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Selective Impairment as Evidence for Mental Organisation of Arithmetical Facts: BB, A Case of Preserved Subtraction?
NOTE SELECTIVE IMPAIRMENT AS EVIDENCE FOR MENTAL ORGANISATION OF ARITHMETICAL FACTS: BB, A CASE OF PRESERVED SUBTRACTION? Mauro Pesenti*, Xavier Seron and Martial Van Der Linden (Unite de Neuropsychologie Cognitive, Universite Catholique de Louvain)
Number processing and calculation have become during the past 20 years an active field of investigation to study the nature of mental representations. Research in cognitive arithmetic deals with the mental representation of numbers and arithmetic and tries to build models for the processes that access and use numerical knowledge. As regards simple mental arithmetic, basic arithmetical facts (e.g., 6 X 5 =?) have been considered in turn as the result of mental counting strategies (Groen and Parkman, 1972; Parkman, 1972; Parkman and Groen, 1971; Suppes and Groen, 1967), mental processing ofanalogical representations (Aiken, 1971 ; Aiken and Williams, 1968, 1973; Moyer and Landauer, 1967; Resile, 1970), retrieval from semantic networks (Ashcraft, 1982, 1987 ; Ashcraft and Battaglia, 1978; Ashcraft and Stazik, 1981; Campbell, I 987a, 1987b; Campbell and Graham, 1985; Geary, Widaman and Little, 1986; Miller, Perlmutter and Keating, 1984; Siegler and Shrager, 1984), and use of procedural rules (Baroody, 1983, 1984, 1985). Procedural strategies are important in children, especially in using backup counting strategies (either counting fingers or verbal counting) when memory retrieval fails to produce a satisfactory answer (Geary and Burlingham-Dubree, 1989) and are even still used by adults for some kinds of problems (McCloskey, Aliminosa and Sokol, 1991 ; McCloskey, Harley and Sokol, 1991; Sokol, McCloskey, Cohen et a\., 1991). However, evidence from studies on arithmetical facts in normal adults and brain-damaged patients generally supports the view of retrieval from memory, i.e. arithmetical information would be 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; McCloskey et a!., 199Ib); when the stored fact is not immediately found, it could be retrieved using procedural knowledge. However, despite a large number of studies, there is still no general agreement about the organization of such a network. Recently, Dagenbach and McCloskey (1992; hereafter referred to as D&M) addressed specifically the question of the organization of arithmetical facts in memory through neuropsychological data reporting the case of RG, a 78-year-old brain-damaged patient with impaired arithmetic performances. Three important dissociations were observed. In the production of arithmetical fact tasks, a first dissociation was found among operations: while addition and multiplication were grossly impaired (respectively 75 and 78% of errors), subtraction was less severely impaired (39% of errors) . Furthermore, in production tasks, a second dissociation was observed among problems potentially solvable by rules (e.g., any number multiplied by 0 equals 0: N X 0 = 0) and problems requiring retrieval of specific facts (e.g. , 6X 5): for all operations, .problems solvable by rules were better performed. Finally, a third dissociation was found among production and verification tasks. The dissociation observed among the three operations in production was not present in verification: RG's verification accuracy was nearly 70% for all operations. For the authors, these results suggest that stored arithmetical fact representations are segregated by arithmetical operations, that a
• Research assistant, Nation al Fund for Scientific Research (Belgium).
Cortex, (1994) 30, 661-671
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M. Pesenti and Others
distinction has to be drawn between arithmetical rules and arithmetical facts, and that production and verification tasks involve different retrieval processes. The case reported here exhibits some converging evidence with the dissociations found by D&M. Our patient showed a clear dissociation among operations and a similar dissociation among rules and retrieval problems. However, the pattern of performance was similar between production and verification multiplication tasks. CASE HISTORY
BB, a 39 year-old woman, began to show in 1989 a mood change characterized by retiring within herself and episodes of sadness. In February 1991, she had troubles of balance, gait and motor coordination, and a general alteration of cognitive abilities. A CT scan and IRM examination in May and June 1991 both revealed an important cortico-subcortical atrophy without signs of focalization. Neuropsychological examinations in January 1992 corroborated the general cognitive impairment of BB. Due to the precocious cortical atrophy and the cognitive impairments, a precocious evolving dementia was diagnosed. BB was a graduate secretary and had successfully completed an accountancy course some years before her illness began. She claimed to be an expert knitter before her troubles, a hobby in which she used to perform easily complex calculations. As mentioned above, the neuropsychological assessment in January 1992 showed a general cognitive impairment with some moderate deficits in language, memory and intellectual functions. She was well oriented in space and time. By that time, her deficits appeared stronger for arithmetical performances than for other cognitive abilities. Language Assessment
BB's spontaneous speech was very poor. She was quite helpless in conversation and answered the examiner's questions just with very short but grammatically correct sentences. Nonetheless, comprehension of conversational speech was normal. Oral and written comprehension were perfect. She made no errors in repetition and reading of words, nonwords and sentences, ended easily and correctly short incomplete sentences, and correctly executed simple orders. She was able to give definitions (6/6) and opposites (11112) of words; her only error in the latter task was a recurrent perseveration of a previous item. The construction of sentences with target-words was correct (l1112). She was nearly perfect in semantic categorization tasks (70/72), both with pictorial and verbal material. Her spelling was good for short spoken words and non-words (4/5), but worse for long ones (0/6). Her performance in a sentence-picture matching task was moderately impaired (32/48), especially when having to deal with sentences in the passive voice (0/4). In a pictures naming task, she made 40% errors (18/45); for 60% (11118) of these errors, her responses suggested that she did not at all visually recognize the items. The production of automatized verbal sequences was correct for the days of the week and the alphabet, but impaired for the months of the year and the vowels; her counting from 1 to 21 was correct. Although her writing was slowed down, BB was able to write her name and address on request, as well as letters, words, non-words and sentences to dictation. Her writing was severely altered, with distorted grapheme production and an important dysrothography, but it remained understandable. Memory Assessment ~B's short-term memory was moderately impaired. Her digit span in auditory modality was weak (span of 4), but her spatial span was normal (span of 4 on the block-tapping test; Milner, 1971). For long-term memory, her performance on a free-recall I cued-recall I recognition task (a French language adaptation of the test devised by Grober and Buschke, 1986) was normal, but it was below percentile 5 in a face recognition task (Warrington, 1984).
