NOTE SUBTRACTION BUGS IN AN ACALCULIC PATIENT*

Luisa GirellP and Margarete Delazer2

e Department of Psychology, University College London; 2 Department of Neurology, University Clinic Innsbruck)

ABSTRACT

We report a patient, MT, who presented a specific, though not isolated, deficit in written calculation. Despite a preserved knowledge of simple arithmetic - single-digit addition and subtraction - he failed systematically in multi-digit subtraction. The nature of errors was consistent across problems and reflected the application of a disturbed underlying algorithm. Moreover, the pattern of error observed mimics a very common finding in developmental studies on arithmetical procedure acquisition (Fuson, 1990, 1992; Young and O'Shea, 1981; VanLehn, 1986, 1990). The data suggest that, within calculation skills, syntax may exist as a system of stable, but inappropriate, rules which are independent of any underlying conceptual knowledge.

INTRODUCTION

The execution of a complex calculation, such as 2453 + 879, is a multi-step process. In addition to number comprehension and production mechanisms, three specific cognitive operations are required. First, the arithmetical sign (e.g., +) has to be recognised in order to identify the operation to be performed; second, the single-digit operations have to be answered or retrieved from memory (e.g., 3 + 9) and finally, the sequence of single-digit operations has to be organised as specified by arithmetical procedures (e.g., write the Ones, carry the Tens).

These distinct cognitive mechanisms are thought to constitute selectively vulnerable components of the calculation system (McCloskey, Caramazza and Basili, 1985; McCloskey, 1992).

Patients with a selective inability to process operational symbols have been occasionally reported (Ferro and Botelho, 1980; Grewel, 1969). While there is strong evidence that the retrieval of arithmetical facts can be selectively impaired after cerebral lesion (e.g., Warrington, 1982; Sokol, McCloskey, Cohen et a!., 1991; Cohen and Dehaene, 1994; Hittmair-Delazer, Semenza and Denes, 1994), there are few reports of selective impairment in calculation procedures (e.g., McCloskey, Caramazza and Basili, 1985; Lucchelli and De Renzi, 1993).

Benson and Weir (1972) reported a patient who was unable to perform multi-digit multiplication and division; however, it is not clear from their report the extent to which the performance in multi-digit calculation was affected by a defective knowledge of arithmetical facts.

McCloskey et a!. (1985) reported four patients whose deficit in multi-digit calculations were attributed to defective knowledge of arithmetic procedures. However, no background data were given for these cases (except for HY, McCloskey, Sokol and Goodman, 1986) and the authors limited their description to a total of 15 errors for the four patients. Grafman,

* The paper has been presented at the British Neuropsychological Society in London, April 1995.

Cortex, (1996) 32, 547-555

548 Luisa Girelli and Margarete Delazer

Kampen, Rosenberg et al. (1989) reported a patient affected by progressive dementia whose initial complaint was a problem in calculation. The patient presented an orderly dissolution of arithmetical skills; initially, while still able to solve complex addition and subtraction, he failed in executing multiplication and division problems (this patient, however, was also unable to answer single-digit multiplication and division). Lucchelli and De Renzi (1993) described the case of a young student who suffered an infarct in the territory of the left anterior cerebral artery. He lost knowledge of procedures while the retrieval of arithmetical facts was only mildly impaired.

More evidence for a double dissociation between impairments in arithmetical facts and arithmetical procedures comes from a study on developmental dyscalculia (Temple, 1991). While a 17 year old boy accurately answered arithmetical facts but failed to apply arithmetic procedures, a 19 year old girl showed the reverse pattern with intact procedures and disturbed arithmetical facts.

Furthermore, dissociations can occur within the procedural component of the calculation system. Recently, a patient whose calculation skills were limited to simple and complex substraction, has been reported by Lampl, Eschel, Gilad et al. (1994). This evidence strongly suggests that individual operation procedures are independently represented in the calculation system. So far, very few cases of deficit in arithmetic procedures despite intact number facts retrieval within the same operation have been reported; moreover, no systematic analysis of the errors was ever performed.

