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Noncommutability of The N + 0 Arithmetical Rule: A Case Study of Dissociated Impairment
NOTE NONCOMMUTABILITY OF THE N + 0 ARITHMETICAL RULE: A CASE STUDY OF DISSOCIATED IMPAIRMENT Mauro Pesenti1, Nathalie Depoorter2 and Xavier Seron1 (1Unité de Neuropsychologie Cognitive, Université Catholique de Louvain; 2Service de Réadaptation Fonctionnelle; CHU Vésale, Montignies-le-Tilleul, Belgique)
ABSTRACT It has been shown that some arithmetical problems are stored in the form of individual facts representations (e.g., 3 × 4 = 12) whereas others are solved by general stored rules (e.g., 0 × N = 0). We describe the performance of a brain-damaged subject who presented a mild impairment in arithmetical fact retrieval. Although her performance was almost perfect for rule-based problems in all arithmetical operations, she was severely impaired for 0 + n problems in contrast with her relatively good performance for the corresponding n + 0 problems. This dissociation extends to addition the noncommutability of arithmetical rules described in multiplication. Key words: arithmetical facts, arithmetical rules INTRODUCTION It is widely accepted that the answers to simple one-digit arithmetical problems (e.g., 3 + 4 or 3 × 4) are stored as declarative knowledge in a semantic network or in an associative structure reflecting the associative strength between a given problem, its correct solution, and competing false solutions (for a review, see Ashcraft, 1992). Such problems, which are usually retrieved directly without any actual computation, are called arithmetical facts. Evidence from studies in children, normal adults and brain-damaged subjects supports the view that, in healthy adults, direct retrieval is the dominant strategy for such problems. However, when memory retrieval fails to provide a satisfactory answer, compensatory procedural strategies (counting, decomposition into known problems, etc.) can be applied to compute it. It has been shown that performance greatly varies across problems and individuals and that direct memory retrieval, in fact, accounts for only 80% of the reported strategies for simple addition and multiplication even in educated adults (LeFevre, Bisanz, Daley et al., 1996; LeFevre, Sadesky and Bisanz, 1996). Besides these backup procedural strategies occasionally applied when memory retrieval fails to provide an answer, it has been proposed that some classes of problems, essentially those involving 0s or 1s as operands, are readily solved by applying general procedural rules instead of memory retrieval. On the basis of healthy, as well as brain-damaged subjects’ performance, a distinction has thus been drawn between fact-based problems requiring complete retrieval and such potential rule-based problems. Firstly, speeded response times in healthy subjects usually show differences for rule-based problems as compared to the others (departure from the general principle that latencies of response increase with the magnitude of the operands, shorter latencies in production and longer ones in verification paradigms, higher error rates; Miller, Perlmutter and Keating, 1984; Stazyck, Ashcraft and Hamann, 1982). Secondly, dissociations are often observed between rule- and fact-based problems in brain-damaged subjects’ performance, with rule-based problems being more or less impaired than the fact-based ones (McCloskey, Aliminosa and Cortex, (2000) 36, 445-454
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Sokol, 1991). Moreover, the nature of the impairment itself is different. Whereas it is nonuniform for fact-based problems within a given operation, with error rates higher for some problems than for others suggesting individual representation for each fact (e.g., 3 × 4 = 12, 3 × 5 = 15, etc.), a uniform impairment/unimpairment is commonly observed within a given subset of rule-based problems. For example, multiplication problems sharing a null operand can all be solved using a general rule such as Any number multiplied by 0 equals 0; hence, an impairment/unimpairment of this rule will similarly affect all the 0s multiplication problems. This uniformity of (non)impairment goes beyond the error rate and extends to the amount and timing of improvement during testing or training as shown by the patient PS (Sokol, McCloskey, Cohen et al., 1991). In this patient, all the 0s multiplication problems were initially impaired to the same extend (97.5% errors) and they all improved to the same extent (94.8% correct) and at he same moment, whereas the error rate, amount and timing of improvement substantially differed across fact-based problems. Across the various operations, several subsets of problems may be candidates for a rule-based resolution. The N × 0 rule1 in multiplication is the most extensively studied rule but other less documented ones cover problems involving 0 and 1 in multiplication (N × 1 = N), additions (N + 0 = N), subtraction (N – 0 = N; N – N = 0) and division (0 : N = 0; N : 0 = not valid; N : 1 = N; N : N = 1). So far, however, rule-based problems have received much less attention than fact-based problems, and are often not even tested at all. Table I reviews the few cases reported in the literature for which the performance in rule-based problems is documented. McCloskey and his co-workers presented and discussed the performance of several brain-damaged patients for the N × 0 rule as compared to N × 1 and M × N problems (McCloskey et al., 1991; Sokol et al., 1991). They observed that the N × 0 rule was more disrupted than the N × 1 rule for most of their patients and never less disrupted, and that the performance was dramatically worse for N × 0 problems than for M × N problems in about sixty percent of the patients but better for the others. In nearly all cases, the errors took the form N × 0 = N (e.g., 0 × 3 = 3), suggesting that the patients were wrongly generalising the N + 0 = N addition rule2. Unfortunately, the N + 0 problems were not tested in these patients, except in GE who fully preserved the appropriate rule. Patient BE (Hittmair-Delazer, Semenza and Denes, 1994), who had lost the ability to retrieve most of the multiplication fact-based problems, exhibited a complete preservation of conceptual knowledge and other procedural rules; interestingly, he presented no impairment for the N × 0 and N × 1 rules. The N × 1 problems never appeared more impaired than M × N problems (Dagenbach and McCloskey, 1992; Dehaene and Cohen, 1997; Hittmair-Delazer, Sailer and Benke, 1995; McCloskey et al., 1991; Pesenti, Seron and Van der Linden, 1994; Sokol et al., 1991). Although not fully preserved, the N + 0 problems were less impaired than the M + N problems in two patients in which they were investigated. The N – 0 problems were less impaired than the M – N problems in one (RG; Dagenbach and McCloskey, 1992) of these two patients and fully preserved in the other (BB; Pesenti et al., 1994). Finally, the only case for which division was briefly investigated showed a complete impairment of the 0 : N rule despite a good preservation of the corresponding N × 0 rule; the N : 1 and N : 0 rules also appeared completely impaired (Cipolotti and de Lacy Costello, 1995). An interesting observation is the nonuniform impairment within the subsets of N × 0 problems. Indeed, two patients in the sample presented a uniform performance across the n × 0 problems and across the 0 × n problems, but differed between these two subsets (McCloskey et al., 1991). In patient FW, the error rate was high for n × 0 but low for 0 × n, whereas the opposite pattern was observed in JB. Moreover, both patients presented sudden changes of performance across the testing blocks. For FW, performance on 0 × n problems was consistently good across the various blocks of trials, whereas it was severely use the notation N × 0 to refer to all the multiplicative problems involving 0, whatever the order of the operands (e.g., 3 × 0 and 0 × 3); 0 × n and n × 0 refer to all the problems for which zero is, respectively, the first (e.g., 0 × 2, 0 × 3, etc.) or the second operand (e.g., 2 × 0, 3 × 0, etc.). We use the letters M and N to refer to numbers in the range 1-9. Similar conventions apply for other rules. 2Accordingly, the problem 0 × 0, which usually appears 100% correct, is generally not included in the analyses because a correct answer can stem from the application of the correct as well as the wrong rule. 1We
Pesenti et al. (1994) Hittmair-Delazer et al. (1994) Dehaene and Cohen (1997) McCloskey et al. (1991) Hittmair-Delazer et al. (1995) McCloskey et al. (1991) Sokol et al. (1991) McCloskey et al. (1991) McCloskey et al. (1991) McCloskey et al. (1991) Dehaene and Cohen (1997) McCloskey et al. (1991) Sokol et al. (1991) Dagenbach and McCloskey (1992) McCloskey et al. (1991) McCloskey et al. (1991)
Authors
48 – 6 – 50 – 5 – – – 32 – – 90 – –
fact-based 8 – – – – – 0 – – – – – – 13 – –
N+0 11 – – – – – 0 – – – – – – 16 – –
n+0
Addition
5 – – – – – 0 – – – – – – 11 – –
0+n
–-: not tested; ?: not mentioned; * sudden change from 2.5% to 95% correct during the testing.