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Selective impairment ofarlthmetiml facts
Intellectual Functions and Praxis Assessment
BB's intellectual functions were relatively preserved as shown hy her score on the MiUHill vocabulary test (23134, a weak score but still within the normal range; Gerard, 1983). BB also showed a severe oonstructive apraxia, but had no clear sign of:a visual :agnosia. Despite her weak performance in the drawing naming task mentioned above, she was nearly perfect in real picture naming and matching tasks (1071110). Numerical Processing Assessment
In February 1992, an evaluation Qf BB's difficulties with numbers was carried out through the Ca1culationand Number .Processing battery I of Ddoche and Seron(l991). The scores for the :subtests :of the battery and for the arithmetical fact relr·ieva! tasks are presented in Table 1. TABLE 1
SCOI:es of BB at the Calculation and Number P{:()ce5ring Battery
Correct/Items Subtests 1. Counting 2. Dots ,counting 3. Transcoding phonological .... phonologic,al phonological -4 Arabic phonological -4 ,alphabetical Arabic ..... phonological Arabic ..... alphabetical alphabetical ..... phonological alphabetical ..... Arabic 4. Arithmetical signs 5. Writing down an operation 6. Written calculation 7. Estimation of the result of a n operation K Number positioning OR an analogical scale 9. Perceptive estimation of quantities 10. Contextual magnitude judgements II. Precise numerical knowledge 12. Magnitude comparison 13. Grammaticality judgements Arithmetical facts addition multiplication
217 14115 'fJ/6
29130
14115 26132 8115 32/32 20129 5/8 3/4
0/6 5/8 7110 016 5/5 516 70170 72/72
315 10/20
Numeral Comprehension
The comprehension of numerals appeared to be unimpaired: magnitude comparison and grammaticality judgement tasks were perfectly performed in all the presentation formats (spoken verbal, written Arabic and written verbal numerals). Numeral Production
The production of numerals raised few problems. Arabic numeral production was good; only verbal numeral writing was severely impaired, but this impairment was due mainly to
'I The battery administered to BB was in fact a modified version of the Deroche and Seran battery. The transcoding and magnitude comparison subtests included more items than the original version;. moreover, 11 grammaticality judgement task was added, In this later task, the patient had to decide whether spoken verbal or written numerals were correct or not· (~xample of non grammatical items: ten-two, eight-twenty or forty hundred).
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M. Pesenti and Others
the words' graphic realisation than to the use of alphabetical numeral lexicon per se. At the syntactic level, systematic lexicalizations of the zero in items of the form "thousand + units" or "thousand + tens" were noted (e.g., 1019 was read and written "mille zero dix-neuf'2, an uncommon tum of phrase but which could be accepted in spoken Belgian French). Finally, no intrusion of response other than numbers was ever observed.
Arithmetic Identification and wntmg to dictation of arithmetical signs were slightly impaired. Complex written calculations were totally impaired: carrying and borrowing mechanisms as well as spatial positioning were preserved, but the resolution algorithms (order of steps to the solution) were severely altered, and this, paired with calculation errors, led to incorrect solutions for all problems (0/15). Finally, on simple mental arithmetical problem testing, retrieval of. basic table facts was altered for addition (3/5) and multiplication (10120), mainly for large problems in both cases. In conclusion, on the one hand, BB's numeral comprehension was good and her numeral production was only slightly altered, whilst on the other hand, her performance in arithmetic was impaired. As BB presented this arithmetical alteration with a general number processing preservation, a follow-up study of her performance in arithmetical facts retrieval was decided. A main point of interest is, as we shall see later, that BB presented a retrieval deficit of arithmetical facts which was not homogeneous among operations but with a selective sparing of subtraction. This case is interesting because it adds convergent information to a similar case presented by D&M and support the hypothesis-of segregated organization of arithmetical facts in memory, depending on the operation. Another point of interest is that our case, contrary to D&M's case, presented no dissociation of performance between production and verification multiplication tasks. The possibile theoretical pertinence of such a "no dissociation" will be discussed after the presentation of the experimental investigations. STUDY
I:
ARITHMETICAL FACTS PRODUCTION
Method From February to August 1992, BB was asked four times (each session spaced by a two month interval) to complete subtraction, addition and multiplication table facts (i.e., 0+0 through 9 + 9 for addition, 1 - 0 through 10 - 10 for subtraction, 0 X 0 through 9 X 9 for multiplication); both problems and responses were spoken. The three operations were presented by alternating series of 10 or 15 trials; the series were separated by other numerical tasks (numeral transcoding, magnitude comparison, etc.) and pauses.
