Quarterly Journal of Experimental Psychoiogy (1982)34A, 31-51

THE FRACTIONATION OF
ARITHMETICAL SKILLS: A SINGLE
CASE STUDY
ELIZABETH K. WARRINGTON

National Hospital
Queen Square, London WCI, England
A single case study of a patient with a selective impairment of arithmetical
calculation is reported. DRC became severely acalculic after sustaining a left
parietal intra-cerebral haematoma. His numeracy skills were documented as
fully as possible. It was found that his knowledge of the significance of individual
numbers and his concept of quantity appeared to be intact, but simple addition,
subtraction and multiplication were all performed laboriously and inaccurately
and his performance was inconsistent from trial to trial. Further it was shown
that it was his knowledge of arithmetical facts not his knowledge of arithmetical
operations which was impaired. The findings are discussed in relation to models of
arithmetical calculation and it is concluded that for DRC there was a deficit in
accessing one category of verbal semantic knowledge systems, namely arithmetical
facts.

Introduction
T h e component processes of numeracy have been studied in normal adults in whom
these skills are well established and in young children in whom these skills are still
being acquired. A further source of evidence derives from the neurological
literature. It is now generally accepted that acalculia, the impairment of the
ability to perform arithmetical calculations, can occur as a relatively selective
deficit. Although difficulties with calculation had been previously recorded in the
context of more generalised aphasic symptomatology (Lewandowsky and Stadelmann, 1908; Peritz, 1918), Henschen (1919)first identified the syndrome
“acalculia”, which he described as a disturbance of calculating produced by a focal
lesion of the brain. H e collated a large series of patients in whom there was a
primary impairment of arithmetical calculations, a deficit he considered to be
largely independent of the frequent concomitant syndromes, aphasia, alexia and
agraphia (Henschen, 1919,1920).
Arithmetical calculation is a complex accomplishment and as such is likely to
comprise many subcomponents. T h e neurological case descriptions of disorders
of numeracy provide some pointers to the variety of functional systems subserving
arithmetical calculation. Henschen (1919)himself drew a distinction between
disturbances of number recognition, in which he implicates number sense and

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31

32

E. K. WARRINGTON

number meaning, and a more restridted deficit in carrying out arithmetical operations. Lange (1933) drawing upon the observation that written calculations were
often performed worse than oral calculations emphasised the patient’s inability to
manipulate the position or direction of numbers in space. He extended this
formulation to single digit notation; these too were conceptualised as markers on a
visuo-spatial scale. Following Lange, Kleist (1934) and Krapf (1937) at much the
same time noted the vulnerability of the patient’s ability to manipulate spatially
(both horizontally and vertically) the position of numbers.
Guttman (1937) described a patient who knew the multiplication tables and
could carry out simple arithmetical calculations, yet had an unexpectedly grave
difficulty with number estimation and number knowledge. Cohn (1961) distinguished between two quite distinct syndromes in a series of 8 patients in whom
acalculia was the major presenting symptom. He tested these patients’ knowledge
of their multiplication tables and their ability to do written multiplication. Two
major types of deficits emerged (I) the misalignment and misordering of digits and
(2) faulty memory of “tables”.
It is of some interest to note that the former type
was in every case associated with a right hemisphere lesion and that the latter type,
faulty memory of tables, occurred either with a diffuse lesion or a left hemisphere
lesion.
T h e evidence for selective impairment of arithmetical calculation reported in the
neurological literature is at best qualitative and often merely anecdotal. I n no
instance are control data reported and no study to date reports systematic and
quantitative data of a patient’s numeracy capacities and incapacities. The aim of
the present communication is to report a comprehensive and quantitative investigation of a single patient whose abilities to carry out arithmetical calculations appeared
to be selectively impaired. I n particular, assessment of the component processes
of arithmetical skills is attempted. T h e findings are considered in relation to
current models of arithmetical calculation.
Case report

DRC, a 61-year-old, right-handed consultant physician was admitted to the
National Hospital on 18 October 1979 for investigation of headache and speech
disturbance, which had developed suddenly on awakening 3 days previously. On
the day of admission he had a further episode of severe headache and transient loss
of consciousness, following which there was a significant worsening of his dysphasic
symptoms. He had been treated for high blood pressure for the previous 2 years.
On examination he had a macula-splitting right homonymous hemianopia. H e
was clearly dysphasic, expressive, receptive and nominal speech functions being
impaired, and it was noted that he was unable to perform simple calculations; for
example when asked to solve 5+7 he replied, “13 roughly”. T h e neurological
examination was otherwise normal.
The CAT scan (18 October 1979) showed an area of increased attenuation in the
left posterior parieto-occipital region and the trigone was displaced anteromedially ;
the diagnosis of an intra-cerebral haematoma was made (see Figure I). A repeat
CAT scan ( 3 0 October 1979) showed that there had been a slight resolution of the

FIGUREI .
haematoma.

CAT Scan (18.10.79) showing a left posterior parieto-occipital intra-cerebral

33

FRACTIONATION OF ARITHMETICAL SKILLS

intracerebral haematoma. There was no evidence of more widespread changes and
in particular the very mild atrophic changes were considered to be normal for his
age. He was treated conservatively and his symptoms improved. DRC was tested
once or twice daily, at his own request, between 12 November 1979 and 23
November 1979 (the day of his discharge).