The data indicate that knowledge of arithmetical facts and knowledge of procedures are functionally independent; however, while the first side of the double dissociation (i.e., intact procedures, disturbed knowledge of facts) seems well established by several reports, the second (i.e., disturbed procedures, intact fact knowledge) clearly needs further support.

The acquisition of arithmetical procedures and the difficulties typically encountered by young students when performing multi-digit calculation have been extensively investigated (Fuson, 1990, 1992; VanLehn, 1986, 1990). Since Brown and Burton's seminal study (1978) it has been consistently reported that most of the errors in children's written calculations are due to the use of incorrect strategies rather than to the incorrect recall of number facts (Young and O'Shea, 1981; Resnick, 1982).

Systematic errors are considered to be determined by an incomplete or misleading understanding of the procedures: namely they are result of misconceptions (VanLehn, 1983). According to VanLehn (1983, 1986) these misconceptions can be formally represented and precisely described. That is, children's errors can be precisely reproduced by making one or more perturbations to the formalised representation of the correct algorithm. These perturbations are called 'bugs' (Brown and Burton, 1978). A generative theory of bugs that simulates learning processes has been advanced (Brown and VanLehn, 1980) and its predictions concerning the types of procedural errors accurately match a large data base collected from several thousand students.

According to VanLehn (1986, 1990) the successful application of arithmetical procedures does not require teleological/conceptual understanding but simply represents a schematic knowledge.

Similarly, Resnick (1982) distinguished between the syntax and the semantics of a calculation algorithm. The syntax includes a set of rules indicating how to proceed when solving a problem (e.g., right to left, write the ones etc.). The semantics consists in the actual understanding of each single step in the procedure (e.g., in multi-digit subtraction, the 1 inserted as part of borrowing really represents a 10 or a power of ten depending in which column the borrow is required). Though the syntax reflects an underlying semantics no explicit reference to the semantics is needed to perform the algorithm successfully.

The lack of teleological (conceptual) understanding (VanLehn, 1986), as well as the disconnection between syntactic and semantic knowledge (Resnick, 1982), are thought to determine systematic errors.

In the present study, we report a single-case study that supports the selective vulnerability of arithmetical procedures. The patient showed a specific breakdown in multi-digit subtraction. Moreover, we describe the systematic nature of the patient's procedural errors and the dissociation between preserved and lost sub-procedures.

Subtraction bugs in an acalculic patient

TABLE I

Language Assessment. Aachener Aphasie Test (Luzzatti, Willmes and De Bleser, 1991)

Token Repetition Written language Naming Comprehension Auditory Written Classification: Global aphasia

Raw score

31160 56/150

0/90 4/120

74/120 53/60 21160

CASE REpORT

T-value

44 18 5 9 4

Deficit

medium severe severe severe

mild medium-severe

549

MT is a 55 year old, right-handed man, with 5 years of education. Before his illness he used to work as a retailer. He indicated he used to carry out calculations as part of his daily work. In 1987 he suffered a CVA in the territory of the left middle cerebral artery. He developed severe aphasia and weakness of the right arm. A CT scan showed a wide area of hypodensity involving the fronto-temporo-parietal regions of the left hemisphere. In terms of the motor impairment a complete recovery was made over the following months, but there was no improvement of the language deficit. Since the onset of the CV A the patient had undergone intensive language therapy; however, at the time of the present testing - 4 years after the onset - his aphasic pattern was still of global type.

Neuropsychological Assessment

Spontaneous speech was non-fluent and limited to single words and a few stereotype phrases. MT was submitted to the Italian version of the Aachener Aphasie Test (Versione Italiana: Luzzatti, Willmes and De Bleser, 1991). He scored very poorly in all subtests (see Table I).

He showed a severe deficit in the Token Test and in Written comprehension whereas Auditory comprehension was only mildly impaired. Repetition was severely compromised as well as naming. MT could not read nor write, either to dictation or when attempting to assemble printed letters. His visuo-spatial span, as assessed by the Corsi test, was 4 (mean=5.11; SD= 1.01; Spinnler and Tognoni, 1987). In the Raven Matrices test he scored 25/36.