Results BB's percentage of errors remained stable (except for the last block of additions) over time (see Table II; this stability was also observed for other numerical tasks not reported here); hence, results are averaged over the 4 blocks 3 •
Dissociation among Operations Table II presents the percentage of errors for the 4 blocks of subtractions, additions and multiplications. Taken globally, the results show a strong dissociation among the three operations: mean percentage of errors was 33% for addition, 57% for multiplication, but only .8% for subtraction. Addition and multiplication differed significantly from one another
"Thousand zero nineteen". An extended qualitative analysis of the errors would fall beyond the scope of this article. So, we will confine ourselves to a quantitative analysis. 2
3
Selective impairment of arithmetical facts
665
TABLE II
Percentage of Errors by Operation and Blocks Operations Blocks
I 2
3
4 Mean
Subtraction
Addition
Multiplication
All problems Size constant (65 items) (55 items)
All problems Size constant (100 items) (55 items)
All problems Size constant (100 items) (55 items)
9 9 9 6 8
11 11
7
28 25 29 49
9
33
7
29 20
33
44.5 31
53 57 61 56 57
49 51 63.5 56 55
(t3 = - 23.75; p<.03). Moreover, for subtraction, the statistical analysis while keeping the size of the problems constant (the 55 problems sharing the same operands, e.g. all m - n, m + nand m X n problems with O::'Sm,n::'S9 and m?n) confirmed the strong dissociation among operations: subtraction differed significantly from addition (t3 = - 3.9; p<.03) and from multiplication (t3 = -15.3; p<.OOl).
Dissociation among Kinds of Problems Whereas the retrieval of stored answers is considered to be the main strategy, some sets of problems might be solved by reference to general rules. These potential rules are N + 0 = N, N-O=N, N-N=O, NXO=O, and Nxl=N (see McCloskey et aI., 199Ia). Moreover, it seems to us that two other sets of problems might be considered differently from those requiring fact retrieval: N + 1 and N - I problems might indeed be solved using very simple procedural strategies (such as forward and backward counting by one). For the three operations, a strong dissociation was observed among these kinds of problems (Table III shows the percentage of errors for each kind). Except for the NX 0 rule, problems potentially solvable by rules were performed better than problems requiring retrieval of specific facts. As expected, performances for the N + I and N - 1 problems differed from fact-based problems: N + 1 problems appeared to be less impaired (9% errors) than factbased additions (48% errors), and N-] problems more impaired (17% errors) than factbased subtraction (12.5% errors). Moreover, the dissociation among the three operations was still present when only the errors for the fact-based problems were considered (subtraction: 12.5%, addition: 48%, multiplication: 59%). Paired comparisons indicated that addition and multiplication did not differ from one another (t3 = - 1.2; p> .3), and, for the 28 fact-based problems sharing the same operands, that subtraction differed significantly from addition (t3 = - 4.9; p<.02) and multiplication (t3 = - 12.9; p<.OOI). TABLE III
Percentage of Errors by Operation and Kind of Problem Problems Problems with "0" Problems with "I" N-N IO-N Fact-based problems Fact-based problems size constant (28 items)
Operations Subtraction 0 16.7 2.5 4.5 12.5 12.5
(0/40) (6/36) (1/40) (2/44) (141112) (141112)
Addition 2.6 (2/76) 8.8 (6/68) 48 (123/256) 48.2 (541112)
Multiplication 88.2 (67/76) 11.8 (8/68) 59.4 (152/256) 60.7 (681112)
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Discussion A strong dissociation among operations was thus observed, with a selective preservation of subtraction. Usually, dissociations among operations take the form of consistently worse performance for multiplication than for .addition or subtraction, .and sometimes worse for subtraction than for addition (McCloskey et aI., 1991a). However, such patterns of results ~ould reflect differences among operations in premorbid strength of the stored facts rather than ~segregation of the representations of arithmetical facts by operation in memory. The preservation of subtraction reported by D&M and the one presented here rule out this interpretation since a presumably easier or better learned operation (e.g., addition) showed greater impairment than a more difficult or less well-learned operation (e.g., subtraction). Nevertheless, a critical point has to be discussed before certifying BB' s preservation for subtraction. D&M used 100 subtraction problems such that for each m + n ;= p problem, .a complementary p-n;=m problem was tested (e.g., 15-7;=8 for 8+7;=15). In this study, however, subtraction never presented operands and consequently responses greater than 10, and the comparison analysis was carried out on what we considered to be real arithmetical faet, namely, holding constant the size of the operands across operations, implying that subtraction never presented operands and responses greater than 9. From this, one could object that subtraction and addition are not comparable in the way this was done, and even that subtraction appeared less impaired because addition involved greater or more difficult problems. Several considerations argue against this objection. From a theoretical point of view, we decided not to use D&M's complementary subtraction set because of the somewhat unclear cognitive status of this set. Let us recall that we are not considering addition or subtraction in general, but rather addition and subtraction arithmetical facts. In this sense, it is not clear whether 15 - 7 =; 8 belongs to any stored subtractive network. Such subtraction problems are generally not included in the "simple subtraction set" (Geary, Frensch and Wiley, ]993; Siegler