Psychological test findings
DRC was first tested on a shortened version of the WAIS on 26 October 1979
and again on 12 November 1979. His pro-rated IQ scores and the individual agescaled scores for each subtest obtained on the two occasions are given in Table I.
There had been some improvement in all the subtest scores during this period, with
the exception of arithmetic which remained at the defective level. During this
2-week period DRC’s language functions largely recovered. The following test
results were recorded on 12 November 1979: his spontaneous speech was somewhat ponderous but his sentence construction was almost always correct as to word
choice and syntax. On a graded-difficulty naming test he scored 26/30 which is
equivalent to a vocabulary age-scaled score within the superior range (see McKenna
and Warrington, 1980). On a shortened version of the Token test of comprehension his score was 12/15 which is at the lower limits of normal performance
TABLE
I
WAIS intelligence quotients and WAIS individual subtest scores
__

~

26 October 1979

12November 1979

90

103
I10

~~

Verbal IQ
Performance IQ
Arithmetic
Similarities
Digit span
Vocabulary
Picture completion
Block design
Picture arrangement

97
5
9
9
I2
I0
I0

8

4
11

I4
I4
I0
I2
12

(Coughlan and Warrington, 1978). On the Nelson reading test he read 43\50
words correctly, equivalent to a reading IQ within the superior range (Nelson and
O’Connell, 1978). His recognition memory for words, 45/50 correct, was average
(Warrington, I 974).
DRC was retested on the arithmetic subtest of the WAIS on 15 November 1979
and he obtained an age-scaled score of 5 . However given unlimited time he solved
9/14 problems correctly, including the last three most difficult items. He
reasoned aloud while attempting these calculations and it was quite apparent that
he knew the appropriate sequence of arithmetical operations for the solution and
that his difficulty was one of actual computation. For example, Q.13 (A man earns
E6olweek; if 15% is withheld for taxes, how much does he receive each week?)

34

E. K. WARRINGTON

was correctly solved in I min 10 s. He verbalised as follows “SO we do it 10%
first, is E6, half of that is E3, equals E9, 60 minus nine is
He had no difficulty in reading either single numbers or compound numbers.
For example, 897,201 was correctly read as eight-hundred-and-ninety-seventhousand, etc. Similarly he had no difficulty writing similar numbers to dictation.
His very satisfactory digit span (eight forward and five backward) could hardly
have been achieved if he had not been able to perceive or articulate number names
efficiently.
DRC was encouraged to introspect about his difficulties with arithmetical
calculations. In his view the process of addition and subtraction were no longer
automatic”, and he was adamant that this had been the case previously. He
claimed he often knew the approximate solution but not the exact solution and
indeed before attempting a calculation he would often say “approximately . . . ‘d ”.
He would sometimes comment “it must be an odd (or an even) number.” Given
time he could add and subtract accurately by counting forwards or backwards in
units of I or 2. He himself considered that there was a trade-off between speed and
accuracy. In his view if he were to calculate slowly and also check his solution, his
performance would be error-free. He claimed that his understanding of arithmetic operations was unimpaired, and indeed his verbalisation of arithmetical
problems and his definitions of the four basic operations which are quoted below
bear this out. Though not strictly correct, they are impressive as definitions
attempted without prior warning.
( I ) Addition - “Adding is a process of, well if I use the word summing, it is a
process of summing in which the number of certain magnitude is increased
by a number of another magnitude by the numbers which have been defined
in the case for each of these magnitudes.”
(2) Subtraction - “Subtraction is the reduction of the magnitude of a particular
number by the magnitude stated of another number and that is it.”
(3) Multiplication - “The operation of multiplication is to increase by equal
magnitudes a stated number.”
(4) Division - “Division is the apportioning of a number of a certain magnitude
into division such as each number is then of the same magnitude, is the same
fraction so the number is fractioned and a proportion of that fraction is then
removed. I am looking for a number now. A proportion of a fraction is
removed, or rather the proportion is fractioned and the fraction is then
divided and a decision is made as to which of the fractions is to be kept and
which rejected.”
In summary, DRC’s earlier dysphasic symptoms had almost entirely resolved,
the only residual deficit of any significance being an acalculia. This selective
impairment of arithmetical calculation is the subject of the following experimental
investigation.

E~I.”

14

Experimental investigation
DRC’s ability to carry out arithmetical calculations and tests of numeracy was
documented as fully as possible. Five matched controls (consultant physicians of
the National Hospital, age range 61 to 66) were tested on some of the following

FRACTIONATION OF ARITHMETICAL SKILLS

35

procedures. These subjects volunteered to act as controls for a consultant
physician whom they knew to have an acalculia but they were not given prior
knowledge of the tests to be administered.
Test I - addition