BB BE BOO CM DA FW GE HM IE JB MAR MD PS RG SB TM
Patient
TABLE I
59 6 30 17 53 18 8 3 29 4 31 27 19 89 22 50
fact-based 7 0 – 0 0 30 100 100 1 79 – 100 41* 97 3.5 –
N×0 11 0 – 0 0 55 100 100 2 57 – 100 41* 98 4 –
n×0 3 0 – 0 0 7 100 100 1 100 – 100 41* 95 3 –
0×n
Multiplication
Percentage of Errors by Patient and Type of Problem and Rule for Addition and Multiplication
12 0 6 1 0 1.5 12 1.5 1 0 11 1 1.5 19 0 1
N×1
19 0 0 ? 0 1 10 0 2 0 11 1 1 16 0 2
n×1
5 0 11 ? 0 2 14 3 1 0 11 1 2 22 0 0
1×n
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impaired in the first blocks but perfect for the last ones on n × 0 problems. For JB, performance on n × 0 problems was completely impaired across blocks, whereas it was completely impaired in the first blocks but perfect for the last ones on 0 × n problems. This double dissociation was interpreted by the authors as suggesting the existence, at least for some individuals, of two separate rules. Does such a dissociation hold only for the N × 0 rule or also for the others? Indeed, the zero rules for addition and subtraction have a different status since, contrary to the zero rules for multiplication and division, they both leave the non-null operand unchanged. There might thus be separate rules for multiplication, but only a general unique one for addition and subtraction. However, since the other zero rules are far less documented, it is not easy to answer this question. The present study aims at investigating the precise status of the zero rules in the various arithmetical operations. We report the case of a brain-damaged patient exhibiting some moderate deficits in arithmetical fact retrieval. Whereas most arithmetical rules were virtually fully preserved, this patient presented a striking dissociation between preserved n + 0 and impaired 0 + n problems. CASE REPORT DF, a left-handed woman, was born in 1953 and worked as a medical technician in a neurophysiological laboratory. In December 1998, she underwent a kidney transplant. Three weeks after the transplant, in January 1999, DF woke up one night and found herself “lost and confused”. She needed her husband’s help to turn on the light and to walk in the bedroom. In the morning, she still felt “insecure and more dependent upon her family members”. Two days later, she was eventually admitted to an emergency ward. A CT-scan examination revealed several regions of hypodensity bilaterally in the temporoparietal areas. The neurological assessment carried out the same day revealed oculomotor exploration deficits, optic ataxia, dysgraphia and dyscalculia, left-right confusion, finger agnosia and some signs of apraxia, all of which suggested both Balint and Gerstmann syndromes. The results of the neuropsychological assessment carried out five days after the initial episode are reported below. Language Assessment DF’s spontaneous speech was fluent and presented no articulatory, phonological or word-finding difficulties. Comprehension of conversational speech was normal. She made no errors in repetition of words, non-words and sentences, ended easily short incomplete sentences, and correctly executed simple orders. The production of automated verbal sequences (alphabet, musical notes, days, months, counting from 1 to 21) was normal. She performed correctly in a word-picture matching task. The naming of black and white line drawings out of various semantic categories (bathroom and kitchen utensils, sewing instruments, etc.) was weak but within normal range (86/90 correct). Her reading aloud was correct for short words (2-8 letters) but difficult for longer ones; sentence and text reading was laborious because of a fragmentary decoding. Her writing presented some abnormalities characteristic of right posterior lesions (undulation of the baseline, duplication of downstrokes and letters, neglect of the left part of the sheet, etc.). Short-Term Memory Assessment DF’s performances in short-term memory tasks were poor but within the normal range for simple digit spans (forwards: 4; backwards: 3). In a storage-plus-processing span3, DF scored 3For
a description of the task, see Pesenti, Seron, Samson et al. (1999). The patient is presented with sequences of simple arithmetical problems written in Arabic notation on single cards; on each card, the second digit is presented within a small frame. After a sequence of problems has been presented, she recalls all the second (framed) numbers in order. In the storage-alone condition, she just has to remember the targets without paying attention to the problems. In the storage-plus-processing condition, she has to give aloud the correct answer to each problem while at the same time remembering the second number.
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4 for the storage-alone condition which corresponds to her forward span, but could not perform the storage-plus-processing condition even with series of length 2: she either memorised the framed number or performed the operation but could not manage both at the same time. Finger Gnosis and Left-Right Discrimination Assessments Finger gnosis was tested in three situations combining the modality (tactile or verbal) of stimulation and response without visual control (eyes closed). DF presented a few errors in each condition (tactile-motor: 28/30; verbal-motor: 19/20; tactile-verbal: 27/30 correct). Left-right discrimination was tested in two conditions. With her eyes closed, DF was asked to designate specific parts of her body without or with specification of the hand to be used (e.g., Touch your left knee or Touch your left knee with your right hand). She performed perfectly the first condition (16/16 correct) but made several errors in the second (24/32 correct). In all cases, the hand was always correct but the opposite part of the body was designated; the majority of these errors (7/8) consisted in designating the right instead of the left part of the body. DF reported that she had presented this left-right confusion since she was a child. Praxis Assessment Buccofacial praxis was unimpaired. DF showed a moderate constructive apraxia, clear signs of optic ataxia and neglect for the left hemifield. These problems regressed progressively and had almost disappeared when the number processing and calculation investigation was carried out. Inhibitory Processes Assessment DF presented a discrete impairment of inhibitory processes. She performed correctly the Hayling test (control part: 15/15; inhibition part: 13/15; Burgess and Shallice, 1996), she was correct but slow in the Trail Making test (Reitan and Wolfson, 1995), and she was very laborious on the interference condition of the Stroop test (colours: 91/100, words: 94/100, interference: 88/100; Stroop, 1935). Numerical Processing Assessment The numerical processing assessment was carried out in March 1999. DF’s scores for various tests with spoken verbal and Arabic numerals are presented in Table II. Her performances ranged from good to perfect (95-100% correct) in numerical comparison (pair and standard comparison), number positioning on a scale, completion of intervals (e.g., Which number comes between 4 and 6?), oral counting and parity judgement tasks. She also answered correctly questions concerning personal (e.g., phone number, weight, etc.) and nonpersonal number facts (e.g., The year World War II ended, etc.). She performed less well (85-90% correct) in proximity judgement (e.g., Which number is closer to 4: 5 or 9?) and ordering of triplets (e.g., Put the following numbers in ascending order: 6, 2, 5) tasks, probably because of the greater short-term memory demand. She made 23/237 errors (90% correct) in transcoding tasks, among which 15 were lexico-syntactic errors and 8 consisted of intrusions of the source notation into the production notation (Tegner and Nyback, 1990). Complex written calculations were impaired: carrying and borrowing mechanisms as well as spatial positioning were preserved, but DF made calculation errors at intermediary steps of the solution, which led to incorrect responses for half of the problems. She was only 50% correct when asked to grossly estimate the result of multidigit operations in a multiple-choice task (e.g., 275 × 4 = 50, 2300, 600 or 1200?). Globally, the comprehension of numerals thus appeared largely unimpaired and the errors observed in production could all be due either to mild perceptual, memory or inhibitory (for the notation intrusion errors) problems. Moreover, these errors concerned multi-digit numbers and never affected numbers in the range of arithmetical fact operands and answers.