and Schrager, 1984; Woods, Resnick and Groen, 1975) and are more likely solved using procedllml (counting, decomposition ... ) than direct retrieval strategies (Siegler, 1989; Svenson and Hedenborg, 1979). And even if it were to be considered as a fact-based problem, it is not certain that it is really comparable to 8 + 7;= J5 which D&M supposed to be its .counterpart. Does the mathematical complementarity between addition and subtraction mean such a cognitive correspondence? We do not tbink that problems with answers greater than 10 arec.omparable to problems startin,g from operands greater than 10. Thus, rather than matching addition and subtraction problems in the way used by D&M, we JIllltched them with respect to the size of their .operands. This is, in many respects, the way everywhere used t.o compare addition and mUltiplication. It remains that, matching them this way, subtraction and addition differ in the range of numbers to be produced as answers Cnumbersgreater than 10 for addition, below 10 for subtraction), and that the dissocation m~ght reflect some numeral production deficit. This interpretation, however, does not hold for BB. First, her performance in numeralcornprehensionand production tasks was nearly perfect a.nd she could easily produce numbers in the range of addition as well as subtraction responses4. Furthermore, restricted to the 21 fact-based addition and subtraction problems with answers in the range 2-9, her rate of errors was lower for subtraction (15.5%) than for addition (32%), and for the 10 addition and subtraction problems with 10 as answer or operand, it was still lower for subtraction (5%) than for addition (35%). It thus seems legitimate for us to conclude that subtraction is relatively preserved. Concerning arithmetic rules, the results confirm the distinction between rule-based and fact-based problems. Except for the NX a rule, problems potentially solvable by rules were better performed than problems requiring retrieval of specific facts. Such a dissociation supports the distinction between the mental representation of arithmetical rules and arithmetical facts (see for convergent evidence McCloskey et aL, 1991a; McCloskey et aI., 1991b; Sokol et aL., 1991). Moreover, the pattern of results for N+] and N -] problems suggests that
BB'~~rrors in the transcoding tasks concerned numbers greater than 100; all her errors in spoken verbal numerals were of the form described at note 2, Moreover. at the I.as! session, BBcompletea lwo time5 :Ii yerbaJ fluency task for the numbers (she was asked to' onllly produce numbers duting I minute). Produced numbers ranged from! to 41 the lirs! time, and from I to 100 .the second lime, with numbers under 10 and between 10 .and 20 equally frequent. 4
Selective impairment oj arithmetical facts
661
these sets of problems, without appealing to formal rules, could be treated differently than fact-based problems. This, of course, needs further investigations.
STUDY
2:
MULTIPLICATiON VERIFICATION
Experiment 1: Simple Verification Task Method In October 1992, BE was proposed a verification task for multiplication in which she had to judge problems presented with correct or' incorrect answers. Each of the 100 table multiplication problems was presented twice, once with the correct answer (true items) and once with an incorrect answer (false items). Five kinds of incorrect answers: were randomly assigned5 : operand distractor (an answer that would result from randomly increasing or decreasing one of the problem operands by 1 of 2, "4 X 9 = 28"), table distractor (a correct response for a problem that shares no common operand with the problem, "7 X 8 = 45"), operation distractor (the correct response for the same operands in addition, "2X 8 = 10"), nontable distractor (an impossible multiplication response, "5X4=29"), and frequent error distractor (an erroneous response frequently given by BB in the multiplication production task6 , "8 X 4 = 2j.'');. Incorrect answers outside the range of correct answers for multiplication were exclUded. Both, problems and answers were written on sheets of papltc (10 problems, per sheet); BE responded bY' ti<>:king off the correct or incorrect boxes:.
Results The percentage and pattern of errors were similar'to those observed in the multipl'i'cati'on production task. Overall, BB was 46% wrong (901197 7 errors, 27.5% for the true items and 63% for false'items). As in the production task, NX 0 problems were more impaired (89% errors, true items: 83%, false items: 64%; production: 89.5%), and NX 1 problems less impaired (20.5% correet, true items: 12%, false items: 29%; production: 12%) than factbased problems (52% correct, true'items: 16%, false items: 87.5%; production: 59%). For false items, the reverse of the classical split effect (Ashcraft and Battaglia, 1978) was observed: BB rejected more easily false items with small splits than with large splits (split of I and 2: 62.5% correct, split of 3 to 10: 36.5% correct, split larger than 10: 15.6% correct; X2 (3, N = 198) = 4 1.2; p< .000 I}. Finally, BB took very little advantage of the parity rule (Krueger, 1986; Krueger and Hallford, 1984): she performed just a little better with respect to false items violating the parity rule than to non-violating ones (violating: 43% correct; non-violating: 26.5% correct; X2 (I, N = 100) = 3. I.; p<.08).
Discussion Although these results strikingly resemble those observed in the production task and thus suggest no dissociation between verification and production in multiplication, they are harMy interpretable due to the fact that chance performance is 50% correct in a verification. task. The results, mainly for the facts-based problems, do not say much about BB's real level of impairment. In order to reduce this "chance bias", a multiple choice verification task was designed for mUltiplication.