On an initial numeracy screening test his performance was error free in single-digit addition (717) and substraction (717) but he showed impairment in simple multiplication (4/ 15). In multi-digit written calculation only addition problems were solved correctly (4/4). He was able to count from 1 to 30 in written form and to write down a multi-digit operation in the correct vertical alignment. MT was unable to score on a graded difficulty mental calculation test (Jackson and Warrington, 1986).

Numerical Skills

A more detailed investigation of MT's number processing and calculation skills was carried out using the Miceli and Capasso Battery (1991) (Table II). Not surprisingly, the overall performance indicated a severe deficit in encoding and producing spoken and written verbal numerals.

MT's performance was error free in quantitative tasks involving dots. He was able to select the larger of two Arabic numbers (magnitude comparison: 33/34), but the same task was poorly performed when the stimuli were either spoken or written verbal numerals. Consistent with the outcome of the language assessment, he could neither repeat spoken numerals nor read aloud written verbal numerals; his reading of Arabic digits was limited

550 Luisa Cirelli and Margarete Delazer

TABLE II

Number Processing and Calculation (Miceli and Capasso, 1991)

Non-numerical tasks Magnitude comparison Dot pattern seriation

Magnitude comparison Visual-digits Visual-number words Verbal

Transcoding tasks Repetition Reading arabic numerals Reading number words Writing arabic numerals Writing (from words to digits)

Recognition of arithmetical signs Visual Verbal

Mental calculation

Written calculation Addition Subtraction Multiplication

10110 5/5

Total score

33/34 11/34 11/34

0/55 II/55 0/55

24/55 11155

10110

0/20 Id/ld

10/10 10110 3/10

1-2 digit

19/20 1I120 10/20

11125

24/25 11/25

2d/2d

4/5 4/5 0/5

3d/3d

3/5 2/5 0/5

to single numbers (ones: 8110; teens 3/5) as well as transcoding from written verbal numerals to Arabic digits (ones: 8/10; teens 3/5). However, he was able to write Arabic numerals to dictation up to two-digits (24/25).

He correctly recognised the arithmetical signs when they were visually presented. MT showed a preserved ability to solve written single-digit addition and subtraction problems. However, he failed to retrieve simple multiplication facts (3/10). Interestingly, most of the errors involved non-table numbers, for example, 6 X 7 = 37. When the problems were presented auditorily his overall performance dropped drastically .

Performance in multi-digit written calculation varied according to the type of operation. In the case of addition problems, few calculation errors were observed whereas in subtraction problems MT failed to use the borrow procedure. Moreover, 9 out of 10 multiplications were solved as if they were addition, which were, it is worth noting, correctly carried out. For example, presented with the multiplication 78 X 26 he worked out the sum of the two factors and answered 104. Since no operation errors were observed in single-digit multiplication, this pattern is unlikely to stem from a failure to recognise arithmetical signs. It is possible that "switching to addition" was an undeliberate procedure the patient adopted to overcome either his lack of memory for the multiplication algorithm, or his difficulty in retrieving multiplication facts. This interpretation, however, is speculative rather than conclusive.

Finally, the patient was able to compose a given number, ranging from two to three digit Arabic numerals (e.g., 120) from tokens of different numerical value (i.e., 1, 2, 5, 10, 20, 50, 100) (23/24).

In summary, the data indicate that MT's calculation deficit cannot be attributed to comprehension or production difficulties. He was able to pelform a magnitude comparison task as well as to compose a given number from token, which shows a good comprehension of Arabic numerals. Moreover, he solved written addition and single-digit subtraction and he correctly wrote, to dictation, single digit numbers. This indicates a preserved ability to produce Arabic digits.

In order to investigate MT's impairment in the execution of calculation procedures a

Subtraction bugs in an acalculic patient 551

series of multi-digits operations were administered. Due to MT's severe deficit in retrieving multiplication facts, only multi-digit additions and subtractions were presented.