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Scores Obtained by DF in the Calculation and Number Processing Tasks Tasks
Notation
Transcoding
Counting Numerical comparison Comparison to a standard (55) Number positioning on a scale (0-100) Completion of intervals Proximity judgement Ordering of triplets Estimation of results Number facts Parity judgement Written multi-digit calculation Addition Subtraction Multiplication
correct / items
Arabic to spoken verbal Arabic to written verbal Written verbal to spoken verbal Written verbal to Arabic Spoken verbal Spoken verbal Spoken verbal Arabic Spoken verbal Arabic Spoken verbal Spoken verbal Arabic Spoken verbal Arabic Spoken verbal Spoken verbal
68 / 70 67 / 75 16 / 17 63 / 75 6/7 114 / 119 32 / 32 32 / 32 6/6 6/6 14 / 14 41 / 48 43 / 48 18 / 20 4/8 22 / 22 21 / 22 2/2 1/2 0/2
ARITHMETICAL FACTS INVESTIGATION Materials and Methods In March 1999, DF was asked several times (each session spaced by a two-day interval) to complete addition, subtraction, multiplication and division table facts (i.e., 0 + 0 through 9 + 9 for addition, 0 – 0 through 9 – 9 for subtraction, 0 × 0 through 9 × 9 for multiplication, and the corresponding 64 divisions with divisor and answer ≥ to 2). The four operations were presented in unmixed series separated by other numerical tasks or pauses. Both problems and responses were spoken. No feed-back was given to DF. Results The percentage of errors remained stable over sessions (Table III); hence, results are averaged over the various blocks. Globally, DF made 15, 2, 10 and 19% of errors TABLE III
Percentage of Errors by Operation and Sessions. Operations Sessions
Addition (100 items)
Subtraction (55 items)
Multiplication (100 items)
Division (64 items)
1 2 3 4 5 6 Mean Post-test
11 13 19 14 17 – 15 4
0 5.5 – – 0 2 2 2
11 10 – 9 11 – 10 6
– – – – 19 19 19 6
– : not tested.
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respectively for addition, subtraction, multiplication and division, showing a good preservation of subtraction as compared to the mild impairment for the other operations. N + 0 problems were much more impaired than any other type of problem4 (Table IV). TABLE IV
Percentage of Errors by Operation and Type of Problem Operations
Type of problems
Addition
0s problems * 1s problems N – N, ties fact-based problems
48 1 0 9.5
Subtraction
(43/90) (1/90) (0/40) (30/320)
0 3 0 3.5
Multiplication
(0/36) (1/36) (0/36) (4/112)
0 1.5 6 16
Division
(0/72) (1/72) (2/32) (40/256)
– – 19 (3/16) 19 (24/128)
* see note 3; –: not tested.
Addition N + 0 problems appeared severely impaired in comparison to M + N problems (Tables IV and V). Within N + 0 problems, there was a striking difference between n + 0 problems which were only mildly impaired (9% errors) and 0 + n ones which were dramatically impaired (87% errors). All errors had the form N + 0 = 0. On many occasions, DF verbalised these problems by saying “Nothing plus N makes nothing”. Although she was not completely confident with this statement, she did not realise that it was wrong. When she verbalised the n + 0 problems, she would say “N plus nothing makes N”. For M + N problems, 70% of the errors were at a distance of ± 1 from the correct answer and 23% at a distance of 2; all errors but one concerned problems with an answer larger than 10. TABLE V
Percentages of Errors Made by DF on Additions (5 blocks) Second operand
First operand
0
1
2
3
4
5
6
7
8
9
0+… 1+… 2+… 3+… 4+… 5+… 6+… 7+… 8+… 9+…
0 0 0 0 0 0 40 20 0 20
60 0 0 0 0 0 0 0 0 20
100 0 0 0 0 20 0 0 0 0
80 0 0 0 0 0 0 0 20 0
80 0 0 0 0 0 0 0 0 0
100 0 0 0 0 0 0 60 80 0
80 0 0 0 0 0 0 40 20 20
100 0 0 0 20 0 20 0 80 20
80 0 0 40 0 0 0 40 0 20
100 0 0 0 0 0 20 60 20 0
Subtraction DF was slow but made only a few errors for subtraction, none of them for M – 0 problems. All errors concerned problems with operands larger than 7. Multiplication DF was virtually perfect on rule-based multiplications (1 error on 8 × 1; see Table IV). She made about 10% of errors for fact-based problems including ties, essentially on large 4The
problems 0 + 0, 0 – 0 and 0 × 0 were not included in the analyses (see note 2).
