantity and their interrelations (Piaget, 1965).

The process of the development of the meaning of number — from a concrete representation to a concept — is likewise understood differently: some authors hold that the meaning of number becomes conceptual owing to the accumulation of an ever-greater number of varied connections standing behind one and the same number (Menchinskaya, 1957); others believe that this development is possible owing to the word, which makes it possible to abstract from the real quantity represented in the sensory material (Galperin, 1966); still others maintain that mastering the concept is possible owing to an action performed with a special "abstract object" — number (Davydov, 1962); a fourth group does not investigate the path of the formation of number, but instead determines the reason why the concept of number has not yet arisen in the child: ordination is not in the required coordination with cardination (Piaget, 1965).

Thus, an analysis of the history of the development of numeral systems allows us to conclude that a numeral system constitutes a model of number (and not merely a designation), necessary for the objectification of number, which is an abstract object. It is precisely thanks to this objectification that the concept of number becomes possible (Tsvetkova, Oliva, 1978). The method proposed by V.V. Davydov of laying off numbers on an axis to the left and right of an initial point in accordance with an adopted measure (used to familiarize children with number and its content) presupposes the presence of a spatial factor in the concept of number, in operations with it, and in its use in teaching a person number and counting in ontogenesis. This method appears to us to be highly effective, since it reflects the psychological content of number.

Despite the differences among these studies, all the authors point to the complex psychological structure of number, its connection with spatial perception and with speech, and to the necessity of higher forms of analysis and synthesis for the formation of a full-fledged concept of number. Having passed through a complex path of formation and development, the concept of number in an adult becomes highly stable and flexible, while operations with number become contracted and automatized.

So then, what are numbers, what is the concept of number, and what is its psychological content? The concept of number in an adult may be conditioned by at least four parameters: the direct representation of the quantity standing behind the number; the position of the number within the system of other numerical signs, i.e., its position in the place-value grid (its place in the series of digits composing the number, and its place within the class); awareness of the internal composition of the number and its connections with other numbers; and understanding of the complex, indirect connection between the digital notation of a number and its expression in verbal form. Therefore, the simple quantitative notions that arise when they are designated verbally ("five", "seven", "nineteen", etc.) are always mediated by the well-known place-positional structure of the notation of the number.

Indeed, the digit five has a different quantitative characteristic depending on the place it occupies in the notation of a number: in one case this will be the number 5000 (five thousand), in another — 500 (five hundred), in a third — 500,000 (five hundred, but now of thousands). Here the dependence of the magnitude of the number not only on the place of the digit within the number (position), but also on its place within the class (order), is evident. However, in all these cases the natural digit 5 will appear, and its concrete value each time depends on the positional-place-value structure of the multi-digit number. This complex knowledge of the meaning of a digit in connection with its place in the place-value grid can be formed only on the basis of a person's visuospatial representations.

Furthermore, any number presupposes the presence of multiple connections with other numbers, which can be discovered by breaking it down into its constituent numbers. Thus, the number 25 is not perceived by a person as a group consisting of separate units; in his perception it breaks down into tens and units, and the meaning of this number is grasped only through the perception of its place-value structure. Potentially, however, it can also be perceived as 5 x 5, 30 - 5, 20 + 5, etc., i.e., the operation of decomposing a number into its constituent elements creates the possibility of obtaining one and the same number in various ways. All this speaks to the enormous richness of the potential connections standing behind a number. Naturally, the preservation of numerical concepts must be revealed not in the preservation of the external visual representations of the notation of a number — its digital composition — but in the preservation of those most complex connections of number with logical operations, spatial representations, speech, etc., which introduce number into a complex and coherent system of knowledge.

On the basis of an analysis of the history of the development of numeral systems and of the genesis of number, it can be asserted that the formation of the concept of number is connected with mastering a numeral system. The latter constitutes a model of number (and not merely a designation), necessary for the objectification of number, which is itself an abstract object. The state of the concept of number in a given subject is connected with the mastery and assimilation of the modern numeral system. This conclusion is based on the study of the history of the development of number and of counting operations, and on the results of the study of patients with "acalculia".

Thanks to the objectification of the concept of number, modern man does not need to traverse the entire historical path of its development in order to master this concept. Therefore we believe that the activity of mastering the place-positional numeral system is an activity the product of which is the concept of number. Of decisive importance for this conclusion is the fact that, in the process of learning, children can master the numeral system and the concept of number only with the help of an adult. The activity of mastering the numeral system and the concept of number develops in the same way as all other higher mental functions, gradually acquiring, in the course of interiorization, a "mental" and contracted form, the "folded" character of which does not allow its complex structure to be seen. J. Piaget wrote on this account: "The fundamental properties of the number system, the nature and behavior of numbers, are so deeply rooted that to the average adult they seem self-evident".(FOOTNOTE: Piaget J. The Child's Conception of Number. Moscow: Prosveshchenie, 1965.)