DRC attempted to add all combinations of numbers between I and 9. This
matrix (45cells) was completed in random order, each addition being spoken by the
experimenter at a I word/s rate, the larger of the two numbers being given first
(e.g. 4 add 2). His accuracy and speed of response (timed by a stop watch to the
nearest second) were recorded. “Immediate” responses were recorded as 0’’s.
(These conditions of presentation and recording were also adopted for Tests 2 to 6).
This test was administered on three occasions (IS, 20 and 21 November 1979).
His overall error rate was 11%. He made no errors on ties (e.g. 5f5); his error
rate for sums 10 and under was 1‘3% and for sums above 10, it was 23.3%;
otherwise there did not appear to be any noticeable pattern of errors. In view of
the overall low error rate on this test (and also on tests, 2, 6) a fine grain analysis of
the consistency of his error responses over different occasions was not attempted.
His overall average speeds of response for correct and incorrect solutions were 2
and 3.1 s respectively. However there was a wide range of response latencies; from
less than I s to 14 s for a correct solution and from I to 10 s for an incorrect
solution (see Table I1 for the overall distribution of his response latencies on this
test).
Inspection of the response times of the control group on tests 2,s and 6 (described
below) indicated that the greater number of response times were “bunched”
between o and 2 s, the occasional response having a much longer latency (see Table
11). In this and in the following analyses response latencies of over 2 s will be
called long latency responses.
The percentage of long latency responses as a function of the minimum addend,
the smaller of the two numbers to be added, are shown for DRC in Figure 2, but no
very clear pattern emerges. On the other hand his long latency rate for sums over
10 was considerably higher than for sums up to and including 10 (33’3% and 17‘3%
respectively). His long latency rate for ties was very low (3%).
An indication of the consistency of the pattern of short and long response
latencies over his three attempts at this test was computed as follows : First, in this
and in the following consistency computations, cells in which there were ties (e.g.
3 f 3 or 3-3) and cells in which the addend or subtrahend was I or 2, were
eliminated as the long latency rates were very low; and more important a solution
by counting could be achieved within the time limits of the short latency responses.
It was then assumed that his performance on each trial for each remaining cell was
independent (his overall long latency rates for each trial of each test were fairly
constant). The expected distribution of long latency responses for each cell of the
matrix (irrespective of whether the solution was correct or incorrect) was then
derived from the overall probability of a long latency response occurring. Thus
the chance occurrence of cells in the matrix with consistent responses (three long
latencies or three short latencies) and inconsistent responses (one long and two

Test 6 multipfiation

Test 2 addition
1-9/11-19
Test 4 subtraction
1-911-9
Test 5 subtraction
I - ~ / I1-19

1-911-9

Test I addition

Latency

34
149
73
99

151

DRC
3
Controls(n=5) 153
DRC
16
Controls ( n = 5 ) 197

51
72

46

21

I

DRC
7
Controls (n=5) 208
DRC
I4

DRC

0

TA~LE
I1

2

I7

2
0

z
I

3
2

10

4

20

11

23

0

6

I

10

4
10
0

I

2

22

0

13

7
4
8

I

6

34
6

29

39
61
8

29

5

0

55
35
I

4

5

12

4

26
8

~~

5
26

3
13

~

21

2

Distribution of response latencies

0

2

0

I3

2

0

10

2

8

0

0

I

4

0

I

0

4

0

0

0

0

4

I

0

0

0

3

0

0

2

1 0 1 1

2

I

9

0

2

0

5

I

0

2

2
I

0

37

2

0

4

z

I Z I Z +

2

;

2

%

y
w

i

FRACTIONATION OF ARITHMETICAL SKILLS

37

short latency responses or two long and one short latency responses) can be
compared (using a Chi-squared test) with the observed occurrence of consistent
and inconsistent responses. DRC’s consistent and inconsistent response rate
together with the expected values are given in Table 111.
It is clear that very simple additions are no longer “automatic” for DRC. A low
but not insignificant error rate was observed and his speed of response was slow and
variable. It is assumed that the control group, who were not given this test, would
have responded accurately, without a significant number of long latency responses.
Test 2 - addition

DRC attempted to add all number combinations of I to 9 and 1 1 to 19. This
matrix (81 cells) was completed in random order, the larger of the two numbers
being given first (e.g. 11 add 3). His accuracy and speed of response (as for Test
I) were recorded. DRC attempted this test on three occasions (15, 16 and 19
November 1979). The control group attempted this test once. DRC’s overall
error rate was 8% (the controls made no errors). For sums up to 20 and over 20
his error rate was 10% and 7% respectively. His overall mean speed of response
for correct and incorrect solutions was 3.1 and 5.8 s respectively. The distribution
of response latencies for DRC and for the control group are given in Table I1 and
DRC’s percentage of long latency responses plotted as a function of the minimum
addend are given in Figure 2. Not only is DRC more inaccurate than the control
100

P

Minimum addend

FIGURE
2.
Addition: The percentage of DRC’s long latency responses is shown as a function of
the minimum addend for Test I (0)
and for Test 2 (a).

group on this test, but the proportion of long latency responses (which donot
conform to any clear-cut pattern) are of a totally different order of magnitude than
for the control group.
DRC’s consistent and inconsistent response rate together with the expected
values are given in Table 111. As before his degree of response consistency does
not differ from chance.

E. K. WARRINGTON

38

TABLE
I11
Analysis of consistency of short and long latency responses
Test I
E
O
Consistent
cells

Test 2
E
O

Test 4

E

O

Test 5
E
O

Test 6
E
O

LLL

2'0

3

12'2

I2

I

3

31

35

3'5

3

sss

3'3

4

5'3

6

5'3

8

0.6

2

37'5

37

7-2

6

25-9

25

5-3

4

24-8

17 22-7 24

8.4

8

20.4

20

9-2

6

65.0

9

(or LL)
(or SS)

