Developmentol Dyscalculia -...

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Developmentol Dyscalculia

Ladislav Kosc, Ph.D.

SUMMARY A definition of developmental dyscalculia, stressing the hereditary or congenital affection of the brain substrate of mathematical func-tions, is put forth. This disorder is clearly distinguished from other forms of disturbed mathematical abilities. A classification of devel-opmental dyscalculia is then outlined, dis-tinguishing the following forms: verbal, practognostic, lexical, graphical, ideognostical and operational developmental dyscalculia. Finally an investigation is presented of mathe-matical abilities and disabilities in eleven-year-old pupils from normal schools in Bratislava, Czechoslovakia. A number of tests measuring symbolic functions were applied to 66 sus-pected dyscalculics with normal IQs who had neurological examinations. The tests are characterized and the results briefly described; some examples of concrete pathological solu-tions to test items are given. This investigation suggests that nearly 6% of children of the so-called normal population can be justifiably ex-pected to have symptoms of developmental dyscalculia as defined in this study.

The professions concerned with child devel-opment are relatively well informed about developmental dyslexia and dysgraphia, and many studies have been published throughout

the world. The concept of developmental dyscalculia, however, has so far remained nearly unknown, even though it actually belongs to the nervous system and even though it occurs at least as frequently as the other disorders. If one considers the great importance attributed today to mathematical knowledge in general and the amount of time devoted to the instruction of this subject in schools, it is hard to understand why this question has not been given more attention. Cohn (1968) explains this by suggest-ing that mathematical disability is not con-sidered as socially disabling as reading and writing disability. But failure in mathematics is not at all unusual, and some cases present rather serious problems. A more thorough study of this question seems justified.

MATHEMATICAL ABILITIES, DISPOSITIONS AND SKILLS For a solid understanding of developmental dyscalculia as a disorder of mathematical abili-ties, some fundamental concepts must be made clear. Mathematical abilities are "qualities which are a condition for the successful study and performance in mathematics" (Rican 1964, p. 367), or the ability "to comprehend the nature of the mathematical (and similar) prob-lems, methods and verifications; to learn, memorize, and reproduce them; to combine

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them with other problems, symbols, methods and verifications; to use them when solving mathematical (and similar) problems" (Verdelin 1958, p. 13).

A great number of neurological investiga-tions give proof of the existence of special dispositions (aptitudes) for mathematics. When special centers in the brain are damaged, dis-orders in the field of mathematical abilities result. These areas are therefore considered to be the anatomico-physiological bases of these abilities. The existence of genetic dispositions for mathematics is supported by the findings of Barakat (1951, p. 154f) who proposes the fol-lowing arguments:

(a) An investigation of the family history of individuals with both high and low levels of mathematical abilities shows that both math-ematically "gifted" and exceptionally sub-normal children are no exception in these families.

(b) An investigation of monozygotic twins shows correlations between results in arithmetic (r=.76 in 7- to 9-year-old and r=.61 in 11- to 13-year-old children), correlations much higher than between any other two individuals.

(c) Studies concerningdifferences in mathe-matical performances between boys and girls show clearly that boys perform at a higher level

Volume 7, Number 3, March, 1974

than girls at all ages and in all social and educa-tional (professional) groups.

(d) Research on children exceptionally gifted in mathematics shows that they possess surprising mathematical knowledge from earliest childhood, relatively irrespective of ex-ternal influences.

All this suggests that the individual is born with certain dispositions for mathematics. Even good equipment, however, can be impaired in the course of development. When this happens within the first year, while the child's brain is still malleable, the result is a practically ir-reparable disorder of mathematical abilities similar to when these dispositions are genetical-ly imperfect. In both these cases it is a matter of developmental dyscalculia.

Teaching can promote the acquisition of mathematical skills, but when the disposition is impaired, the child is not able to acquire certain skills and knowledge despite intensive, system-atic training.