Written Multi-digit Calculation

The task was completed over two sessions and did not involve any time pressure. The patient carried out the task with sustained attention and did not use overt strategies.

Addition

The patient was presented with 15 multi-digit (two and three-digit Arabic numerals) addition problems. 9 of them required the carry procedure. Problems were presented in random order.

MT's performance was both fast and accurate (100% correct).

Subtraction

36 multi-digit problems (two and three-digit Arabic numerals) were visually presented. Stimuli were chosen so that some problems did not necessitate borrowing (e.g. 78-26; N = 7) while others did (e.g. 84-57; N = 29). Problems were presented in random order.

The patient correctly solved all the problems without borrowing (7/7), but made as many as 25/29 errors in the problems that required borrowing.

Error Analysis

The nature of the errors was consistent across problems. MT systematically subtracted the smaller digit from the larger one regardless of their location in the top or bottom numbers (Figure la). However, in some cases he still applied the borrow procedure. Examples of his errors are illustrated in Figure 1.

In order to clarify MT's procedure when solving multi-digit subtraction, the problems were regrouped according to the magnitude of the numbers involved and the column where the borrow procedures was required (see Table III).

TABLE III

MT's Peiformance in Multi-digit Subtraction

MT's incorrect procedures

Example borrowing from S from L still borrow

84 - the tens in 2d-2d 100% 57 =

33

501- the tens and/or hundreds in 3d-3d 100% 322=

221

854- the tens and/or hundreds in 3d-2d 100% YES 98=

744

138- the hundreds in 3d-2d* 0% 74=

64

* Problems with no 0 in the top line.

552 Luisa Girelli and Margarete Delazer

a) b)

c) d)

Fig. 1 - Illustration of MT's procedures in multi-digit subtraction; (a) Smaller-from-Larger; (b) Smaller-from Larger-with-borrow; (c) linkage strategy example; (d) Dif.f-O-N=N.

In all problems requiring the borrow procedure MT consistently applied the "Smaller­from-Larger" bug with one exception: when a two digit number was subtracted from a three digit number and borrowing was required in the tens column (e.g., 138-74). In fact, when a two-digit number is subtracted from a three digit number, so that the leftmost column is blank in the bottom line, the standard borrow procedure can be bypassed if the hundred and the ten units in the top line are linked together to be perceived as a whole number. Thus, in the example (see Figure lc), 1 is linked to 3 so that 7 is directly subtracted from 13. These were the only problems requiring the borrow procedure correctly answered (4/4).

In some problems with the same structure (3d-2d with borrowing required in the tens column) MT still applied the Smaller-from-Larger bug as if the linkage between hundreds and tens units was not preceived. These problems, however, presented a further feature: the top digit in the tens column was zero. When MT encountered a column of the form O-N, he consistently wrote N as the answer (Figure ld; 3/3). This bug has been described by VanLehn as the "Diff-O-N=N" bug (1986) and it constitutes a special-case of the Smaller­from-Larger bug. It is possible that, in such a case, the O-bug would preclude the execution of the "linkage strategy".

Furthermore, in a sub-group of problems where the Smaller-from-Larger bug was applied, MT still used the borrow procedure (Figure lb). Interestingly, the Smaller-from-Larger and the Smaller-from-Larger-with-borrow bugs were not used randomly. The former was observed in all problems where minuend and subtrahend have the same number of digits (9/9 problems of which 4 of the form 2d-2d and 5 of the form 3d-3d). The latter was applied when a 2 digit number was subtracted from a 3 digit number (16/16).

Subtraction bugs in an acalculic patient 553

DISCUSSION

The patient we described presented a specific, though not isolated, deficit in written calculation. He was fast and accurate in the solution of single-digit addition and subtraction problems, but, while he mastered the computational procedures of addition appropriately, he systematically failed when presented with multi-digit subtraction problems.