'(Except for the NxO and Nx I prOblems that. were all operation distractors),. 6 Frequent error distractors were also table or nontables distractors. 7 Due to errors in preparing' the' item~, Ox 0;. both for tme and false items, artd 9X 7 for tme ones- were nor inducted in' the analysis,
M. Pesenti and Others
668
TABLE IV
Percentages of Errors for the Multiplications in Production First operand
o I
2
3 4 5 6 7
8 9 Total
Second operand
0 0 25 100 100 100 75 100 100 100 100 80
75 0 75 0 25 25 25 0 0 0 22.5
2
3
4
5
6
7
8
9
Total
100 0 25 0 0 0 25 0 0 50 20
100 25 0 0 25 0 75 75 100 100 50
100 0 50 75 0 25 0 100 100 75 55
100 0 0 0 0 0 100 100 100 75 47.5
100 0 50 75 75 75 50 100 100 100 72.5
100 25 0 0 100 100 100 100 75 100 70
100 0 0 100 100 100 25 75 100 100 70
100 0 50 100 75 100 100 100 100 100 82.5
87.5 7.5 35 45 50 50 60 75 77.5 80 57
Experiment 2: Multiple Choice Verification Task Method
In November 1992, BB was proposed a multtple choice verification task for multiplication. For each problem (from 0 X 0 through to 9 X 9), she had to select the correct answer among 6 different responses (see above for definitions; examples are given for "6 X 5"): the correct answer ("30"), an operand distractor ("20"), a table distractor ("16"), an operation distractor ("11"), a nontable distractor ("43"), and a frequent error distractor ("41"). Both problems and answers were written on sheets of paper (8 problems per sheet); for each problem, the positions of the correct response and distractors were randomly assigned. TABLE V
Errors in the Multiple Choice Verification Task for the Multiplications First operand
0
1
2 3 4 5 6 7 8 9
Total
Second operand
0 E E E E E E E E E
9
E E E E
2
3
4
5
6
7
8
9
E
E
E
E
E
E
E
E
9 2
E E
E E E
E E E
E E E
5 5 2 5 4 8 8 49
E
E
E E
5
E E
5
E E E
4
E
3
E E E E
5
E E E
4
6
7
Total
1
Results
Again, the percentage and pattern of errors were very similar to those observed in the multiplication production task. Dissociation among production and verification tasks? Tables IV and V clearly show that, both for production and multiple choice verification, (I) NX 0 problems were severely impaired (respectively, 89.5% and 95% errors in the production and verification tasks), with Ox 0 being the only preserved product; (2) NX 1 problems were relatively preserved
Selective impairment of arithmetical facts
669
(respectively, 12% and 18%), with the vast majority of errors being the correct answer for addition; and (3) for the fact-based problems, the rate of errors was average (respectively 59% and 42%), but higher for large problems (respectively 89% and 81 %). Finally, the kind of distractors erroneously chosen by BB in verification was similar to the kind of errors she made in production (see Table VI), Discussion The multiple choice verification task confirmed the pattern of performance observed in the simple verification and production tasks: rule-based and fact-based problems presented nearly the same level of impairment in the three tasks and the kinds of errors in verification were highly comparable with production errors. In contrast, the patient RG (D&M) presented different patterns in multiplication production and verification: he was, respectively, 8% but 24% correct for the N X O problems, 81% but 39% for the NX! problems, and 22% but 65 % for the fact-based problems. Moreover, the dissociation among operations observed in production was not found in verification, leading D&M to conclude that production and verification must involve different retrieval processes. For BB, addition and subtraction verification tasks were not administered because her level of performance began to change for these operations at that time. Nonetheless, when comparing verification and production tasks for multiplication, one cannot disregard the strong similarity of error patterns which was not present in RG, and it seems highly implausible to us that this similarity could be TABLE VI
Percentage of Kind of Errors in the Production and Multiple Choice Verification Tasks Errors/distractors Operands Tables Non-tables Operation Frequent error Non-responses
Tasks Production 57 18 15 10
(87/152) (27/152) (23/152) (15/152)
o
Verification 63 (17127) 15 (4127) 11 (3/27) 7 (2127) 4 (1127)
o
just fortuitous . On the contrary, it suggests that the deficit responsible for errors in both tasks is the same: either the same process (or processes) is (are) involved in production and verification, and this process is impaired in BB, or production and verification rely on different processes leading to the same mental representations, and these later are altered in BB. This alternative clearly has to be further investigated. It must be noted that BB's performance did not improve in the two verification tasks, as is classically observed (Campbell, 1987b; Zbrodoff and Logan, 1990), suggesting that the proposed answers did not act as primes, maybe because BB did not take advantage of item structure guiding judgement in normal subjects (split, parity rule).
CONCLUSIONS
A patient suffering from precocious evolving dementia with impaired arithmetic performances is described here. This patient showed during a long period of time specific di ssociations in basic mental arithmetic. First, in arithmetic production tasks, a strong dissociation among operations was observed: BB performed much better in subtraction than in addition and multiplication. From what we know, D&M's case and this one are the unique systematic reports of such a subtraction preservation. This selective preservation suggests the idea of a segregation by operation of the arithmetical facts mental representations.
670.
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S'econd, except for the NXO' rule, rule-based problems were better performed than factbased problems sustaining the distinction of arithmetical rules and arithmetical facts. mental representations. Moreover, the results for N+ 1 and N~ 1 problems suggest that these sets of problems might be solved using other strategies than fact retrieval. Finally, BB's pattern of performance in the production and veri.fication tasks presented a strong similarity, both for the severity of the disorder and the pattern of errors. Seemingly, BB did not use item structure (split, parity rule) to guide her judgements .. This may explain why production and verification lead to the same pattern of results. ABSTRACT
We· report hen: th€· {;ase· ofa patient (BB). suffering from 31 preca.:ious evolving d¢mentiw with impaired arithmetic' performance-, who, showed specilk and theoretical pertinent dissociations: in basic mental arithmetic. First, in a task invoLving production of answers to simple arithmetic. problems, a strong di"s50ciation was, found among operations:. while multiplication was severely, impaired, addition was moderately' and subtra