Inconsistent . LLS
cells
(or LS)
LSS

-

-

(or LS)
XP

P

0.93
>om05

0.13
>0*05

6.8
>o.o5

7'19

0.15

>0.05

>0.05

Test 3 - addition
DRC attempted to add all number combinations of I to g and I I to 19 as in Test
z except that the smaller addend was spoken first (e.g. 8 add 12). His performance
on this test can be compared with his performance on Test 2 (on the trial attempted
on the same day, 16November 1979). His error rate was 24*7y0when the smaller
addend was spoken first compared with 7.4% when the smaller addend was spoken
second. The proportion of long latencies on the former task (smaller addend
spoken first) was 45*7y0compared with 38.3% on the latter task (smaller addend
spoken second). DRC appears to be both less accurate and slower when the
smaller of the two addends is presented first.
The number of errors on this test together with the errors from the 2nd and 3rd
trial of Test z (the actual errors on trial I of Test z were inadvertently not recorded)
were sufficient (n=32) to warrant an error analysis. His error responses are
plotted as a function of the correct solution in Figure 3, the short latency error
responses are indicated separately from the long latency error responses. The
correlation (product moment) between the correct solution and the short and long
latency error responses was 0.64 and 0.89 respectively.
Test 4 - subtraction
DRC attempted to subtract all number combinations of I to 9 from I to 9
(calculations with o as the remainder were included, negative remainders were
excluded). Each subtraction problem was spoken at a I wordjs rate (e.g. eight take
away three) the matrix (45cells) being completed in random order. This test was
administered on three occasions (IS, 16 and 20 November 1979). His overall error
rate was 8%. The distribution of his response latencies is shown in Table 11; his
overall mean response time for correct solutions and incorrect solutions was 2.2 and
2.5 s respectively. DRC's long latency rate as a function of the minimum
manipulation (that is the difference between the whole and the remainder or the
whole and the subtrahend, whichever is the smaller (e.g. 6-2=4 or 6-4=2) is
shown in Figure 4.

FRACTIONATION OF ARITHMETICAL SKILLS
28

-

26

-

39

:/

Addition

2 24C

5!! 2 2 20-

e

5 18C

Correct rolution
20

-

18

-

16

-

u

5

14-

0

E 12e

; 10L

0

-2

8642

2

1

1

1

1

1

1

I

I

I

4

6

8

10

12

14

16

18

20

Correct solution

FIGURE
3. Error analysis: The distribution of incorrect responses is shown as a function of the
correct solution for Test 2 (addition) and Test 5 (subtraction). The short latency responses (0)
are shown separately from the long latency responses (0).

There is a tendency for the long latency rate to increase with the size of the
minimum manipulation (note that with the present procedure the number of data
points per condition decreases with increase in the minimum manipulation).
DRC’s consistent and inconsistent response rate together with the expected values
are given in Table 111, and again his consistency rate is not different from chance.
Test 5 - subtraction

DRC attempted to subtract all possible number combinations of I to 9 and 1 1
to 19 according to the procedure for Test 4. This matrix (81 cells) was attempted
on three occasions (15, 19 and 20 November 1979). The control group attempted
this test once. DRC’s overall error rate was 18% (the control group made no

E. K. WARRINGTON

0

1

2

3

4

5

Minimum manipulation

FIGURE
4. Subtraction: T h e percentage of DRC’s long latency responses is shown as a function
of the minimum manipulation for Test 4 (subtraction).

errors). The distribution of response latencies for DRC and the control group are
shown in Table 11. His average response times for correct and incorrect solutions
were 5.5 and 11.3 s respectively. DRC’s error rate and long latency rate and the
long latency rates of the controls as a function of the minimum manipulation are
shown in Figure 5. DRC’s performance on both measures deteriorates with the

Minlmum monipulation

FIGURE
5. Subtraction: The percentage of DRC’s error responses is shown as a function of the
T h e percentage of long latency responses for DRC (0)
minimum manipulation for Test 5
and for the controls (0)
is shown as a function of the minimum manipulation for Test 5 .

(a).

increase in the size of the minimum manipulation. DRC’s consistent and inconsistent response rate together with the expected values are given in Table 111.
The degree of response consistency does not differ from chance.
DRC’s error responses plotted as a function of the correct solution are given in
Figure 3. All but five of his error responses were of long latency. The correlation
between his error response and the correct solution (combining short and long
latencies) was 0.72.

FRACTIONATION OF ARITHMETICAL SKILLS

41

Test 6 - multiplication

DRC attempted to multiply all combinations of numbers between 2 and 9.
This matrix (64 cells, 4 x 3 and 3 x 4 etc. were both tested) was completed in random
order on two occasions (19 and 22 November 1979). The control group attempted
this test once. DRC’s overall error rate was 8.6% and the distribution of response
latencies is given in Table 11. His overall average speed of response for correct and
incorrect solutions was 2.2 s and 2.4 s respectively. The long latencies rates for
DRC and the control group as a function of the minimum multiplicand are given in
Figure 6. The long latency response rate for DRC was of a totally different order
80

60

c
C

E

i

/-\o

n

-QH

2

3

4

1

I

5

6

I

7

8

Minimum multiplicand

FIGURE
6 . Multiplication: The percentage of long latency responses for DRC (0)and for the
controls (0)
is shown as a function of the minimum multiplicand for Test 6.

of magnitude than for the control group, and furthermore his long latency responses

did not conform to any obvious pattern of task difficulty. DRC’s consistent and
inconsistent response rate together with the expected values are given in Table
111. The degree of response consistency does not differ from chance.