THE CONCEPT OF DEVELOPMENTAL DYSCALCULIA Developmental dyscalculia is defined by Bakwin (1960) as a "difficulty with counting" and by Cohn (1968, p. 651) as a "failure to recognize numbers or manipulate them in an advanced culture." Gerstmann (1957) describes dyscalculia (Gerstmann's syndrome) as an "isolated disability to perform simple or com-plex arithmetical operations and an impairment of orientation in the sequence of numbers and their fractions" (Alexander and Money 1966, p. 286 ). However, as I have explained in relatively great detail elsewhere (Kosc, 1967-68, 1970), I consider dyscalculia to be a much more complicated disorder and define it as fol-lows:

Developmental dyscalculia is a structural disorder of mathematical abilities which has its origin in a genetic or congenital disorder of those parts of the brain that are the direct anatomico-physiological substrate of the maturation of the mathematical abilities adequate to age, without a simultaneous disorder of general mental functions. [Kosc 1970a, P. 192]

From my definition it is evident that: (a) the above definitions of Bakwin, Cohn, Alexander and Money, and Gerstmann do not take into

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consideration the relationship between general mental abilities and special mathematical abili-ties. The term "dyscalculia" refers specifically to a disorder of the special abilities for mathe-matics without a simultaneous defect in general mental abilities.

(b) The developmental aspect of the disorder has not been taken into consideration by these authors, despite the fact that the disorder is called developmental dyscalculia. The other definitions include any disorder of mathe-matical abilities - even those occurring in adulthood due to direct damage of the respec-tive brain centers. Actually, developmental dyscalculia ought to include only those dis-orders of mathematical abilities which are a consequence of hereditary or congenital impair-ment of the growth dynamics of the brain centers, which are the organic substrate of mathematical abilities.

(c) Ignoring the developmental aspect of this disorder, as these authors have done, leads to an obvious diffuseness when attempting to identify developmental dyscalculia in general and its symptoms in particular. Cohn (1968, p. 651) admits this when he states that in a number of children with this problem the devel-opment and use of the numerical concept is similar in many aspects to the development and use of numerical concepts in normal children. According to him, "the only differential char-acteristic is that in certain particular cases chil-dren take a longer time and need more energy to use numbers adequately." The term "devel-opmental dyscalculia," however, applies only to a child showing a distinctly lower-than-average level of mathematical age in relation to his normal mental age. This mathematical quotient (Math. Q.) is calculated by a formula analogous to that for computing IQ:

A Math. Q. lower than 70-75 is considered pathological. It follows that when diagnosing developmental dyscalculia, the child's age must inevitably be considered.

(d) Finally, Bakwin (1960) and Cohn (1968) neglect the structural analysis of mental abili-ties in general and mathematical abilities in particular. Studies that have been made usually

mention only the differences between the levels of scores in arithmetical and other subtests or between two arithmetical subtests of the W-B scale (Alexander and Money 1966). Factor analysis of mathematical abilities (Barakat 1951, Verdelin 1958, Canisia 1962, Kosc 1967), as well as detailed psychological analysis of disorders of mathematical functions of the brain in adults (Grewel 1952, Luria 1946, 1947, 1962, 1966, Schultz 1959, Cohn 1961, Hecaen et al. 1961, Kosc 1967-68), have shown clearly that mathematical ability itself is not simple and compact; as in general mental abili-ties, it is necessary to differentiate between several relatively isolated abilities or functions. These specific abilities seem to be unevenly developed even in normal children and adults, and if disorders such as developmental dyscal-culia are present, not all functions are equally affected.

DEVELOPMENTAL DYSCALCULIA AND OTHER DISORDERS OF MATHEMATICAL ABILITIES From my definition and its explanation, it is clear that developmental dyscalculia — by which I mean a disorder of the maturation of mathematical abilities — must be differentiated from postlesional dyscalculia — which is a re-gression of previously normal mathematical abilities and is acquired mainly in adulthood.

Furthermore, it follows that in addition to structural disorders of mathematical abilities, called dyscalculia, there are also disorders of the global level of mathematical abilities, such as acalculia (a complete failure of the ability) and oligocalculia (a relative decrease of all par-tial mathematical abilities approximately of an equal degree). It is also necessary to distinguish paracalculia as a distinct qualitative disorder of mathematical abilities, occurring in the majority of cases within the frame of reference of mental illnesses (psychoses).