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and medium problems (respectively, 70 and 27.5% of the errors). The vast majority (90%) were operand errors (the erroneous answer is a correct response for another problem sharing an operand with the target problem: e.g., 6 × 8 = 56); she gave the answer 56 for 36% of operand errors. For about 30% (115/400) of the problems, DF either paused, repeated, or asked the examiner to repeat the problem once or several times before answering; she always tried to retrieve the answer from memory and never worked it out using backup strategies. Division DF performed the divisions laboriously and erred in about 20% of the problems distributed over the whole range of the tables. She usually solved the problems by retrieving the corresponding multiplication fact, but, in some instances, she could not manage turning the problem round to decide which operand was the answer of the division (e.g., for 35 : 7, she said “thirty-five divided by seven… five times seven… seven times five… seven [she asked the examiner to repeat the problem]… seven or five… five”). Post-Test Once the whole arithmetical fact investigation was completed, DF received feed-back on her performance and errors. She was reminded of the N + 0 rule and the commutability of n + 0 and 0 + n problems by manipulating concrete material. After a few weeks, she completed once each operation (Table III). At this post-test, DF made only one error on the first 0 + n problem in the series, and her performance slightly improved for multiplication and division. All errors were similar to those observed previously. DISCUSSION We described the performance of DF, a brain-damaged subject who presented a mild impairment in arithmetical fact retrieval. Globally, her performance was better preserved for subtraction than for the other operations5 which adds converging evidence to an already described dissociation (Dagenbach and McCloskey, 1992; Lampl, Eshel, Gilad et al., 1994; McNeil and Warrington, 1994; Pesenti et al., 1994). Such a dissociation had been interpreted as a disruption of segregated memory networks for each operation (Dagenbach and McCloskey, 1992; Pesenti et al., 1994) but it might also reflect selective impairment to different types of calculation procedures (verbally-mediated memory retrieval vs. manipulation of numerical magnitudes; Dehaene and Cohen, 1997). Although DF presented no clear signs of verbal deficits nor better preserved magnitude processing abilities, her relatively mild impairment does not allow one to tackle this problem correctly. The most interesting aspect of her performance was the severe impairment of the 0 + n rule despite a perfect preservation of all the other rules including the corresponding n + 0 one. Moreover, once she had been reminded of the correct rule, DF’s performance showed the uniform improvement which appeared spontaneously in other patients for the N × 0 rule (McCloskey et al., 1991). In discussing the dissociation within N × 0 rules, McCloskey et al. noted that it was difficult to exclude completely an account in terms of disruption of the access to a unique rule from one subset of 0s problems but not the other. This case clearly rules out this alternative explanation: DF spontaneous verbalisations showed that she used two different rules, one correct for n + 0 and one incorrect for 0 + n problems. McCloskey et al. further discussed the potential source of the particular errors occurring on rule-based problems and they distinguished two interpretations. The first one assumes the retrieval of a wrong rule (i.e., partial activation of rules applying to other subsets of 5The
preservation held for larger subtraction problems. At the post-test, DF was presented 4 times with the 45 M – N subtraction problems (with 10 ≤ M ≤ 18 and 1 ≤ N ≤ 9) and she made 4% of errors on average.
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problems; here, 0 + n = 0 because 0 × n = 0), the other the storage of incorrect rules. In this latter account, an incorrect rule such as N + 0 = 0 might have a developmental origin. Since addition and subtraction are encountered during childhood before multiplication, children might come up with an erroneous generalisation that a null operand in an arithmetical problem leaves the other operand unchanged. Then, the correct N × 0 = 0 rule is learned through instruction, but the more general (incorrect) rule might nevertheless remain in memory and come into play in the case of disruption of the correct one. Again, the present case does not support this hypothesis. Indeed, a developmentally-based generalisation is not very plausible for addition and the present dissociation shows that the 0s addition problems (and thus perhaps the N – 0 subtraction problems too) do not refer to a general zero rule of the form Additions (and subtractions) with 0 leave the non-null operand unchanged. On the contrary, it suggests that, at least in some individuals, separate rules exist for mathematically equal but formally different problems. This extends to addition the noncommutability of arithmetical rules already described in multiplication. A weaker form of the stored incorrect rule explanation proposes that the incorrect rule is generated after the correct rule is disrupted by brain-damage (when unable to access a rule, a patient may attempt to infer the correct answer from general knowledge of arithmetic, generate an incorrect rule, store and then apply it). Although attractive, this explanation seems to us very difficult to distinguish from the retrieving the wrong rule account on the basis of the empirical evidence hitherto collected and thus needs further investigation. Acknowledgements. We wish to thank DF who kindly accepted to participate in long testing sessions. MP is supported by the PAI/IUAP Program from the Belgian Government. REFERENCES ASHCRAFT, M.H. Cognitive arithmetic: A review of data and theory. Cognition, 44: 75-106, 1992. BURGESS, P., and Shallice, T. Response suppression, initiation and strategy use following frontal lobe lesions. Neuropsychologia, 34: 263-273, 1996. CIPOLOTTI, L., and DE LACY-COSTELLO, A.L. Selective impairment for simple division. Cortex, 31: 433449, 1995. DAGENBACH, D., and MCCLOSKEY, M. The organization of arithmetic facts in memory: Evidence from a brain-damaged patient. Brain and Cognition, 20: 345-366, 1992. DEHAENE, S., and COHEN, L. Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33: 219-250, 1997. HITTMAIR-DELAZER, M., SAILER, U., and BENKE, Th. Impaired arithmetic facts but intact conceptual knowledge: A single case study of dyscalculia. Cortex, 31: 139-147, 1995. HITTMAIR-DELAZER, M., SEMENZA, C., and DENES, F. Concepts and facts in calculation. Brain, 117: 715728, 1994. LAMPL, Y., ESHEL, Y., GILAD, R., and SAROVA-PINHAS, I. Selective acalculia with sparing of the subtraction process in a patient with left parietotemporal hemorrhage. Neurology, 44: 1759-1761, 1994. LEFEVRE, J., BISANZ, J., DALEY, K.E., BUFFONE, L., GREENHAM, S.L., and SADESKY, G.S. Multiple route to solution of single-digit multiplication problems. Journal of Experimental Psychology: General, 125: 384-306, 1996a. LEFEVRE, J., SADESKY, G.S., and BISANZ, J. Selection of procedure in mental addition: Reassessing the problem size effect in adults. Journal of Eexperimental Psychology: Learning, Memory and Cognition, 22: 216-230, 1996b. MCCLOSKEY, M., ALIMINOSA, D., and SOKOL, S.M. Facts, rules, and procedures in normal calculation: Evidence from multiple single-patient studies of impaired arithmetic fact retrieval. Brain and Cognition, 17: 154-203, 1991. MCNEIL, J., and WARRINGTON , E. A dissociation between addition and subtraction with written calculation. Neuropsychologia, 32: 717-724, 1994. MILLER, K., PERLMUTTER, M., and KEATING, D. Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory and Cognition, 10: 46-60, 1984. PESENTI, M., SERON, X., SAMSON, D., and DUROUX, B. Basic and exceptional calculation abilities in a calculating prodigy: A case study. Mathematical Cognition, 50: 1-52, 1999 (in press). PESENTI, M., SERON, X., and VAN DER LINDEN, M. Selective impairment as an evidence for mental organisation of arithmetical facts: BB, a case of preserved subtraction? Cortex, 30: 661-671, 1994. REITAN, R., and WOLFSON, D. Category Test and Trail Making Test as measures of frontal lobe functions. The Clinical Neuropsychologist, 9: 50-56, 1995.
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STAZYK, E.H., ASHCRAFT, M.H., and HAMANN, M.S. A network approach to mental multiplication. Journal of Experimental Psychology: Learning, Memory and Cognition, 8: 320-335, 1982. SOKOL, S.M., MCCLOSKEY, M. COHEN, N.J., and ALIMINOSA, D. Cognitive representations and processes in arithmetic: Inferences from the performance of brain-damaged subjects. Journal of Experimental Psychology: Learning, Memory and Cognition, 17: 355-376, 1981. STROOP, J.R. Studies of interference in serial verbal reactions. Journal of Experimental Psychology, 18: 643-661, 1935. TEGNER, R., and NYBACK, H. “To hundred an twenty4our”: A study of transcoding in dementia. Acta Neurologica Scandinavica, 81: 177-178, 1990. Mauro Pesenti, Unité de Neuropsychologie Cognitive, Département de Psychologie Expérimentale, Université Catholique de Louvain, Place Cardinal Mercier, 10, B-1348 Louvain-la-Neuve (Belgium). E-mail: [email protected].