Number knowledge
Quite apart from the computations which can be performed with them, numbers
have connotations. Consider the number 5, it is a small number, it is the label
signifying the number of fingers or toes, it represents the number of units in one
of the coins, it is the age of a child just starting school. Similarly, 47 represents
many, say, family members, but few houses in a street; applied to years it signifies
a person of middle age and applied to time it is a post-war year. Two aspects of
individual number knowledge can be differentiated, precise quantity numbers and
estimated quantity numbers (though whether this is a valid dichotomy is not at
present known). Nevertheless, tests were devised to assess both these aspects of
quantity.
Test 7 - estimation of quantity
In this test the subject was asked to state or estimate the number of black dots
randomly arranged on sheets of white paper ( 8 ” 5”).
~ There were 20 stimulus

E. K. WARRINGTON

42

items (2-10, 15, 20, 25, 30, 40,50, 60, 70, 80,90 or IOO dots/stimulus) which were
presented to the subject in random order. Thus the series comprised quantities
which can be readily counted and quantities which could only be estimated. A
5-s inspection period was used in the first trial to allow the subject to become
familiarised with the task. Three further trials using a 2-s exposure duration were
administered. DRC’s results together with the five control subjects expressed as a
median value (for the three trials) are shown in Figure 7. DRC’s performance is

1.9
ii

+)

f

1.7

dtp

-

1.5 1.3 -

z 0.9-: -

U

2

1.1

E
0

0.7

g 0.50.30.Itl I , , I , I I , I
0.10.30.5 0.70.91.1 1.3 1.5 1.7 1.9

Log of actuol number

FIGURE
7. Estimation of quantity: The log of the estimated number of dots is shown as a function
and for the control group
of the log of the actual number of dots for DRC (0)

(e).

clearly unimpaired either for quantities which could be counted (up to 10)or
quantities which had to be estimated (over IO), indeed his number estimation for
the larger numbers was marginally more accurate than that of the control group.
Response latencies were not recorded on this task, but it was the impression of the
experimenter that DRC responded rapidly and as quickly as the control subjects.

Test 8 - rapid asithrnetical estimations
DRC appeared to be able to estimate approximate solutions to arithmetical
calculations. A test was devised to assess this ability, namely to give the approximate solution to arithmetical problems which for most normal subjects the solution
would not be “automatically” known. There were 12 problems, (four divisions/
fractions, four additions and four subtractions), all involving large 2 to 3 digit
numbers (see Appendix I). The subjects were presented with the problem and
instructed to respond quickly (within I to 2 s) giving an approximate solution.
This procedure was not entirely satisfactory in so far as the control subjects would
give slow responses apparently attempting a partial computation in the time available. Nevertheless DRC’s responses were within the normal range for five of
these 12 problems and relatively close to the normal range on the remaining seven
problems (see Appendix I).

FRACTIONATION OF ARITHMETICAL SKILLS

43

Test g - numerical cognitive estimates
Shallice and, Evans (1978) devised a “cognitive estimates” test in which the
subjects were asked to give “reasonable”, “sensible”, responses to questions for
which exact factual information is not generally known (e.g What is the best-paid
job or occupation in Britain today? How tall is the average English woman?).
Following Shallice’s technique, a test comprising 10numerical estimation questions
was devised (Shallice’s items 5, 6 and 9 were included) (see Appendix 11). DRC’s
responses were within the range of responses given by the control group for 9/10
questions (see Appendix 11).

Test 10 - number size judgements
A test was devised in which the subject was asked to “circle” the larger of two
numbers. 20 pairs of two- or three-digit numbers in which difference in size
ranged from one to 12(e.g. 28 to 29,185 to 197)were typed in a column: there were
three sets of 20 number pairs on the page (8”x 10”). The time taken for the subject
to circle the larger of each pair for each of the three sets of numbers was recorded.
DRC’s median response time for the three trials was 37” and the median response
time for the five control subjects ranged from 18”to 32”. DRC’s median response
time is only marginally slower than the slowest control subject and bearing in mind
that DRC was handicapped by a resolving right visual field defect his performance
on this task is considered to be entirely satisfactory.

Test II - number facts
Knowledge of exact number facts was assessed, and the five control subjects
responded correctly to the following questions (which can be answered using
general knowledge available to almost all subjects).
(I) Boiling point of water, (2)feet and inches in I metre, (3)inches in a foot, (4)feet
in a yard, (5) ounces in a pound, (6) centimetres in a metre.
DRC’s only error was on item (5) to which he responded “between one dozen and

20”.
Test 12 - numerical abilities test
Hitch (1978a) devised a test of numerical ability in which there were three
sections :
(I) A test of specd and accuracy of elementary arithmetic with natural numbers.
This test comprised six subtests : (I) adding three-digit numbers, no carrying required; (2) adding three-digit numbers, carrying required; (3)
subtracting three-digit numbers, no carrying required; (4) subtracting
three-digit numbers, carrying required ; (5) multiplication of digits (max
12); (6) division of two-digit numbers by single digit numbers.
(11)A test of manipulation and conversion of decimals and fractions and
percentage where the numerical element was relatively simple. This test
comprised four subtests : ( I ) solving fractions, (2) solving decimal problems; (3) conversions between fractions, decimals and percentages; (4)
further long divisions and multiplications. This test was unpaced.