As above, the problem stated is dyscalculia (or acalculia or oligocalculia) only when a dis-order of mathematical functions without a simultaneous disorder of general mental abili-ties is present. Disorders of mathematical abili-ties within the context of oligophrenia (mental retardation) or dementia are called secondary




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acalculia, dyscalculia or oligocalculia, possibly also paracalculia. (All four, however, are dis-tinguished according to established criteria.)

Finally, except in the case of objectively cerebrally (developmentally or postlesionally) conditioned disorders of mathematical abilities, there are functional deviations in level and quality of mathematical abilities which are con-ditioned psychologically in connection with neurosis or organic illnesses, etc. If a person has had inadequate instruction, or if a child because of neurosis, objective illness, or fatigue is not able to demonstrate his potential abilities or directly acquired knowledge and skills ade-quately, this is not a disorder of these abilities as such but merely a deficit which is called pseudo-acalculia, pseudo-dyscalculia, pseud o-oligocalculia.

In dyscalculic children, one ought to antici-pate a certain degree of secondary deficit in the field of mathematical abilities, that is, a com-bination of developmental dyscalculia with pseudo-dyscalculia. In this sense, even truly dyscalculic children could often reach much higher levels of skill in mathematical operations than they actually do, if their instruction could be appropriately organized and, above all, if their compensational mechanisms could be used to the fullest.

BASIC FORMS OF DEVELOPMENTAL DYSCALCULIA Although the theory and practice of behavior disorders currently differentiates between exogenously and endogenously conditioned dis-orders, between ^function and c/ysfunction, and between individual, isolated disorders of func-tion and those that belong to broader cate-gories, the literature on disorders of mathe-matical abilities still evidences a considerable diffuseness, lack of uniformity, and even direct contradictions in the terminology and the pro-posed categorization. I have proposed a uni-form system of nomenclature and classification of disorders in mathematical abilities (Kosc 1970b), and I am including in this review some ideas from that paper on the symptomatology of disorders of mathematical abilities. All of these symptoms may occur in isolated form or in combination.


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Developmental dyscalculia (as well as acal-culia or oligocalculia) can best be classified and characterized as follows:

(a) Verbal dyscalculia is manifested by the disturbed ability to designate verbally mathe-matical terms and relations, such as naming amounts and numbers of things, digits, numerals, operational symbols, and mathe-matical performances. There are cases of brain-damaged persons who are not able to identify the numbers dictated to them in the form of numerals (for example: to show the dictated number of fingers) although they are able to read or write the respective number or to count the amount of things (sensory-verbal dyscal-culia). Or, on the other hand, persons with verbal dyscalculia are not able to name the amount of presented things or the value of written numbers although they are able to read or write the dictated number (motor-verbal dyscalculia).

(b) Practognostic dyscalculia. In these cases there is a disturbance of mathematical manipu-lation with real or pictured objects (fingers, balls, cubes, staffs, etc.). Mathematical manipu-lation includes the enumeration (single addi-tion) of the things and comparison of estimates of quantity (without their addition). A patient with practognostic dyscalculia is not able to set out sticks or cubes according to the order of their magnitudes, not even to show which of the two sticks or cubes is bigger or smaller, or whether they are the same size.

(c) Lexical dyscalculia. This particular dis-order is concerned with a disability in reading mathematical symbols (digits, numbers, opera-tional signs, and written mathematical opera-tions). By far the most serious form of lexical dyscalculia is when the child is not able to read the isolated digits and/or simple operational symbols (+, - , x, -=-, etc.). In the less serious forms, he cannot read multidigit numbers (especially with more than one zero in the middle), numbers written in a horizontal rather than a vertical line, fractions, squares and roots, decimal numbers, and so on. In some cases he interchanges similar looking digits (3 for 8, 6 for 9, and vice versa), or two digit numbers are read as reversed (12 as 21).

(d) Graphical dyscalculia. This is a disability

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in manipulating mathematical symbols in writing, analogous to lexical dyscalculia. Graph-ical dyscalculia often occurs where there is dysgraphia and dyslexia with letters; eventually, in the most serious disabilities of this kind, the patient is not able to write numbers dictated to him, to write the words for written numerals, or even to copy them. Leischner (1957) presents a patient who wrote " 6 " and " 9 " in a combination of both symbols, like this §. In less severe cases, where a person cannot write numbers with two or three digits, he writes them in the opposite direction or he isolates the separate elements (i.e., 1284 as 1000, 200, 80, 4 or 1000, 200, 84) or he ignores the zeros (i.e., 20073 as 273 or 20730); or he devises his own idiosyncratic manner. This person may not be able to write any mathematical symbol, even though he can write the word for the dictated number, e.g., to the dictated " 8 " he writes "eight."