The data indicate a specific loss of arithmetical procedures despite intact arithmetical fact knowledge within the same operation. Thus, the case completes the double dissociation between fact and procedural knowledge as proposed by current calculation models (McCloskey et aI. , 1985; Dehaene, 1992) and confirms the functional independence of calculation components.

The nature of the observed errors was consistent across problems and reflected the application of a disturbed underlying algorithm. MT systematically subtracted the smaller number from the larger one irrespective whether the larger digit was in the top or in the bottom line. Surprisingly, in a group of problems, MT preserved the knowledge of the borrow­sub-procedure. He correctly decreased the one column left when the larger digit was in the bottom line. Thus he recognised the situations where borrowing is required and so did not apply the sub-procedure randomly. Despite this, he subtracted the smaller from the larger number. This result clearly shows that a sub-procedure can be preserved and applied without being completely understood.

We thus find a dissociation within the subtraction algorithm, between preserved sub procedures (i.e., borrow when the larger digit is in the bottom line) and lost sub procedures (i.e., subtract the number in the bottom line from the number in the top line).

This pattern of error has never been observed either in acalculic patients or in normal subjects. Indeed, no similar error occurred in the performance of three matched control subjects.

However, the Smaller-from-Larger bug is one of the most common errors observed in children (Fuson, 1990; Resnick, 1982; Young and O'Shea, 1981); within developmental research, it is interpreted to result from the application of the "Swap vertically" repair heuristic (Resnick, 1982). Taking the absolute difference of each column's digits, it makes the borrow procedure no longer necessary. If a child does not know how to borrow, switching the arguments is one of the few possibilities for continuing the calculation.

MT's error pattern in multi-digit subtraction includes three different bugs labelled by VanLehn as "Smaller-from-Iarger", "Smaller-from-larger-with-borrow" and "Diff-O-N = N" (VanLehn, 1990). These bugs violate the non-commutative nature of the subtraction and the constraint that the columns, although handled one a time, cannot be treated as a string of unrelated single-digit problems.

The relation between syntax and semantics in the development of calculation procedures has already been investigated (Resnick, 1982). Taking a wider perspective, the question to be addressed is how concepts and procedures are related in mathematics (Hiebert, 1986; Hittmair-Delazer et aI., 1994). MT's case suggests that this distinction cannot be neglected in the study of acquired procedural deficit.

Evidence for an intact procedural knowledge not supported by conceptual understanding has been previously reported (Sokol et aI., 1991). Sokol et al. (Sokol and McCloskey, 1990, 1991; Sokol et aI., 1991; McCloskey et aI., 1991) described a series of patients showing a deficit in retrieval of multiplication facts including a very low performance in problems involving 0 (e.g., NXO, OXN). However, all patients were virtually error-free in O's in multi-digit multiplication problems. The authors interpreted the observed dissociation postulating "special-case procedures" for processing multi-digit O's problems which would allow one to bypass retrieval of single-digit problems. The dissociation between single-digit and multi-digit O's problems suggests that "special-case procedures" consist in a rote strategy which does not derive from conceptual knowledge.

More revealing about the procedural-conceptual interaction in arithmetic is the development of new algorithms which violate basic arithmetical principles. Evidence for this was found in MT's performance which indicated an impaired schematic knowledge of the subtraction algorithm.

In order to compensate his deficit MT systematically applied an inappropriate algorithm

554 Luisa Girelli and Margarete Delazer

and the nature of this algorithm does not reflect conceptual understanding. This case strongly suggests that, within calculation skills, syntax may exist as a system

of stable - but inappropriate - rules which are independent of any underlying conceptual knowledge.

Acknowledgements. The research reported was supported by Grant SCI*-CT91-0730 from the European Community. We are grateful to Marcus Giaquinto and Brian Butterworth for helpful comments on the first draft of this paper. We wish to thank Prof. Gianfranco Denes for permission to study MT, a patient under his care.

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Luisa Girelli. Department of Psychology, Universi ty College London, Gower Street, London WCIE 6BT, U.K.