(Received 30 August 1999; accepted 2 Dicember 1999)
ABSTRACT It has been shown that some arithmetical problems are stored in the form of individual facts representations (e.g., 3 × 4 = 12) whereas others are solved by general stored rules (e.g., 0 × N = 0). We describe the performance of a brain-damaged subject who presented a mild impairment in arithmetical fact retrieval. Although her performance was almost perfect for rule-based problems in all arithmetical operations, she was severely impaired for 0 + n problems in contrast with her relatively good performance for the corresponding n + 0 problems. This dissociation extends to addition the noncommutability of arithmetical rules described in multiplication. Key words: arithmetical facts, arithmetical rules INTRODUCTION It is widely accepted that the answers to simple one-digit arithmetical problems (e.g., 3 + 4 or 3 × 4) are stored as declarative knowledge in a semantic network or in an associative structure reflecting the associative strength between a given problem, its correct solution, and competing false solutions (for a review, see Ashcraft, 1992). Such problems, which are usually retrieved directly without any actual computation, are called arithmetical facts. Evidence from studies in children, normal adults and brain-damaged subjects supports the view that, in healthy adults, direct retrieval is the dominant strategy for such problems. However, when memory retrieval fails to provide a satisfactory answer, compensatory procedural strategies (counting, decomposition into known problems, etc.) can be applied to compute it. It has been shown that performance greatly varies across problems and individuals and that direct memory retrieval, in fact, accounts for only 80% of the reported strategies for simple addition and multiplication even in educated adults (LeFevre, Bisanz, Daley et al., 1996; LeFevre, Sadesky and Bisanz, 1996). Besides these backup procedural strategies occasionally applied when memory retrieval fails to provide an answer, it has been proposed that some classes of problems, essentially those involving 0s or 1s as operands, are readily solved by applying general procedural rules instead of memory retrieval. On the basis of healthy, as well as brain-damaged subjects’ performance, a distinction has thus been drawn between fact-based problems requiring complete retrieval and such potential rule-based problems. Firstly, speeded response times in healthy subjects usually show differences for rule-based problems as compared to the others (departure from the general principle that latencies of response increase with the magnitude of the operands, shorter latencies in production and longer ones in verification paradigms, higher error rates; Miller, Perlmutter and Keating, 1984; Stazyck, Ashcraft and Hamann, 1982). Secondly, dissociations are often observed between rule- and fact-based problems in brain-damaged subjects’ performance, with rule-based problems being more or less impaired than the fact-based ones (McCloskey, Aliminosa and Cortex, (2000) 36, 445-454
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Sokol, 1991). Moreover, the nature of the impairment itself is different. Whereas it is nonuniform for fact-based problems within a given operation, with error rates higher for some problems than for others suggesting individual representation for each fact (e.g., 3 × 4 = 12, 3 × 5 = 15, etc.), a uniform impairment/unimpairment is commonly observed within a given subset of rule-based problems. For example, multiplication problems sharing a null operand can all be solved using a general rule such as Any number multiplied by 0 equals 0; hence, an impairment/unimpairment of this rule will similarly affect all the 0s multiplication problems. This uniformity of (non)impairment goes beyond the error rate and extends to the amount and timing of improvement during testing or training as shown by the patient PS (Sokol, McCloskey, Cohen et al., 1991). In this patient, all the 0s multiplication problems were initially impaired to the same extend (97.5% errors) and they all improved to the same extent (94.8% correct) and at he same moment, whereas the error rate, amount and timing of improvement substantially differed across fact-based problems. Across the various operations, several subsets of problems may be candidates for a rule-based resolution. The N × 0 rule1 in multiplication is the most extensively studied rule but other less documented ones cover problems involving 0 and 1 in multiplication (N × 1 = N), additions (N + 0 = N), subtraction (N – 0 = N; N – N = 0) and division (0 : N = 0; N : 0 = not valid; N : 1 = N; N : N = 1). So far, however, rule-based problems have received much less attention than fact-based problems, and are often not even tested at all. Table I reviews the few cases reported in the literature for which the performance in rule-based problems is documented. McCloskey and his co-workers presented and discussed the performance of several brain-damaged patients for the N × 0 rule as compared to N × 1 and M × N problems (McCloskey et al., 1991; Sokol et al., 1991). They observed that the N × 0 rule was more disrupted than the N × 1 rule for most of their patients and never less disrupted, and that the performance was dramatically worse for N × 0 problems than for M × N problems in about sixty percent of the patients but better for the others. In nearly all cases, the errors took the form N × 0 = N (e.g., 0 × 3 = 3), suggesting that the patients were wrongly generalising the N + 0 = N addition rule2. Unfortunately, the N + 0 problems were not tested in these patients, except in GE who fully preserved the appropriate rule. Patient BE (Hittmair-Delazer, Semenza and Denes, 1994), who had lost the ability to retrieve most of the multiplication fact-based problems, exhibited a complete preservation of conceptual knowledge and other procedural rules; interestingly, he presented no impairment for the N × 0 and N × 1 rules. The N × 1 problems never appeared more impaired than M × N problems (Dagenbach and McCloskey, 1992; Dehaene and Cohen, 1997; Hittmair-Delazer, Sailer and Benke, 1995; McCloskey et al., 1991; Pesenti, Seron and Van der Linden, 1994; Sokol et al., 1991). Although not fully preserved, the N + 0 problems were less impaired than the M + N problems in two patients in which they were investigated. The N – 0 problems were less impaired than the M – N problems in one (RG; Dagenbach and McCloskey, 1992) of these two patients and fully preserved in the other (BB; Pesenti et al., 1994). Finally, the only case for which division was briefly investigated showed a complete impairment of the 0 : N rule despite a good preservation of the corresponding N × 0 rule; the N : 1 and N : 0 rules also appeared completely impaired (Cipolotti and de Lacy Costello, 1995). An interesting observation is the nonuniform impairment within the subsets of N × 0 problems. Indeed, two patients in the sample presented a uniform performance across the n × 0 problems and across the 0 × n problems, but differed between these two subsets (McCloskey et al., 1991). In patient FW, the error rate was high for n × 0 but low for 0 × n, whereas the opposite pattern was observed in JB. Moreover, both patients presented sudden changes of performance across the testing blocks. For FW, performance on 0 × n problems was consistently good across the various blocks of trials, whereas it was severely use the notation N × 0 to refer to all the multiplicative problems involving 0, whatever the order of the operands (e.g., 3 × 0 and 0 × 3); 0 × n and n × 0 refer to all the problems for which zero is, respectively, the first (e.g., 0 × 2, 0 × 3, etc.) or the second operand (e.g., 2 × 0, 3 × 0, etc.). We use the letters M and N to refer to numbers in the range 1-9. Similar conventions apply for other rules. 2Accordingly, the problem 0 × 0, which usually appears 100% correct, is generally not included in the analyses because a correct answer can stem from the application of the correct as well as the wrong rule. 1We
Pesenti et al. (1994) Hittmair-Delazer et al. (1994) Dehaene and Cohen (1997) McCloskey et al. (1991) Hittmair-Delazer et al. (1995) McCloskey et al. (1991) Sokol et al. (1991) McCloskey et al. (1991) McCloskey et al. (1991) McCloskey et al. (1991) Dehaene and Cohen (1997) McCloskey et al. (1991) Sokol et al. (1991) Dagenbach and McCloskey (1992) McCloskey et al. (1991) McCloskey et al. (1991)
Authors
48 – 6 – 50 – 5 – – – 32 – – 90 – –
fact-based 8 – – – – – 0 – – – – – – 13 – –
N+0 11 – – – – – 0 – – – – – – 16 – –
n+0
Addition
5 – – – – – 0 – – – – – – 11 – –
0+n
–-: not tested; ?: not mentioned; * sudden change from 2.5% to 95% correct during the testing.