In the history of the theory of methods of teaching arithmetic, different views on the concept of number, and correspondingly on the methods of teaching computational operations, are also noted. One of these views, on the basis of which the so-called method of studying numbers was implemented, is connected with an understanding of number as something contemplated, something that can be represented. In this method, mastering the concept of number was proposed to be achieved by memorizing the numerical series (such a view was held, for example, by the German methodologist A.W. Grube).

Proponents of another, opposing direction (in particular, A.I. Goldenberg) maintained that the teaching of arithmetic should proceed not from "number to number", but from operation to operation. In their view, the concept of number, like every concept, is not subject to contemplation or representation. A very important argument against the method of studying numbers, advanced by D.D. Galanin, is that a fact retained in memory as the mere memorization of the composition of a number is immobile, capable of neither deformation nor development.

Mastery of the complex structure of number, of its concept, is a necessary prerequisite for the transition from the concept of number to operations with it. Counting operations, like the concept of number, are complex in their psychological structure, are included within the decimal numeral system, and depend on it. The complexity of counting operations is conditioned by a multitude of different factors, and above all by the presence of the decimal system and of abstract numbers with which a person must operate, by the nature of the computational operation itself and the quantities participating in it, by the ways in which the operation is carried out, by the participation of speech in it, etc. Thus, the processes of addition and subtraction have a different psychological structure depending on whether these operations take place within a ten or involve carrying over through it. Operations within a ten are performed using ready-made numerical groups, whereas an operation involving carrying over through a ten constitutes a complex chain of interconnected intermediate operations (for example, 33 + 28).

The operation of subtraction is more complex. Even counting down by ones — the system of reverse ordinal counting — is a difficult process, and these difficulties increase when one must count down not by ones but by small groups of units. The greatest complexity of the computational process is connected with those operations of subtraction that can be carried out only in a mediated way, involving a series of auxiliary techniques, for example, in subtraction with carrying over through a ten (for example, 55 - 8). In this case, subtraction becomes a mental activity that includes several sequential operations in its structure. Here the subject is required to have clear knowledge of the place-value structure of the number, the ability to divide the number appropriately and carry out the intermediate operations, and to retain the intermediate links in working memory, and all of this must take place against a background of stability of the overall program of activity, of activity level, and of the regulation of actions. In operations of subtraction, no less important a factor is the preservation of spatial representations, which allow the subject to establish, in the intermediate operations, the required direction of counting, which is expressed either in adding or in subtracting the intermediate results; for example, in subtracting 17 from 35 (35-17), in some cases the intermediate number must be added, and in others — subtracted:

No less psychologically complex and difficult are the processes of multiplication and division. Multiplication, in those cases when it goes beyond the limits of the well-known "multiplication table", automatized through past experience, can be represented as consisting of a series of sequential operations. Like other arithmetic operations, it requires firm preservation of the place-value structure of the number, the ability to find the internal composition of a number, taking into account the direction and sequence of the arrangement of the numbers, retaining the intermediate results obtained in memory, etc. Division, likewise requiring all the listed factors to be taken into account, is a more conscious process compared with multiplication, since multiplication within the limits of a previously well-established table can proceed automatically.

All this becomes possible only on the basis of instruction and the specially organized formation of the concept of number, which in an adult becomes firmly established, while operations with numbers become automatized. The psychological structure of counting and of counting operations is revealed in studies of their genesis in the child. It has been established that the formation of these mental actions passes through a series of stages — from their visual-active form to an abstract one, unfolding "in the mind". However, even at the highest stages of the formation of these mental actions — the concept of number and counting operations — they retain components of spatial number in determining its meaning (J. Piaget, 1965; P.Ya. Galperin, 1953; N.A. Menchinskaya, 1955; L.S. Tsvetkova, 1972, 1975, et al.; V.V. Davydov, 1957; N.I. Nepomnyashchaya, 1958; L.S. Georgiev, 1960).

A brief analysis of the history of the development of number, of its psychological content, and of counting operations testifies to the extraordinary complexity of the described type of intellectual activity (IA), which has its own specificity in psychological content, structure, and the regularities of its unfolding. Let us consider number and the performance of counting operations in the context of the structure and unfolding of IA.