Lexical and graphical dyscalculia are also called numerical dyslexia and numeral dysgraphia, and together both are labeled numerical dyssymbolia in the literature.

(e) Ideognostical dyscalculia. This is a dis-ability primarily in understanding mathematical ideas and relations and in doing mental calcula-t i on . Grewel (1952) calls this disability "asemantic aphasia," but it is more accurate to call it "ideognostic dyscalculia." In the most serious cases of this type of dyscalculia a person is not able to mentally calculate the easiest sums. Often, a person with brain dysfunction is able to read or write the written numbers but is unable to understand what he has read or writ-ten. For instance, he knows that 9 = "nine" and that "nine" is to be written as 9, but he does not know that 9 or nine is one less than 10, or 3 x 3, or one-half of 18, etc. In these and similar cases, it is not possible to label this kind of dysfunction as numerical dyslexia or dys-graphia, or even as operational dyscalculia (see below). In this case the formation of ideas, a gnostic function, is disturbed; it is appropriate-ly called ideognostic dyscalculia.

(f) Operational dyscalculia. In this case, the ability to carry out mathematical operations is directly disturbed. Hecaen etal. (1961) call this

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form of dysfunction "anarithmetia." A typical occurrence is the interchange of operations, e.g., doing addition instead of multiplication; subtraction instead of division; or substitution of more complicated operations by simpler ones (e.g., 12+12= (10+10) + (2+2); 3 x 7=7+7+7=21; or in serious disturbances: 777). Typical also is a preference for written calcula-tion of sums (tasks) which could be easily cal-culated silently, or calculation by counting on the fingers where the task could be easily solved silently or in writing and without counting fingers.

A disability like operational dyscalculia is most difficult to identify because of the neces-sity for carefully following the subject's pro-cedure in completing the operations — especially when the subject is a patient who cannot verbalize what (and how and why) he is doing according to his partial rules. In the cases of combined symptoms of different types of dyscalculia, especially the combination of ideognostic and operational dyscalculia, it is nearly impossible to discover when and how the wrong achievement was conditioned by one or by the other disability.

In every case it must be pointed out that incorrect results do not reveal what kind of disturbance is really involved in a particular task; it is quite possible to get correct results by means of incorrect procedures. Green and Bus-well (1930) gave an example in which a student approached the task 86 — 4 as follows: "Six and four equals ten; ten and eight equal eighteen." Then he wrote his answer back-wards: 81. His result differs from the correct answer by one, but the procedure was essential-ly wrong. This case clearly presents a symptom of operational dyscalculia.

It is obvious that various combinations of symptomatology of developmental dyscalculia occur mostly in combination with symptoms of other impaired symbolic functions of the brain (especially with dyslexia and dysgraphia with letters) or other disorders of the higher nervous system (within the frame of reference of minimal brain dysfunction), analogous to post-lesional dyscalculia or acalculia in adulthood resulting from brain damage.

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Research 169

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N = 375 4th grade 5th grade 6th grade

Boys 2.21 2.34 2.84

Girls 1.82 2.17

2.66

Total 2.05 2.25 2.76

Two sets of group tests measuring math-ematical abilities were then applied. On the first test, Kalkulia I, which I developed for research purposes, the subject has to mathematically manipulate varying numbers of designed objects and solve simple geometrical problems. Subjects are required to determine the number of black dots, placed in various patterns (for the most part symmetrically arranged along one of the axes) against a standard schematic background composed of 10 by 10 dots (see specimen items, Fig. 1). The space below the pattern of each item can be used by the subject for auxiliary numerical computations; the final re-

FIGURE 1. Specimen items of the test Kalkulia I.