BB BE BOO CM DA FW GE HM IE JB MAR MD PS RG SB TM
Patient
TABLE I
59 6 30 17 53 18 8 3 29 4 31 27 19 89 22 50
fact-based 7 0 – 0 0 30 100 100 1 79 – 100 41* 97 3.5 –
N×0 11 0 – 0 0 55 100 100 2 57 – 100 41* 98 4 –
n×0 3 0 – 0 0 7 100 100 1 100 – 100 41* 95 3 –
0×n
Multiplication
Percentage of Errors by Patient and Type of Problem and Rule for Addition and Multiplication
12 0 6 1 0 1.5 12 1.5 1 0 11 1 1.5 19 0 1
N×1
19 0 0 ? 0 1 10 0 2 0 11 1 1 16 0 2
n×1
5 0 11 ? 0 2 14 3 1 0 11 1 2 22 0 0
1×n
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impaired in the first blocks but perfect for the last ones on n × 0 problems. For JB, performance on n × 0 problems was completely impaired across blocks, whereas it was completely impaired in the first blocks but perfect for the last ones on 0 × n problems. This double dissociation was interpreted by the authors as suggesting the existence, at least for some individuals, of two separate rules. Does such a dissociation hold only for the N × 0 rule or also for the others? Indeed, the zero rules for addition and subtraction have a different status since, contrary to the zero rules for multiplication and division, they both leave the non-null operand unchanged. There might thus be separate rules for multiplication, but only a general unique one for addition and subtraction. However, since the other zero rules are far less documented, it is not easy to answer this question. The present study aims at investigating the precise status of the zero rules in the various arithmetical operations. We report the case of a brain-damaged patient exhibiting some moderate deficits in arithmetical fact retrieval. Whereas most arithmetical rules were virtually fully preserved, this patient presented a striking dissociation between preserved n + 0 and impaired 0 + n problems. CASE REPORT DF, a left-handed woman, was born in 1953 and worked as a medical technician in a neurophysiological laboratory. In December 1998, she underwent a kidney transplant. Three weeks after the transplant, in January 1999, DF woke up one night and found herself “lost and confused”. She needed her husband’s help to turn on the light and to walk in the bedroom. In the morning, she still felt “insecure and more dependent upon her family members”. Two days later, she was eventually admitted to an emergency ward. A CT-scan examination revealed several regions of hypodensity bilaterally in the temporoparietal areas. The neurological assessment carried out the same day revealed oculomotor exploration deficits, optic ataxia, dysgraphia and dyscalculia, left-right confusion, finger agnosia and some signs of apraxia, all of which suggested both Balint and Gerstmann syndromes. The results of the neuropsychological assessment carried out five days after the initial episode are reported below. Language Assessment DF’s spontaneous speech was fluent and presented no articulatory, phonological or word-finding difficulties. Comprehension of conversational speech was normal. She made no errors in repetition of words, non-words and sentences, ended easily short incomplete sentences, and correctly executed simple orders. The production of automated verbal sequences (alphabet, musical notes, days, months, counting from 1 to 21) was normal. She performed correctly in a word-picture matching task. The naming of black and white line drawings out of various semantic categories (bathroom and kitchen utensils, sewing instruments, etc.) was weak but within normal range (86/90 correct). Her reading aloud was correct for short words (2-8 letters) but difficult for longer ones; sentence and text reading was laborious because of a fragmentary decoding. Her writing presented some abnormalities characteristic of right posterior lesions (undulation of the baseline, duplication of downstrokes and letters, neglect of the left part of the sheet, etc.). Short-Term Memory Assessment DF’s performances in short-term memory tasks were poor but within the normal range for simple digit spans (forwards: 4; backwards: 3). In a storage-plus-processing span3, DF scored 3For
a description of the task, see Pesenti, Seron, Samson et al. (1999). The patient is presented with sequences of simple arithmetical problems written in Arabic notation on single cards; on each card, the second digit is presented within a small frame. After a sequence of problems has been presented, she recalls all the second (framed) numbers in order. In the storage-alone condition, she just has to remember the targets without paying attention to the problems. In the storage-plus-processing condition, she has to give aloud the correct answer to each problem while at the same time remembering the second number.
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4 for the storage-alone condition which corresponds to her forward span, but could not perform the storage-plus-processing condition even with series of length 2: she either memorised the framed number or performed the operation but could not manage both at the same time. Finger Gnosis and Left-Right Discrimination Assessments Finger gnosis was tested in three situations combining the modality (tactile or verbal) of stimulation and response without visual control (eyes closed). DF presented a few errors in each condition (tactile-motor: 28/30; verbal-motor: 19/20; tactile-verbal: 27/30 correct). Left-right discrimination was tested in two conditions. With her eyes closed, DF was asked to designate specific parts of her body without or with specification of the hand to be used (e.g., Touch your left knee or Touch your left knee with your right hand). She performed perfectly the first condition (16/16 correct) but made several errors in the second (24/32 correct). In all cases, the hand was always correct but the opposite part of the body was designated; the majority of these errors (7/8) consisted in designating the right instead of the left part of the body. DF reported that she had presented this left-right confusion since she was a child. Praxis Assessment Buccofacial praxis was unimpaired. DF showed a moderate constructive apraxia, clear signs of optic ataxia and neglect for the left hemifield. These problems regressed progressively and had almost disappeared when the number processing and calculation investigation was carried out. Inhibitory Processes Assessment DF presented a discrete impairment of inhibitory processes. She performed correctly the Hayling test (control part: 15/15; inhibition part: 13/15; Burgess and Shallice, 1996), she was correct but slow in the Trail Making test (Reitan and Wolfson, 1995), and she was very laborious on the interference condition of the Stroop test (colours: 91/100, words: 94/100, interference: 88/100; Stroop, 1935). Numerical Processing Assessment The numerical processing assessment was carried out in March 1999. DF’s scores for various tests with spoken verbal and Arabic numerals are presented in Table II. Her performances ranged from good to perfect (95-100% correct) in numerical comparison (pair and standard comparison), number positioning on a scale, completion of intervals (e.g., Which number comes between 4 and 6?), oral counting and parity judgement tasks. She also answered correctly questions concerning personal (e.g., phone number, weight, etc.) and nonpersonal number facts (e.g., The year World War II ended, etc.). She performed less well (85-90% correct) in proximity judgement (e.g., Which number is closer to 4: 5 or 9?) and ordering of triplets (e.g., Put the following numbers in ascending order: 6, 2, 5) tasks, probably because of the greater short-term memory demand. She made 23/237 errors (90% correct) in transcoding tasks, among which 15 were lexico-syntactic errors and 8 consisted of intrusions of the source notation into the production notation (Tegner and Nyback, 1990). Complex written calculations were impaired: carrying and borrowing mechanisms as well as spatial positioning were preserved, but DF made calculation errors at intermediary steps of the solution, which led to incorrect responses for half of the problems. She was only 50% correct when asked to grossly estimate the result of multidigit operations in a multiple-choice task (e.g., 275 × 4 = 50, 2300, 600 or 1200?). Globally, the comprehension of numerals thus appeared largely unimpaired and the errors observed in production could all be due either to mild perceptual, memory or inhibitory (for the notation intrusion errors) problems. Moreover, these errors concerned multi-digit numbers and never affected numbers in the range of arithmetical fact operands and answers.