44

E. K. WARRINGTON

(111) A test of appreciation of the magnitude of numbers and the evaluation of
arithmetical expressions. This test comprised four subtests :( I ) magnitude
judgements; (2)approximation problems ; (3) comprehension of numerical
expressions; (4)solving ratio, proportion and percentage problems. This
test was unpaced.
DRC attempted this test on 21 and 22 November according to the procedure
prescribed by Hitch for testing a group of 95 industrial trainee apprentices aged
between 17 and 18. The mean and s.d. of this control sample (kindly provided by
Hitch) on each of the three sections of Hitch‘s test, together with DRC’s score on
each of the 16 subtests are given in Table IV. DRC’s performance expressed in
terms of the standard deviation from the control group mean, together with the
significance of the difference is also given in Table IV. It is reasonable to suppose

TABLE
IV
Scores on Hitch’s test of numerical abilities

DRC’s
score

Section I
(I) Addition - no carrying
(2) Addition - with carrying
(3) Subtraction - no borrowing
(4) Subtraction - with borrowing
(5) Multiplication tables
(6) Division
Total
Section 11
(I) Fraction arithmetic
( 2 ) Decimal arithmetic
(3) Conversions
(4) Long multiplication and division
Total
Section 111
(I) Magnitude
(2) Approximation
(3) Rules for expression
(4) Ratio, proportion and percentages
Total

7
5
8

Controls’
mean
score
I 6-66

20

8.77
14-82
4‘83
23-72

8
48

93-30

0

8
8
9

24-20

2

5’56
7-88
6-15
2-46

27

22-05

I1

8-59
5’32
5 ’40
7.10
26.40

7
6
7
31

DRC’s s.d.
Controls’ from
s.d.
mean
4‘27
2.93
3.68
2-58
8.7 I
7-93
25‘73

-2*26*
- 1-29
- I *85*
- I -87*
-0’43
-2-04*
- 1 ‘75*

3’23
2.82
3 *98
1’37
9’36
1.41
2’02

1.56
I *89
5’27

+

1.71
+0*83
fo.38
-0.05

+0.87

*Significant at 5 % level.

that premorbidly DRC’s performance would have been superior to Hitch’s sample,
on all sections of the test. DRC’s performance on section I demanding speed and
accuracy of elementary arithmetic is significantly worse than that of the control
group. In contrast on sections I1 and I11 of the test his performance was better
(though not significantly) than the control group. Considering the individual
subtests, on four out of six of the subtests in section I his performance was signifi-

FRACTIONATION OF ARITHMETICAL SKILLS

45

cantly worse than that of the control group and in sections I1 and I11 his performance was slightly higher than that of the control group on six out of eight of the
subtests.
Discussion
DRC lost his ability to perform arithmetical calculations with any semblance of
efficiency after sustaining a left posterior haemorrhage. His knowledge of the
significance of individual numbers and his concept of quantity appeared to be
intact, but simple addition, subtraction and multiplication were all performed
laboriously and inaccurately. Two major questions of interest arise. First, which
subcomponent or subcomponents of the processes involved in arithmetical calculation were impaired? Second, do the findings favour any particular model of
arithmetical calculation proposed for the normal person?
However first it is necessary to consider whether it has been established that
DRC had a selective impairment of arithmetical skills which was not secondary to
any other cognitive deficits. There were only minor degrees of impairment on
verbal intelligence tests, on a verbal memory test, and on tests of dysphasia. I n
particular his word comprehension and word retrieval skills were within normal
limits. As his digit span was above average it can be assumed that his ability to
perceive and articulate digits was normal; more important he was not handicapped
by a short-term memory deficit for digits on any of the tests of numeracy. Indeed
on all the tests of numeracy DRC attempted, it seems clear that it was the demands
of the task rather than a non-specific language impairment which determined his
performance.
Calculation is a highly complex skill and it will be argued that it was his arithmetical skills not his number knowledge which were impaired, and further that it
was his knowledge of arithmetical facts not his knowledge of arithmetical operations
which was impaired.
Two aspects of number knowledge or significance can be differentiated, exact
quantity facts and approximate quantity facts. DRC comprehended numbers
both absolutely as number facts (e.g. 12inches= I foot) and relatively (e.g. 12) I I).
He could use numbers to make direct estimates of quantity (as in the visual stimulus
arrays used in Test 8) and also indirect estimates of quantity applying general
principles of numerical knowledge (as in the cognitive estimates used in Test 9).
Thus it is argued that the central representations (semantic entries) of individual
numbers are intact and accessible.*
Consider a simple addition of the form x+y=z.
It has been shown that DRC
comprehended the numbers (i.e. had knowledge of their absolute and relative
quantity). There would appear to be two further stages in achieving a correct
solution, first the instruction add must be understood, and secondly the sum must
be computed (whether by counting or direct access will be considered later). It
has been demonstrated that DRC’s comprehension of the arithmetical operations
was intact. He had no difficulty in explaining verbally the appropriate strategy
*It is not to be assumed that this is a typical feature of all patients presenting with the syndrome
acalculia. Shallice and I investigated a patient STH who was unable to comprehend the significance
of individual numbers and individual number facts and consequently the simplest arithmetical
calculation could not be done. A brief case description of STH is given in Appendix 111.