AN INVESTIGATION OF MATHEMATICAL DISABILITIES IN BRATISLAVA In this study, an attempt was made to identify cases of dysfunction of mathematical abilities from a normal sample population of 11-year-old children in regular classes. The sample con-sisted of 375 pupils (199 boys and 176 girls) selected at random from 14 fifth grade classes in 14 Basic schools in Bratislava. In the first phase of the investigation, the boys' ages ranged from 10.2 to 11.7 (mean = 10.5), the girls' ages from 10.2 to 11.8 (mean = 10.8). All children older than 11.8 were eliminated in the first screening. The mean school marks in mathe-matics at the end of the school year were:

%

^D FIGURE 2. Specimen items of the test PFB.

suit of this calculation is entered in the box to the right of the pattern.

In the investigation summarized here an experimental version of this test was applied, which later was standardized in a new form according to the principles of psychological test construction. This new test, Kalkulia I I , as well as a manual containing instructions for its ad-ministration and interpretation and norms for an age range between 8.6 and 17.6 years was published by the national enterprise Psycho-diagnostica in Bratislava, the test in 1969, its manual in 1972.

In addition a paper-and-pencil group test PFB, a modification of the Minnesota Paper Form Board (also published by Psychodiag-nostica in Bratislava) was administered. The subject draws lines to divide various geometrical figures into the same shapes and number of pieces as those printed beside the figure. Cor-rectly assembled, these fit together into the complete figure (see sample items, Fig. 2). This test was standardized on a Czechoslovak popula-tion and has separate norms for boys and girls ranging in age from 7 to 16 years and 11.6 to 18.6 respectively.

The second set of tests consists of solving

Volume 7, Number 3, March, 1974 51



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common arithmetical tasks of addition, subtrac-tion, multiplication, and division of numbers up to 100, as well as tests of mathematical sequences and mathematical symbols. I developed the tests and so far they have not been standardized on a wider population.

The test of mathematical sequences consists of the following kinds of sequences:

2 4 6 8 10 12 10 5 9 6 8 7 30 6 24 12 18 18

The subject is required to find a rule or prin-ciple in order to complete the subsequent two members of the sequence.

In the test of mathematical symbols, the subject has to mark each of the nine randomly chosen and assembled letters of the alphabet with that number (from 1 to 9) which is writ-ten below the proper letter in the code table (similar to Wechsler's "digit symbols" with the exception that in the test used in this study the subject fills in the number to the symbol and not vice versa).

All the tests were used as screening instru-ments and were not considered indicative of developmental dyscalculia in the children studied. The children studied further were only those who scored at or below the lower 10th percentile of the score distribution on the group tests and who were thus considered failures. The other tests used, described below, were administered individually and are different from those applied in the screening procedure.

Following the screening, three groups of children were delineated from the original sample: those who failed (1) only in the first set of tests, (2) only in the second set of tests, and (3) in both sets of tests. Children with an IQ lower than 90, as tested on three individual general mental ability tests (the Kohs Block Design Test, Goodenough Draw-a-Person Test, and the Terman-Merrill Scale) were eliminated.

There then remained in the experimental group 66 children with a normal level of general mental abilities but with suspected special dysfunctions of mathematical abilities. This experimental group was submitted to a detailed individual psychological and neurological examination.

The psychological assessment was concerned

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FIGURE 3. The appropriate solution to the test "Numerical Triangle."

with determining the child's level for all factors which were so far unambiguously identified by factor analysis — namely, the general, numeri-cal, spatial, verbal, memory, reasoning and school factors. This battery also comprises tests for assessing the level and quality of other symbolic functions, especially reading and writing letters.

The most important tests used in this phase of the investigation were:

(1) Numerical triangle. The subject was asked to write the dictated digits, one beneath the other (see Fig. 3). Then he was asked to add the first two digits in the column and write their sum in the second column at the position half way between the two digits which were summed. The subject was told that when the sum of two numbers was more than 9, only the number in the units column should be re-corded. He then proceeded down the first column adding subsequent pairs of digits and writing their sum in the column to the right, again in a position between the added digits. The subject proceeded in this manner working with each column of numbers across the page until a column containing only one digit was produced. The primary purpose of the test was to determine whether a subject could produce




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FIGURE 4. Rey-Complex-Figure.

the appropriate numerical matrix from the in-

structions, and secondarily to determine the

subject's basic addition skills. The time needed

to accomplish the task and the number of mis-

takes were recorded. "Numerical tr iangle" is a

modification of the well-known Remplein's

tests. There are no norms for its quantitative

evaluation. This test was used solely to note

extremely inappropriate solutions.

