Lecture
To this day, not only in our country but abroad as well, books on disorders of counting, and especially on methods for overcoming them, can be counted on one's fingers. The scarcity of research on acalculia and of the development of methods for overcoming it is due to a number of causes. One of them is the fact that acalculia is part of the syndrome of practically all forms of aphasia, in which the speech impairment often masks the defects of counting and of counting operations. For this reason, the task of restoring speech — which, as things have historically developed, has been handled by speech therapists — has always been regarded as the top priority. Psychologists and neuropsychologists became involved in this work and in scientific research in this direction considerably later. For lack of sufficient time to work with patients in the clinic, it was natural that patients were taught, first and foremost, to speak and understand speech, to write and to read, and no time was left over for sessions devoted to overcoming difficulties in counting. For this reason, the research and methodological aspects of acalculia failed to receive the attention they deserved. Another cause may be considered the difficulty of the subject of study itself — the psychology of counting and counting operations, the concept of the place-value structure of number, the types and causes of their impairment, the connection of counting with thinking, speech, cognitive processes, and so on. As a consequence of these difficulties, obstacles also arose to the development of methods for overcoming this most complex form of intellectual activity.
Neuropsychology itself, as a science, must take on the task of a deeper investigation of the types and forms of acalculia, of the syndromes, and of their systemic influence on the formation of its different forms. The complexity of acalculia lies also in the fact that it belongs not only to the syndromes of speech impairment (in aphasia), manifesting itself differently in each syndrome and requiring different methods of diagnostic investigation and of overcoming it, but also to the gnostic syndromes — spatial, optic-spatial, optic, and others.
We shall not dwell further on the reasons for the lack of a sufficient number and quality of books on acalculia, but will only say once more how important such books are, especially in the practice of restorative training (in adult patients) and of corrective and formative training in children with a developmental delay in the psyche or a mild deviation of it.
This book will not answer every question, but the reader will find in it the answers to the most important ones: "What are the types and causes of the impairment of counting and counting operations?" and "What are the methods for overcoming this defect?"
The author has decided to reissue this book because of the ever-growing need for such literature among practicing specialists working with adult patients suffering from local brain lesions and with children having problems in the development of mental activity. The need for such books grows as the number of patients with local brain lesions increases, along with the number of children with deviations in the development of the psyche who are studying in mainstream schools and experiencing insurmountable difficulties in mastering counting and counting operations. The book is being reissued without substantial changes or additions, apart from minor editorial corrections and an updating of the list of references to reflect current requirements. Over the years it has demonstrated its demand and is therefore now unavailable in libraries and shops. In connection with this, the author has begun further work on deepening the problem of the impairment of counting and counting operations, on developing new concepts, technologies, and techniques for restoring this complex and most important function of the human psychic sphere and of human life activity. The results of this work have already been partly published.
Before analyzing the psychological content of number and of counting operations, their structure, as well as the impairment of counting, we shall give a brief analysis of the history of the development of the concept of number and of counting, with the aim of achieving a deeper understanding of the whole complexity of this form of intellectual activity (IA). Counting has a complex history of origin and development. Thus, F. Engels held that the concept of number was borrowed exclusively from the external world and did not arise out of pure thought.
Number and counting are a product of human culture; their appearance is owed in large measure to the development of trade and of agricultural work. The history of the development of counting began with the ability to establish a correspondence between the number of objects (or parts of an object) requiring enumeration and the number of fingers on the hand. The ten fingers of the hands were the very first and most natural tool and means of counting. Later, notches on wood, pebbles, and the like began to be used as tools of counting. This is why, in Latin, counting was denoted by the word "calculus," which means "counting with pebbles." This word has survived to the present day and is also used in modern Russian, for example in калькулятор (kal'kulyator, "calculator"). During this period of the development of counting, speech, the word, did not yet play its specific role. The word served to designate the relation between groups of objects: the objects being counted and the tools of counting ("equal," "fewer," "more"). Later, distinguishing words appeared — "this," "that," "other" — which can be called the germs of "counting" words, of number words: "first," "second," and so on. Already in these first words the dual nature and function of number is expressed: they expressed the idea of quantity and the idea of order.