46

E. K. WARRINGTON

for solving a complex arithmetical problem such as the WAIS arithmetic
subtest) : he could state the operations required for solution and the order in which
they should be performed. In no instance did he produce an impermissible
response (e.g. a sum less than either addend) and indeed his incorrect responses
were shown to approximate closely to the correct response. There was a very clear
dissociation on Hitch‘s numerical abilities test which specifically compares number
operations with number computations. DRC’s performance was significantly
impaired on the strictly timed tests of arithmetical computations; in striking
contrast his performance was slightly (non-significantly) above average on those
tests which maximise comprehension of arithmetical operations. Particularly
impressive was his good ability to give approximate solutions to more difficult
arithmetical problems involving large numbers. These skills could not be achieved
were the comprehension and application of arithmetical operations not intact.
The dissociation between arithmetical processing in general and accurate arithmetical computations is the main point of interest to emerge from this investigation
of DRC’s acalculia. This distinction, though implicit in the work of Hitch (1978a)
is not made explicit in any study of calculation in the child or normal adult.
Restle (1970) has argued that simple addition is achieved in normal adults by an
analogue operation. DRC’s very satisfactory performance on test 10,in which he
was required to make number size judgements at speed, suggest that such operations may be preserved. If this supposition were correct, then to account for
normal calculation in terms of analogue processing would be insufficient, at least it
would appear to be insufficient to account for not merely accurate calculations but
rapid calculations. It will be argued that access to arithmetical facts is essential
for efficient calculation skills.
Why then is the correct solution inaccessible or only inefficiently accessible?
This can most appropriately be discussed in the context of Parkman and Groen’s
model of the normal process of calculation. They proposed the “counting” model
subsequently elaborated as the retrieval-counting model (Parkman and Groen,
1971; Groen and Parkman, 1972). Their earlier counting model assumed that
mental addition consists of two processes, first a counter is set to the larger addend
and second a process of incrementation occurs so that the number of increments
equals the minimum addend. The strength of this model derived from the
observation that in both children and adults response latencies are a function of the
minimum addend, but it was necessary to revise this simple counting model to take
into account the fact that ties (e.g. 3+3) were all faster and equally faster than other
single digit additions (this was also the case for DRC). The radically different
slopes of the R T functions for children (400 ms/increment) and for adults (20
mslincrement), as has already been acknowledged (Groen and Parkman, 1972), can
hardly command the same explanation. (DRC’s unit incremental times are of the
same order of magnitude as in young children :for example in Test I he achieved an
overall unit incremental time of 0-73 s.) Furthermore, it is difficult to encompass
multiplication and division within the framework of this model.
Therefore a retrieval-counting model was proposed, which states that most
response times reflect direct access to memorised information, such access to each
semantic entry requiring an essentially constant amount of time (all entries being

FRACTIONATION OF ARITHMETICAL SKILLS

47

equipotential) and on those occasions when direct access fails the subject reverts to
the slow “counting” process observed in children (Groen and Parkman, 1972)’
Such a formulation would require a bimodal or skewed distribution of response
times but this has not yet been reported for normal adult performance. However,
inspection of Table z suggests that this may be a feature of DRC’s performance
especially in the more demanding tasks, such as Test 5 . It seems plausible to
suggest that in both young children and in DRC there is a failure of direct access in
the former because there is not yet a direct semantic entry and for DRC because it
is damaged or inaccessible. T h e observation that DRC’s reponse latencies are
inconsistent from trial to trial favours an access deficit, rather than damage to
specific semantic entries. Thus for DRC it is suggested that there is faulty access
to the semantic entries of the facts of arithmetic (e.g. 2+3=5), and that when
direct access fails he resorts to a more inefficient counting strategy by which means
he often arrives at a correct solution, albeit slowly. DRC’s performance on the
multiplication test, the arithmetical operation which would seem to be most
dependent on direct access to arithmetical facts, was inconsistent between the two
trials although his level of performance remained constant. This finding
strengthens the present hypothesis that inaccessibility rather than damage to the
semantic entries of arithmetical facts is at the core of his acalculia. Thus it is
argued that DRC is using the samc processing strategies as a normal adult, the
crucial difference being that the proportion of access failures is much greater.
There is now accumulating evidence from neuropsychological investigations that
semantic memory systems are categorically organised. That broad semantic
categories (e.g. letters, colours, words, objects) can be selectively impaired is
perhaps not unexpected, but there is now evidence that subcategories within a
major system can also be selectively impaired. For example, the double dissociation
of abstract and concrete word comprehension impairment and the evidence
of selective impairment and selective preservation of the comprehension of object
names have been interpreted in these terms (Warrington, 1981). Similarly, it is
suggested that numeracy represents a major category of semantic knowledge and
that in DRC a subcategory of this system is inoperative, namely the accessibility of
arithmetical facts. A more detailed investigation of patients with acalculia could
well provide pointers to the more fine-grain organisation of arithmetic facts than
has been achieved in this investigation.
*Modificationsof this theory have been suggested, for example by Ashcraft and Batlaglia (1978),
but there is as yet insufficient empirical evidence to evaluate their relative merits in relation to DRC.

I am grateful to D r J. N. Blau and to D r C. J. Earl for permission to investigate their
patients and to report my findings. I wish to thank Dr G. Hitch and D r T. Shallice for
their interest in this investigation and for their advice in the preparation of this manuscript.
Richard Hutton very kindly undertook the analysis of the results of Hitch’s numeracy test.
I a m particularly indebted to Pat McKenna for her assistance in testing DRC.