(2) Rey-Osterrieth Complex Figure. This

standardized test, well-known in the U.S., re-

quires copying of a special design (see Fig. 4). It

was placed on a table in f ront of the subject,

and he was instructed to copy it. The time

needed to finish the test and the number of

mistakes were recorded.

(3) The test of arithmetical reasoning was

taken from the Terman-Merrill Intelligence

Scale and contains three mathematical tasks

intended for the 10-year level. The examiner

slowly dictated the text of the three problems

to the subjects, and instructed them to start

working. This test was used to identify dis-

orders in solving verbal reasoning problems and

possible mistakes in spelling words and writ ing

numerals in the dictated sentences (i.e., it was

also designed to reveal graphic performance).


171

The time was not l imited, but was recorded

after each computation was finished.

(4) The digit memory test was taken f rom

common Terman-Merrill test items for the dif-

ferent age levels. The subject was asked to re-

peat numbers f rom 3 to 7 presented by the

examiner at intervals of approximately one

second. The presentation method and the re-

production form were varied successively as

follows, with the same sequence for each

subject:

(a) verbal presentation — verbal reproduction,

(b) verbal presentation — graphic reproduction,

(c) graphic presentation —verbal reproduction,

(d) g raph i c presentation - graphic repro-

duct ion.

The e x p e r i m e n t e r observed each child's

memory level for numbers, and also tried to

separate the subjects into various "computat ion

types" and to discern the frequency of the

different forms of dyscalculia caused by varying

the presentation and reproduction.

(5) The test of successively subtracting 7

from WO is known f rom Luria's techniques of

psychological assessment. This method can dis-

close disorders of counting silently — especially

in subtractions involving transitions over tenths

making it necessary to keep in mind the result

of an operation which becomes the starting

point for the next operation. Each number

given by the subject, any possible corrections,

and the time needed to arrive at the lowest

number were recorded. The test was admin-

istered twice orally and then in wri t ing. It was

first observed whether the subject changed the

quality of his performance in repeating the test

and then, whether this change was revealed

more clearly when passing f rom the oral to the

wri t ten form of recording results.

(6) The numerical square test is designed to

determine the working curve in an attentional

task requ i r i ng manipulation of numerical

material. By a qualitative analysis, it may also

reveal neurotic symptoms. The test material

contains a large white sheet of cardboard where

the numerals from 1 to 25 are distributed in

distinct pr int in random order in 25 five by five

inch squares. The subject has to find and name

the numbers from 1 to 25 in succession as fast

as possible. Since the subjects in this study were

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FIGURE 5. Diminished design to the test "Numerical Square."

children, the procedure was repeated only 15 times instead of 25 times as is usually done with adults. In accordance with Dobrotka (1953), I also introduced an "interval of provocation" after the 10th, 11th, and 12th repetition. This procedure consisted in chal-lenging and encouraging the subject during the performance to work as quickly as he possibly could to find the successive numerals in these particular sets. The test findings and time period required for performing each set are recorded on a graph, forming an achievement curve. From its level and course and from the registered observation of the subject, inferences may be drawn not only as to his "working achievement" in dealing with numerical material but also to a number of personality characteristics of the subjects, (see Fig. 5).

(7) The G-test, developed by the Slovak psychologist M. Milan (1969) is commonly known and widely used in Czechoslovakia. It is designed to determine the ability level of read-ing a comprehensive printed text, possibly also writing speed and spelling accuracy of isolated words, i.e., lexia and graphia - especially im-portant for the purpose of this investigation. The test consists of 186 simple affirmative sentences each of which is followed by a ques-tion to which the subject has to respond in writing with one word (plus a preposition if needed) taken from the sentence. The subject

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FIGURE 6. A. A., boy, 10years, 6 months.

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FIGURE 7. C. O., girl, 10years, 8 months.

has to read and answer the greatest possible number of successive sentences within a time limit of 14 minutes. Three sample items are given and illustrated by the examiner. The test




Research 173

FIGURE 8. B. A, girl, JO years, 6 months.

has been standardized for ages 8 years to adult of the Czechoslovak population.