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CAMPBELL, ll.D. Production, verification and priming of muJtiplication facts. Memory and Cognition, 15:. 349-369, 1987b. CAMPBELL, J.I.D., and GRAHAM, 1.D. Mental multiplication skill s: Structure, process, acquisition. Canadian Journal of Psychology, 39: 338-366, 1985. 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. DELOCHE, G., and SERON, X. EC30l: BaUerie d'evaluation du caJcul et du traitement des nombres chez I'adultes. Glossa, 27: 40-42, 1991. GEARY, D.C., and BURLINGHAM-DuBREE, M. External validation of strategy choiche model for addition. Journal of Experimental Child Psychology, 47: 175-192, 1989. GEARY, D .C., FRENSCH, P.A. , and' WILEY, 1.G. Simple and complex mental subtraction: Strategy choice and speed-of-processing differences in younger and older adults. Psychology and Aging; 8: 242256, 1993. GEARY, D.C. , WIDAMAN, K.F., and UTILE, T.D, Cognitive addition and multiplication: Evidence for a single memory network . Memory and Cognition, 14: 478-487, 1986. GERARD, M. Contribution a I'evaluation de la deterioration mentale chez I'adulte a I' aide du test de vocabulaire de Mill-HilL University of Liege: Unpubli shed thesis , 1983. GROBER, E., and BUSCHKE, H. Genuine memory deficits in dementi a. Developmental Neuropsychology, 3: 13-36, 1986. GROEN, G.l., and PARKMAN, lM. A chronometric analysis of simple addition. Psychological Review, 79: 329-343, 1972. KRUEGER, L.E. Why 2 X 2 = 5 lo oks so wrong: On the odd-even rule in product verification. Memory and Cognition, 14: 141-149, 1986. KRUEGER, L.E., and HALLFORD, E.W . Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification. Memory and Cognition, 12: 171- I 80, 1984 .. MCCLOS KEY , M., AUMINOSA, D., and SOKOL, S.M. Facts, rutes, and procedures in normal calculation: Evidence from multiple single-patient studies of impaired arithmetic fact retrieval. Brain and Cognition, 17: 154-203, 199 1a. MCCLOSKEY, M., HARLEY, W·., and SOKOL, S.M . Models of arithmetic fact retrieval: An evaluation in light of findings from normal and brain-damaged subjects. Journal of Experimental Psychology: Learning, Memory and Cognition , 10: 46-60, 1991b. MILLER, K., PERLM UTIER, M., and KEATING, D. Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory and Cognition, 10: 46-60, 1984. MILNER, B. Interhemispheric differences in the localization of psychological processes in man. British Medical Bulletin, 27: 272-277, 1971. MOYER, R.S., and LANDAUER, T .K. Time required for judgements of numerical inequality. Nature, 215: 1519-1520, 1967. PARKMAN, I.M. Temporal aspects of simple multiplication and comparison. Journal of Experimental Psychology, 95: 437-444, 1972. PARKMAN, I.M., and GROEN, G.1. Temporal aspects of simple additi on and comparison. Journal of Experimental Psychology, 89: 335-342, 1971. RESTLE, F. Speed of adding and compari ng numbers. Journal of Experimental Psychology, 83: 274278, 1970. SI EGLER, R.S. Hazards of mental chronometry: An example from children's subtraction. Journal of Educational Psychology, 81: 497-506, 1989. STEGLER, R.S. , and SHRAGER, J. A model of trategy s choice. In C. Sophian (Ed.), Origins of Cognitive Skills. Hillsdale, NY: Erlbaum, 1984. SOKOL, S.M., MCCLOSKEY, M.,. COHEN, N.J., and AUMINOSA, D. Cognitive representations and processes in arithmetic: Evidence from the performance of brain-damaged patients. Journal of Experimental Psychology: Learning, Memory and Cognition, 8: 320-335, 199 L SUPPES, P., and GROEN, GJ. Some counting models of first grade performance data on simple addition facts . In 1.M. Scandura (Ed. ), Research in Mathematics Education . Washington D.C.: National Council of Teachers of Mathematics, 1967. SVENSON; 0. , and HEDENBORG, M .L. Strategies used by children when so lving simple subtractions. Acta Psychologica, 43: 477-489, 1979. WARRINGTON, E.K. Recognition Memory Test. NFER: Nelson Publishing Company Ltd. , 1984. WOODS, So8. , RESNICK, L.B ., and GROEN, GJ. An experimental test of five process models for subtraction. Journal of Educational Psychology, 67: 17-21 , 1975. ZBRODOFF, NJ ., and LOGAN, G.D. On the relation between production and verification tasks in the psychology of simple· arithmetic. Journal of Experimental Psychology: Learning, Memory and Cognition, 16: 83-97 , 1990. Dr. Mauro Pesenti, Cognitive Neuropsychology Research Unit, Experimental Psychology. Department , Catholic University of Louvain, 20 vo ie du Roman Pays, 1348 Louvain-I a-Neuve. E-mai l: [email protected].
Number processing and calculation have become during the past 20 years an active field of investigation to study the nature of mental representations. Research in cognitive arithmetic deals with the mental representation of numbers and arithmetic and tries to build models for the processes that access and use numerical knowledge. As regards simple mental arithmetic, basic arithmetical facts (e.g., 6 X 5 =?) have been considered in turn as the result of mental counting strategies (Groen and Parkman, 1972; Parkman, 1972; Parkman and Groen, 1971; Suppes and Groen, 1967), mental processing ofanalogical representations (Aiken, 1971 ; Aiken and Williams, 1968, 1973; Moyer and Landauer, 1967; Resile, 1970), retrieval from semantic networks (Ashcraft, 1982, 1987 ; Ashcraft and Battaglia, 1978; Ashcraft and Stazik, 1981; Campbell, I 987a, 1987b; Campbell and Graham, 1985; Geary, Widaman and Little, 1986; Miller, Perlmutter and Keating, 1984; Siegler and Shrager, 1984), and use of procedural rules (Baroody, 1983, 1984, 1985). Procedural strategies are important in children, especially in using backup counting strategies (either counting fingers or verbal counting) when memory retrieval fails to produce a satisfactory answer (Geary and Burlingham-Dubree, 1989) and are even still used by adults for some kinds of problems (McCloskey, Aliminosa and Sokol, 1991 ; McCloskey, Harley and Sokol, 1991; Sokol, McCloskey, Cohen et a\., 1991). However, evidence from studies on arithmetical facts in normal adults and brain-damaged patients generally supports the view of retrieval from memory, i.e. arithmetical information would be 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; McCloskey et a!., 199Ib); when the stored fact is not immediately found, it could be retrieved using procedural knowledge. However, despite a large number of studies, there is still no general agreement about the organization of such a network. Recently, Dagenbach and McCloskey (1992; hereafter referred to as D&M) addressed specifically the question of the organization of arithmetical facts in memory through neuropsychological data reporting the case of RG, a 78-year-old brain-damaged patient with impaired arithmetic performances. Three important dissociations were observed. In the production of arithmetical fact tasks, a first dissociation was found among operations: while addition and multiplication were grossly impaired (respectively 75 and 78% of errors), subtraction was less severely impaired (39% of errors) . Furthermore, in production tasks, a second dissociation was observed among problems potentially solvable by rules (e.g., any number multiplied by 0 equals 0: N X 0 = 0) and problems requiring retrieval of specific facts (e.g. , 6X 5): for all operations, .problems solvable by rules were better performed. Finally, a third dissociation was found among production and verification tasks. The dissociation observed among the three operations in production was not present in verification: RG's verification accuracy was nearly 70% for all operations. For the authors, these results suggest that stored arithmetical fact representations are segregated by arithmetical operations, that a
• Research assistant, Nation al Fund for Scientific Research (Belgium).