48

E. K. WARRINGTON

Appendix I
Test 8 - rapid arithmetical estimation
Problem

DRC response

Controls
response range
~

~

I35
212-250

23-38
6-80

I200

990-1000
1150-I200

1000

I 160-1200

400

381-400

900

30

20-25

60
800

50-5 5
350-800

3000

2500-3000

Appendix I1
Test g - numerical cognitive estimates
Question

DRC response

Controls
response range

109
5' 5"
48"

106-1I I
5' 4"-5' 6"
25"-39"

(I)Age of oldest person alive in Britain
( 2 ) The height of average Englishwoman
(3)The length of an average man's spine
(4)The distance from London to Paris
( 5 ) The distance from London to New York
(6)The distance from London to Edinburgh
(7)The temperature on a hot day in England
(8) Population of Great Britain
(9) Population of Ireland
(10) The number of people who can board on a
double-decker bus

200

100-~00

3000

1500-3000

400
80"
8,000,000

350-600
80" F
50-70 million
1-13 million

80

60-80

~0,000,000

Appendix I11
Case report

STH a 25-year-old left-handed housewife with a long history of hemiplegic
migraine was admitted to the Middlesex Hospital one month after an acute episode
in which she became dysphasic and hemiplegic. On examination there was a
marked right-sided hemiparesis with no movement at all in the upper limb, and
very little in the face. Sensation was diminished in the right limbs. There was a
dense right homonymous hemianopia. These neurological signs showed gradual
recovery over the following 2 months. Her language was severely impaired, her

FRACTIONATION OF ARITHMETICAL SKILLS

49

spontaneous speech being restricted to very short non-grammatical phrases, and
all other aspects of language function were impaired. Her language skills did not
show the same degree of improvements as her other neurological deficits.
A C T scan on 18 March 1980 indicated diffuse isodense swelling of the posterior
part of the left hemisphere with compression of the left lateral ventricle. On a
repeat C T scan on 22 April 1980,the left lateral ventricle had become normal in
shape and in retrospect it was considered that there had been some low attenuation
in the posterior temporal lobe and the occipito-temporal gyri of the left hemisphere
previously.

Psychological findings
STH was tested on a shortened version of the WASS and obtained a verbal IQ
of 80 and a performance IQ of 73. T h e individual age-scaled scores were:
arithmetic, 2; similarities, I I ; digit span, 6; picture completion, 9;block design, 7 ;
picture arrangement, 2. Her sentence construction was extremely halting and
sparse, but her object naming was relatively well preserved in that she scored I 2115
and 13/15 on tests of naming to confrontation and naming from description (see
Coughlan and Warrington, 1978)and on a graded difficulty naming test she scored
9/30 (see McKenna and Warrington, 1980). Her ability to comprehend single
spoken words appeared to be intact insofar as she obtained an SQ equivalent of I 12
on the Peabody picture vocabulary test. I n contrast she was severely dyslexic and
dysgraphic. She read only 14words on the Schonell graded word list (testing was
discontinued after 10 consecutive errors). She failed to spell aloud or write
correctly any words from the Schonell graded spelling test. Of particular interest
in the present context was her apparent greater difficulty in reading single digits
(15124 correct) than letters (23/24correct). She also had difficulty in writing
single digit numbers to dictation (5110 correct). The following tests of numeracy
were administered :
(I) T h e numbers I to 9 were spoken in random order and S T H was asked to
match the spoken to the written number (3 x 3 random array). Her score
was 17/20correct.
(2) A set of numbers greater than 10were selected (1000,
100,50,40,30, 20, 15,
12, 11) and STH was asked to match the spoken number to the written
number (in a 3 x 3 random array). Her score was 12/20correct. She was
only able to read correctly 2/9of this set of numbers.
(3) Pairs of numbers (range I to 12)were selected such that one of the pair was
larger by either I , 2 or 4 (e.g. 1-11, 7-9, and 4-8,etc.). There were 10of
each type of stimulus pairs and each pair was spoken slowly and S T H was
asked to name the larger of the two numbers. She clearly understood the
task and she was able to repeat the message (note her digit span was five).
She scored 23/30 correct and her errors were not obviously related to the
magnitude of the difference. With visual presentation, i.e. pointing to the
larger of the two printed numbers, her performance was error-free.
(4)Cognitive estimates: she attempted a shortened version of the Shallice and
Evans (1978) cognitive estimates test; not one of her responses to the

50

E. K. WARRINGTON

numerical items was satisfactory (e.g. Q. What is the average length of a man's
spine? R. 6 ft). Her number estimation was clearly very impaired.
( 5 ) Number facts: STH was able to count from I to 10fairly efficiently and at
reasonable speed. Questions were devised for which there was an exact
precise number response. She was able to answer the simplest questions
(e.g. How many eggs in 4 doz? How many legs has a bird?), but she was
incorrect in stating the number of toes on one foot, the number of days in
a week and the number of fingers on two hands. It seems likely from her
performance on the WAIS similarities test that her language comprehension
was adequate to the task.
(6) Arithmetical facts: I n view of STH's impairment in the comprehension of
spoken numbers, her oral arithmetical skills were not systematically tested.
It was noted informally that very simple addition, subtraction and multiplication were quite impossible for her.

Comment

STH has an acalculic syndrome quite different from that recorded in DRC. It
would appear that individual spoken numbers have lost their significance for her;
the spoken numbers had no connotations. It is assumed that her difficulties with
oral arithmetic calculations was secondary to a more basic failure at the level of
number knowledge.
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Manuscript received 18 December 1980