RESULTS OF THE INVESTIGATION In the numerical triangle test the groups with global failures differed significantly (at the .01

FIGURE 9. C. O., girl, 10years, 8 months.

level) from both experimental groups in the screening stage of the research in the number of incorrect steps and the testing time. The sub-jects who were unsuccessful in the numerical tests only and not in the performance tests applied in the screening procedure were closest in performance to the group of failures. The numerical triangle test differentiated between the groups to a greater extent inasmuch as the numerical component of mathematical abilities was involved. However, as far as individual cases of children who completely failed this test are concerned, some appeared to be more spatially dyscalculic than others since these children also failed the Rey-Osterrieth Complex Figure test. Samples of failures in the numerical triangle are given in Figures 6, 7, 8 (compare with the correct solution in Fig. 3).

The Rey-Osterrieth Complex Figure did not distinguish between the experimental groups. A few individual children who showed definite failures (having a maximal percentage of mis-takes and the longest testing time) belonged, without exception, either to the group of general failures or to the group who failed only in the performance tests applied in the screen-ing stage of the investigation. Samples of such inadequate performance on this test from three spatially dyscalculic children are given in Figures 9, 10, 11.




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FIGURE 10. A. E., boy, II years, 2 months.

On the arithmetical reasoning problems the lowest average scores and the longest mean time were obtained by the experimental groups who failed in the performance tests (the difference between the groups of global failures and the groups failing only in the numerical tests were statistically significant at the .05 level). Some of the children did not correctly solve any of the three items and further analysis revealed them as verbal or ideognostic dyscalculics.

In the digit memory test all experimental groups were found to have a higher average number of incorrect reproductions on both sets in which the numbers were presented verbally (statistical significance at the .01 level). The greatest number of failures were again recorded in the group of globally failing children. The other two groups manifested approximately the same level in all forms of exposition and repro-duction, which is significantly higher (at the .01 level) than the group of global failures in the screening stage of investigation. Differences were found in the reproduction of four num-bers and were considerable when six numbers had to be reproduced. However, some children failed even when asked to reproduce only a few presented digits (especially when the presenta-tion was verbal). This could be explained not by disorders of concentration or attention but

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FIGURE II. C. F, 10years, 9months.

only by a disturbance of the memory factor of mathematical ability structure. Most of these children also failed in the test of successive subtraction of 7 from 100.

The test of successively subtracting 7 from 100 was given in three forms (two verbal and one written performance). As far as number and range of faulty steps are concerned, the poorest results were achieved by the two ex-perimental groups who failed all the numerical tests applied in the screening. Both of these groups attained an approximately identical, significantly lower standard (at the .01 level) than the third group who failed only in the performance tests. The poorest results were achieved in the first (verbal) set. The findings were found to be analogous when the average time required for the performance of each test set was employed as criterion for the estima-tion. That some children performed so in-adequately on this test and showed such sur-prising perseverance of failure was considered evidence supporting the existence of a disorder of mathematical abilities in the sense of ideognostic dyscalculia.

The numerical square holds a special posi-tion in the test battery used insofar as the poorest results were not attained by the group of global failures but by the group that failed




Research

only the tests of numerical character used in the screening stage of this investigation. This can be accounted for by the fact that among those who failed in numerical tests alone there was a minimal number of children with dyscal-culia in the proper sense of the word; on the other hand, there were many failures due to secondary causes, such as lack of mathematical knowledge or neurosis. (Some of the individual achievement curves, found in children in this group, revealed a typically neurotic course when evaluated according to Dobrotka's criteria [1953].) In any case the numerical square test does not contribute in a decisive way to the identification of dyscalculia as such.

The G-test cited above was included in the test battery chiefly to determine to what extent dyscalculia (numerical dyslexia and dysgraphia) occurs simultaneously with literal dyslexia and dysgraphia. The lowest scores were obtained by both experimental groups who failed in all numerical tests employed in the screening stage of the investigation. From this it was inferred that lexia and graphia are closely related, re-gardless of whether the material involved con-tains numbers or letters. A more exact determi-nation of the nature of the relationship requires further research.