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Scores Obtained by DF in the Calculation and Number Processing Tasks Tasks
Notation
Transcoding
Counting Numerical comparison Comparison to a standard (55) Number positioning on a scale (0-100) Completion of intervals Proximity judgement Ordering of triplets Estimation of results Number facts Parity judgement Written multi-digit calculation Addition Subtraction Multiplication
correct / items
Arabic to spoken verbal Arabic to written verbal Written verbal to spoken verbal Written verbal to Arabic Spoken verbal Spoken verbal Spoken verbal Arabic Spoken verbal Arabic Spoken verbal Spoken verbal Arabic Spoken verbal Arabic Spoken verbal Spoken verbal
68 / 70 67 / 75 16 / 17 63 / 75 6/7 114 / 119 32 / 32 32 / 32 6/6 6/6 14 / 14 41 / 48 43 / 48 18 / 20 4/8 22 / 22 21 / 22 2/2 1/2 0/2
ARITHMETICAL FACTS INVESTIGATION Materials and Methods In March 1999, DF was asked several times (each session spaced by a two-day interval) to complete addition, subtraction, multiplication and division table facts (i.e., 0 + 0 through 9 + 9 for addition, 0 – 0 through 9 – 9 for subtraction, 0 × 0 through 9 × 9 for multiplication, and the corresponding 64 divisions with divisor and answer ≥ to 2). The four operations were presented in unmixed series separated by other numerical tasks or pauses. Both problems and responses were spoken. No feed-back was given to DF. Results The percentage of errors remained stable over sessions (Table III); hence, results are averaged over the various blocks. Globally, DF made 15, 2, 10 and 19% of errors TABLE III
Percentage of Errors by Operation and Sessions. Operations Sessions
Addition (100 items)
Subtraction (55 items)
Multiplication (100 items)
Division (64 items)
1 2 3 4 5 6 Mean Post-test
11 13 19 14 17 – 15 4
0 5.5 – – 0 2 2 2
11 10 – 9 11 – 10 6
– – – – 19 19 19 6
– : not tested.
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respectively for addition, subtraction, multiplication and division, showing a good preservation of subtraction as compared to the mild impairment for the other operations. N + 0 problems were much more impaired than any other type of problem4 (Table IV). TABLE IV
Percentage of Errors by Operation and Type of Problem Operations
Type of problems
Addition
0s problems * 1s problems N – N, ties fact-based problems
48 1 0 9.5
Subtraction
(43/90) (1/90) (0/40) (30/320)
0 3 0 3.5
Multiplication
(0/36) (1/36) (0/36) (4/112)
0 1.5 6 16
Division
(0/72) (1/72) (2/32) (40/256)
– – 19 (3/16) 19 (24/128)
* see note 3; –: not tested.
Addition N + 0 problems appeared severely impaired in comparison to M + N problems (Tables IV and V). Within N + 0 problems, there was a striking difference between n + 0 problems which were only mildly impaired (9% errors) and 0 + n ones which were dramatically impaired (87% errors). All errors had the form N + 0 = 0. On many occasions, DF verbalised these problems by saying “Nothing plus N makes nothing”. Although she was not completely confident with this statement, she did not realise that it was wrong. When she verbalised the n + 0 problems, she would say “N plus nothing makes N”. For M + N problems, 70% of the errors were at a distance of ± 1 from the correct answer and 23% at a distance of 2; all errors but one concerned problems with an answer larger than 10. TABLE V
Percentages of Errors Made by DF on Additions (5 blocks) Second operand
First operand
0
1
2
3
4
5
6
7
8
9
0+… 1+… 2+… 3+… 4+… 5+… 6+… 7+… 8+… 9+…
0 0 0 0 0 0 40 20 0 20
60 0 0 0 0 0 0 0 0 20
100 0 0 0 0 20 0 0 0 0
80 0 0 0 0 0 0 0 20 0
80 0 0 0 0 0 0 0 0 0
100 0 0 0 0 0 0 60 80 0
80 0 0 0 0 0 0 40 20 20
100 0 0 0 20 0 20 0 80 20
80 0 0 40 0 0 0 40 0 20
100 0 0 0 0 0 20 60 20 0
Subtraction DF was slow but made only a few errors for subtraction, none of them for M – 0 problems. All errors concerned problems with operands larger than 7. Multiplication DF was virtually perfect on rule-based multiplications (1 error on 8 × 1; see Table IV). She made about 10% of errors for fact-based problems including ties, essentially on large 4The
problems 0 + 0, 0 – 0 and 0 × 0 were not included in the analyses (see note 2).
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and medium problems (respectively, 70 and 27.5% of the errors). The vast majority (90%) were operand errors (the erroneous answer is a correct response for another problem sharing an operand with the target problem: e.g., 6 × 8 = 56); she gave the answer 56 for 36% of operand errors. For about 30% (115/400) of the problems, DF either paused, repeated, or asked the examiner to repeat the problem once or several times before answering; she always tried to retrieve the answer from memory and never worked it out using backup strategies. Division DF performed the divisions laboriously and erred in about 20% of the problems distributed over the whole range of the tables. She usually solved the problems by retrieving the corresponding multiplication fact, but, in some instances, she could not manage turning the problem round to decide which operand was the answer of the division (e.g., for 35 : 7, she said “thirty-five divided by seven… five times seven… seven times five… seven [she asked the examiner to repeat the problem]… seven or five… five”). Post-Test Once the whole arithmetical fact investigation was completed, DF received feed-back on her performance and errors. She was reminded of the N + 0 rule and the commutability of n + 0 and 0 + n problems by manipulating concrete material. After a few weeks, she completed once each operation (Table III). At this post-test, DF made only one error on the first 0 + n problem in the series, and her performance slightly improved for multiplication and division. All errors were similar to those observed previously. DISCUSSION We described the performance of DF, a brain-damaged subject who presented a mild impairment in arithmetical fact retrieval. Globally, her performance was better preserved for subtraction than for the other operations5 which adds converging evidence to an already described dissociation (Dagenbach and McCloskey, 1992; Lampl, Eshel, Gilad et al., 1994; McNeil and Warrington, 1994; Pesenti et al., 1994). Such a dissociation had been interpreted as a disruption of segregated memory networks for each operation (Dagenbach and McCloskey, 1992; Pesenti et al., 1994) but it might also reflect selective impairment to different types of calculation procedures (verbally-mediated memory retrieval vs. manipulation of numerical magnitudes; Dehaene and Cohen, 1997). Although DF presented no clear signs of verbal deficits nor better preserved magnitude processing abilities, her relatively mild impairment does not allow one to tackle this problem correctly. The most interesting aspect of her performance was the severe impairment of the 0 + n rule despite a perfect preservation of all the other rules including the corresponding n + 0 one. Moreover, once she had been reminded of the correct rule, DF’s performance showed the uniform improvement which appeared spontaneously in other patients for the N × 0 rule (McCloskey et al., 1991). In discussing the dissociation within N × 0 rules, McCloskey et al. noted that it was difficult to exclude completely an account in terms of disruption of the access to a unique rule from one subset of 0s problems but not the other. This case clearly rules out this alternative explanation: DF spontaneous verbalisations showed that she used two different rules, one correct for n + 0 and one incorrect for 0 + n problems. McCloskey et al. further discussed the potential source of the particular errors occurring on rule-based problems and they distinguished two interpretations. The first one assumes the retrieval of a wrong rule (i.e., partial activation of rules applying to other subsets of 5The
preservation held for larger subtraction problems. At the post-test, DF was presented 4 times with the 45 M – N subtraction problems (with 10 ≤ M ≤ 18 and 1 ≤ N ≤ 9) and she made 4% of errors on average.