Indeed, the words "this" and "that," on the one hand, signify quantity — two, and on the other — order, i.e., first "this," then — "that." With the passage of time, special (verbal) names were found for each number of the natural series, i.e., during this period of the development of counting, speech had already become incorporated into its structure and had become the organizer of this process.
The modern system of numeration has traveled an enormous path of development before a correspondence and mutual conditioning was established within it between quantity and order, between real quantity and the sign denoting it, between the writing of a number in its own proper notation and its designation in speech. The history of the development of number and counting knows many systems of numeration, invented by various peoples. These systems differed in the structure by which the number was built, in the rules for writing it, and in the role of the word in forming the concept of number. However, all systems of numeration developed in one direction — toward creating an economical, generalized notation for number, the possibility of "reading" a number without the context accompanying it, and the formation of the concept of number.
The improvement of systems of numeration proceeded chiefly in the direction of developing the concept of number and the rules for operations with it. Many systems of numeration died out; those survived in which a synthesis of the two main qualities of number — order and quantity — was possible. Thus, the Ionic system, which applied the positional principle of writing and reading a number, was in its time the most perfected system; in it, for the first time, a combination of the place-value principle and the positional principle of constructing a number was outlined.
The positional principle, which reflects the dependence of the value of a number on the place it occupies in a series of numbers, and the place-value principle, which reflects the dependence of the value of the magnitude of a number on its place not only within the series but also within the rank itself, appear at the same time as the formation of logical thinking (Arzhanikov, 1917; Galanin, 1910; Galperin, 1960; Georgiev, 1960; Davydov, 1959; Menchinskaya, 1956; Nepomnyashchaya, 1960, and others).
The study of the history of the development of the concept of number and of operations with numbers made it possible to reveal how the process of the "objectification" of number took place, how the concept of number developed, and what role is played, in the formation of the concept of number, by the mastery of the historically elaborated means of reflecting number (mastery of the system of numeration).
The problem of the formation of the concept of number is bound up with the appropriation of a product of human culture, which the modern system of numeration represents, while the state of the concept of number in the subject is bound up with mastery of this system.
Number — that objective characteristic of all objects of the surrounding world — is separated from these objects thanks to the system of numeration, and thereby itself becomes a model of number and its objectification. The system of numeration therefore becomes, as it were, an instrument for mastering the very concept of number. The concept of number developed only thanks to, and together with, the development of the system of numeration and of concrete actions with numbers. It is not the verbal or the written designation of numbers, but precisely the place-value structure of the system of numeration, that promotes the reflection of the numerical characteristics of the surrounding objective world.
Our modern system of numeration makes use of all the achievements of previous systems for designating number. The decimal place-value principle is coordinated in the most rational way with the positional principle through the use of only nine digits, which represent the evolution of the first nine Egyptian (hieroglyphic) signs, and through the use of zero to designate empty places, which facilitates not only the reading of numbers but also arithmetic operations with them. The names of numbers are formed on the additive principle, as among the Greeks, and on the principle of subtraction, characteristic of Latin. The names of the numbers of the second ten in many European languages, including Russian, are derived from the names of the first ten and from the Greek word deka — ten, i.e., from the name of the digit and the name of the corresponding decimal place, with zero not being designated in the process: nineteen (9 + 10); nine hundred (9 x 100), and so on. Thus, the modern concept of number must inevitably include, above all, notions of the place-value structure of number and of its abstract and generalized character.
In an adult, however, the concept of number is so firmly established, and actions with it are so reduced in the composition of their operations and so automatized, that it is difficult to discern the connection of number with real actuality and its complex psychological structure. Broad possibilities for investigating the structure of this complex form of human knowledge are afforded by the study of the formation of number and counting in children, as well as by the study of the pathology of the function of counting, i.e., by genetic and neuropsychological methods of investigation.
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