Neurological examination was carried out by a child neurologist. Dyscalculia or acalculia occur in patients at a neurological clinic mainly w i th in the framework of the Gerstmann syndrome. The neurological examination of the children delineated by screening included, apart from gaining essential anamnestic data and performing routine procedures, the assessment of (a) laterality, (b) right-left orientation, (c) spatial orientation, and (d) finger gnosis. Un-fortunately EEG records were not taken.

Fourteen indicators of laterality were taken into consideration, and the laterality quotient was computed according to Kucera's proposed formula (1961):

A

LQ=5-tLr—r * 1°0 v R+ L + A

R = number of performances with right hand L = number of performances with the left

hand A = number of performances with both hands


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A laterality quotient of 75-100 would indicate right-handedness; 50-74, ambidexterity; and be-low 49 a left-handed individual.

In the assessment of right-left orientation, 14 indicators were considered. Two incorrect responses were regarded as an indication of impaired right-left orientation (Kucera 1961).

The assessment of finger gnosis was per-formed according to Cernacek's (1955) schema. Two incorrect responses were considered to be indicators of gnostic disorder in accordance with Cernacek's criteria.

Out of 15 children of the group of global failures, 11 children showed at least grave suspi-cion of an objective pathological neurological finding; whereas in the group of 21 children that failed in the numerical tests only, neuro-logical deficits were found in just two cases. In the last group (failure in the performance tests) there was not a single case with even a suspicion of an objective pathological neurological finding. In the majority of cases, the presence of a suspected or directly diagnosed minimal brain dysfunction syndrome was inferred from the following symptoms: distinct instability, lack of coordination, speech disorders, impaired concentration of attention, central flaccid paralysis of mild degree, disturbances of right-left orientation, and finger gnosis. The great majority of children considered by the neurolo-gist as gravely suspicious of an objective path-ological neurological finding (10 out of 13 chil-dren) also showed a symptomatology of some type of developmental dyscalculia.

The results of all tests applied as well as the school marks and some important indicators from parents' and teachers' questionnaires and neurological examination were statistically analyzed in detail (including factor analysis) for all the experimental groups in comparison with the whole original sample of studied subjects. Within the framework of this paper it is not possible to quote them all. However, it should be stressed that it was not until detailed analyses of the subjects had been performed and case studies undertaken that each in-dividual case could be delineated and diagnosed as to the presence of developmental dyscalculia. Out of the whole sample of 375 children | studied, dyscalculia was found in a total of 24

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176

(6.4%). This, however, does not mean that each "dyscalculic" child differs in such a distinct way (or differs at all) in all indicators from the whole sample of studied children. In some indicators (interpreted as structural factors of mathematical abilities) dyscalculic children ap-peared to have an entirely normal level ade-quate to their age. The structure of their math-ematical abilities, however, was affected in such a way (with regard to other indicators in which the children showed clear failures) that they could be singled out as dyscalculic (in the sense of the proposed definition of developmental dyscalculia). They were described in detailed individual case histories.

Unfortunately, the group of suspected dyscalculic children identified in our investiga-tion was not followed up, and no remedial, educational or other special corrective method was applied. At present it is our intention to resume the investigation of this group and, if it seems necessary and possible, to submit them to treatment. It is also planned to repeat the whole investigation on other samples of chil-dren residing out of Bratislava, with a particular view to choosing children for special treatment in order to prove whether and to what extent these children are not only "highly suspicious" but really dyscalculic.

As has been mentioned, this investigation was relatively extensive (the research report covers 531 typewritten pages, including tables, graphs, illustrations, and specimen test solu-tions). It is not possible to give a comprehensive report illustrated with detailed statistical data within the framework of this paper. The data have been published elsewhere (in Slovak) and are available from the author on request. My aim here was to draw attention to the described problem and to generate interest in research as well as practical treatment of children who show conspicuous disorders of mathematical abi l i t ies. — Vyskumny ustav detskej psycho logie a patopsychologie, Legionarska 70, Bratislava, Czechoslovakia.

ACKNO WL EDGEMENT This paper was prepared in conjunction with the author's studies at the Institute of Child Development, University of Minnesota, February-May 1971. Trans-lated from Slovak.

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