Cortex, (1994) 30, 661-671
662
M. Pesenti and Others
distinction has to be drawn between arithmetical rules and arithmetical facts, and that production and verification tasks involve different retrieval processes. The case reported here exhibits some converging evidence with the dissociations found by D&M. Our patient showed a clear dissociation among operations and a similar dissociation among rules and retrieval problems. However, the pattern of performance was similar between production and verification multiplication tasks. CASE HISTORY
BB, a 39 year-old woman, began to show in 1989 a mood change characterized by retiring within herself and episodes of sadness. In February 1991, she had troubles of balance, gait and motor coordination, and a general alteration of cognitive abilities. A CT scan and IRM examination in May and June 1991 both revealed an important cortico-subcortical atrophy without signs of focalization. Neuropsychological examinations in January 1992 corroborated the general cognitive impairment of BB. Due to the precocious cortical atrophy and the cognitive impairments, a precocious evolving dementia was diagnosed. BB was a graduate secretary and had successfully completed an accountancy course some years before her illness began. She claimed to be an expert knitter before her troubles, a hobby in which she used to perform easily complex calculations. As mentioned above, the neuropsychological assessment in January 1992 showed a general cognitive impairment with some moderate deficits in language, memory and intellectual functions. She was well oriented in space and time. By that time, her deficits appeared stronger for arithmetical performances than for other cognitive abilities. Language Assessment
BB's spontaneous speech was very poor. She was quite helpless in conversation and answered the examiner's questions just with very short but grammatically correct sentences. Nonetheless, comprehension of conversational speech was normal. Oral and written comprehension were perfect. She made no errors in repetition and reading of words, nonwords and sentences, ended easily and correctly short incomplete sentences, and correctly executed simple orders. She was able to give definitions (6/6) and opposites (11112) of words; her only error in the latter task was a recurrent perseveration of a previous item. The construction of sentences with target-words was correct (l1112). She was nearly perfect in semantic categorization tasks (70/72), both with pictorial and verbal material. Her spelling was good for short spoken words and non-words (4/5), but worse for long ones (0/6). Her performance in a sentence-picture matching task was moderately impaired (32/48), especially when having to deal with sentences in the passive voice (0/4). In a pictures naming task, she made 40% errors (18/45); for 60% (11118) of these errors, her responses suggested that she did not at all visually recognize the items. The production of automatized verbal sequences was correct for the days of the week and the alphabet, but impaired for the months of the year and the vowels; her counting from 1 to 21 was correct. Although her writing was slowed down, BB was able to write her name and address on request, as well as letters, words, non-words and sentences to dictation. Her writing was severely altered, with distorted grapheme production and an important dysrothography, but it remained understandable. Memory Assessment ~B's short-term memory was moderately impaired. Her digit span in auditory modality was weak (span of 4), but her spatial span was normal (span of 4 on the block-tapping test; Milner, 1971). For long-term memory, her performance on a free-recall I cued-recall I recognition task (a French language adaptation of the test devised by Grober and Buschke, 1986) was normal, but it was below percentile 5 in a face recognition task (Warrington, 1984).
663
Selective impairment ofarlthmetiml facts
Intellectual Functions and Praxis Assessment
BB's intellectual functions were relatively preserved as shown hy her score on the MiUHill vocabulary test (23134, a weak score but still within the normal range; Gerard, 1983). BB also showed a severe oonstructive apraxia, but had no clear sign of:a visual :agnosia. Despite her weak performance in the drawing naming task mentioned above, she was nearly perfect in real picture naming and matching tasks (1071110). Numerical Processing Assessment
In February 1992, an evaluation Qf BB's difficulties with numbers was carried out through the Ca1culationand Number .Processing battery I of Ddoche and Seron(l991). The scores for the :subtests :of the battery and for the arithmetical fact relr·ieva! tasks are presented in Table 1. TABLE 1
SCOI:es of BB at the Calculation and Number P{:()ce5ring Battery
Correct/Items Subtests 1. Counting 2. Dots ,counting 3. Transcoding phonological .... phonologic,al phonological -4 Arabic phonological -4 ,alphabetical Arabic ..... phonological Arabic ..... alphabetical alphabetical ..... phonological alphabetical ..... Arabic 4. Arithmetical signs 5. Writing down an operation 6. Written calculation 7. Estimation of the result of a n operation K Number positioning OR an analogical scale 9. Perceptive estimation of quantities 10. Contextual magnitude judgements II. Precise numerical knowledge 12. Magnitude comparison 13. Grammaticality judgements Arithmetical facts addition multiplication
217 14115 'fJ/6
29130
14115 26132 8115 32/32 20129 5/8 3/4
0/6 5/8 7110 016 5/5 516 70170 72/72
315 10/20
Numeral Comprehension
The comprehension of numerals appeared to be unimpaired: magnitude comparison and grammaticality judgement tasks were perfectly performed in all the presentation formats (spoken verbal, written Arabic and written verbal numerals). Numeral Production
The production of numerals raised few problems. Arabic numeral production was good; only verbal numeral writing was severely impaired, but this impairment was due mainly to
'I The battery administered to BB was in fact a modified version of the Deroche and Seran battery. The transcoding and magnitude comparison subtests included more items than the original version;. moreover, 11 grammaticality judgement task was added, In this later task, the patient had to decide whether spoken verbal or written numerals were correct or not· (~xample of non grammatical items: ten-two, eight-twenty or forty hundred).