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problems; here, 0 + n = 0 because 0 × n = 0), the other the storage of incorrect rules. In this latter account, an incorrect rule such as N + 0 = 0 might have a developmental origin. Since addition and subtraction are encountered during childhood before multiplication, children might come up with an erroneous generalisation that a null operand in an arithmetical problem leaves the other operand unchanged. Then, the correct N × 0 = 0 rule is learned through instruction, but the more general (incorrect) rule might nevertheless remain in memory and come into play in the case of disruption of the correct one. Again, the present case does not support this hypothesis. Indeed, a developmentally-based generalisation is not very plausible for addition and the present dissociation shows that the 0s addition problems (and thus perhaps the N – 0 subtraction problems too) do not refer to a general zero rule of the form Additions (and subtractions) with 0 leave the non-null operand unchanged. On the contrary, it suggests that, at least in some individuals, separate rules exist for mathematically equal but formally different problems. This extends to addition the noncommutability of arithmetical rules already described in multiplication. A weaker form of the stored incorrect rule explanation proposes that the incorrect rule is generated after the correct rule is disrupted by brain-damage (when unable to access a rule, a patient may attempt to infer the correct answer from general knowledge of arithmetic, generate an incorrect rule, store and then apply it). Although attractive, this explanation seems to us very difficult to distinguish from the retrieving the wrong rule account on the basis of the empirical evidence hitherto collected and thus needs further investigation. Acknowledgements. We wish to thank DF who kindly accepted to participate in long testing sessions. MP is supported by the PAI/IUAP Program from the Belgian Government. REFERENCES ASHCRAFT, M.H. Cognitive arithmetic: A review of data and theory. Cognition, 44: 75-106, 1992. BURGESS, P., and Shallice, T. Response suppression, initiation and strategy use following frontal lobe lesions. Neuropsychologia, 34: 263-273, 1996. CIPOLOTTI, L., and DE LACY-COSTELLO, A.L. Selective impairment for simple division. Cortex, 31: 433449, 1995. DAGENBACH, D., and MCCLOSKEY, M. The organization of arithmetic facts in memory: Evidence from a brain-damaged patient. Brain and Cognition, 20: 345-366, 1992. DEHAENE, S., and COHEN, L. Cerebral pathways for calculation: Double dissociation between rote verbal and quantitative knowledge of arithmetic. Cortex, 33: 219-250, 1997. HITTMAIR-DELAZER, M., SAILER, U., and BENKE, Th. Impaired arithmetic facts but intact conceptual knowledge: A single case study of dyscalculia. Cortex, 31: 139-147, 1995. HITTMAIR-DELAZER, M., SEMENZA, C., and DENES, F. Concepts and facts in calculation. Brain, 117: 715728, 1994. LAMPL, Y., ESHEL, Y., GILAD, R., and SAROVA-PINHAS, I. Selective acalculia with sparing of the subtraction process in a patient with left parietotemporal hemorrhage. Neurology, 44: 1759-1761, 1994. LEFEVRE, J., BISANZ, J., DALEY, K.E., BUFFONE, L., GREENHAM, S.L., and SADESKY, G.S. Multiple route to solution of single-digit multiplication problems. Journal of Experimental Psychology: General, 125: 384-306, 1996a. LEFEVRE, J., SADESKY, G.S., and BISANZ, J. Selection of procedure in mental addition: Reassessing the problem size effect in adults. Journal of Eexperimental Psychology: Learning, Memory and Cognition, 22: 216-230, 1996b. MCCLOSKEY, M., ALIMINOSA, D., and SOKOL, S.M. Facts, rules, and procedures in normal calculation: Evidence from multiple single-patient studies of impaired arithmetic fact retrieval. Brain and Cognition, 17: 154-203, 1991. MCNEIL, J., and WARRINGTON , E. A dissociation between addition and subtraction with written calculation. Neuropsychologia, 32: 717-724, 1994. MILLER, K., PERLMUTTER, M., and KEATING, D. Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory and Cognition, 10: 46-60, 1984. PESENTI, M., SERON, X., SAMSON, D., and DUROUX, B. Basic and exceptional calculation abilities in a calculating prodigy: A case study. Mathematical Cognition, 50: 1-52, 1999 (in press). PESENTI, M., SERON, X., and VAN DER LINDEN, M. Selective impairment as an evidence for mental organisation of arithmetical facts: BB, a case of preserved subtraction? Cortex, 30: 661-671, 1994. REITAN, R., and WOLFSON, D. Category Test and Trail Making Test as measures of frontal lobe functions. The Clinical Neuropsychologist, 9: 50-56, 1995.
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STAZYK, E.H., ASHCRAFT, M.H., and HAMANN, M.S. A network approach to mental multiplication. Journal of Experimental Psychology: Learning, Memory and Cognition, 8: 320-335, 1982. SOKOL, S.M., MCCLOSKEY, M. COHEN, N.J., and ALIMINOSA, D. Cognitive representations and processes in arithmetic: Inferences from the performance of brain-damaged subjects. Journal of Experimental Psychology: Learning, Memory and Cognition, 17: 355-376, 1981. STROOP, J.R. Studies of interference in serial verbal reactions. Journal of Experimental Psychology, 18: 643-661, 1935. TEGNER, R., and NYBACK, H. “To hundred an twenty4our”: A study of transcoding in dementia. Acta Neurologica Scandinavica, 81: 177-178, 1990. Mauro Pesenti, Unité de Neuropsychologie Cognitive, Département de Psychologie Expérimentale, Université Catholique de Louvain, Place Cardinal Mercier, 10, B-1348 Louvain-la-Neuve (Belgium). E-mail: [email protected].
(Received 30 August 1999; accepted 2 Dicember 1999)