664
M. Pesenti and Others
the words' graphic realisation than to the use of alphabetical numeral lexicon per se. At the syntactic level, systematic lexicalizations of the zero in items of the form "thousand + units" or "thousand + tens" were noted (e.g., 1019 was read and written "mille zero dix-neuf'2, an uncommon tum of phrase but which could be accepted in spoken Belgian French). Finally, no intrusion of response other than numbers was ever observed.
Arithmetic Identification and wntmg to dictation of arithmetical signs were slightly impaired. Complex written calculations were totally impaired: carrying and borrowing mechanisms as well as spatial positioning were preserved, but the resolution algorithms (order of steps to the solution) were severely altered, and this, paired with calculation errors, led to incorrect solutions for all problems (0/15). Finally, on simple mental arithmetical problem testing, retrieval of. basic table facts was altered for addition (3/5) and multiplication (10120), mainly for large problems in both cases. In conclusion, on the one hand, BB's numeral comprehension was good and her numeral production was only slightly altered, whilst on the other hand, her performance in arithmetic was impaired. As BB presented this arithmetical alteration with a general number processing preservation, a follow-up study of her performance in arithmetical facts retrieval was decided. A main point of interest is, as we shall see later, that BB presented a retrieval deficit of arithmetical facts which was not homogeneous among operations but with a selective sparing of subtraction. This case is interesting because it adds convergent information to a similar case presented by D&M and support the hypothesis-of segregated organization of arithmetical facts in memory, depending on the operation. Another point of interest is that our case, contrary to D&M's case, presented no dissociation of performance between production and verification multiplication tasks. The possibile theoretical pertinence of such a "no dissociation" will be discussed after the presentation of the experimental investigations. STUDY
I:
ARITHMETICAL FACTS PRODUCTION
Method From February to August 1992, BB was asked four times (each session spaced by a two month interval) to complete subtraction, addition and multiplication table facts (i.e., 0+0 through 9 + 9 for addition, 1 - 0 through 10 - 10 for subtraction, 0 X 0 through 9 X 9 for multiplication); both problems and responses were spoken. The three operations were presented by alternating series of 10 or 15 trials; the series were separated by other numerical tasks (numeral transcoding, magnitude comparison, etc.) and pauses.
Results BB's percentage of errors remained stable (except for the last block of additions) over time (see Table II; this stability was also observed for other numerical tasks not reported here); hence, results are averaged over the 4 blocks 3 •
Dissociation among Operations Table II presents the percentage of errors for the 4 blocks of subtractions, additions and multiplications. Taken globally, the results show a strong dissociation among the three operations: mean percentage of errors was 33% for addition, 57% for multiplication, but only .8% for subtraction. Addition and multiplication differed significantly from one another
"Thousand zero nineteen". An extended qualitative analysis of the errors would fall beyond the scope of this article. So, we will confine ourselves to a quantitative analysis. 2
3
Selective impairment of arithmetical facts
665
TABLE II
Percentage of Errors by Operation and Blocks Operations Blocks
I 2
3
4 Mean
Subtraction
Addition
Multiplication
All problems Size constant (65 items) (55 items)
All problems Size constant (100 items) (55 items)
All problems Size constant (100 items) (55 items)
9 9 9 6 8
11 11
7
28 25 29 49
9
33
7
29 20
33
44.5 31
53 57 61 56 57
49 51 63.5 56 55
(t3 = - 23.75; p<.03). Moreover, for subtraction, the statistical analysis while keeping the size of the problems constant (the 55 problems sharing the same operands, e.g. all m - n, m + nand m X n problems with O::'Sm,n::'S9 and m?n) confirmed the strong dissociation among operations: subtraction differed significantly from addition (t3 = - 3.9; p<.03) and from multiplication (t3 = -15.3; p<.OOl).
Dissociation among Kinds of Problems Whereas the retrieval of stored answers is considered to be the main strategy, some sets of problems might be solved by reference to general rules. These potential rules are N + 0 = N, N-O=N, N-N=O, NXO=O, and Nxl=N (see McCloskey et aI., 199Ia). Moreover, it seems to us that two other sets of problems might be considered differently from those requiring fact retrieval: N + 1 and N - I problems might indeed be solved using very simple procedural strategies (such as forward and backward counting by one). For the three operations, a strong dissociation was observed among these kinds of problems (Table III shows the percentage of errors for each kind). Except for the NX 0 rule, problems potentially solvable by rules were performed better than problems requiring retrieval of specific facts. As expected, performances for the N + I and N - 1 problems differed from fact-based problems: N + 1 problems appeared to be less impaired (9% errors) than factbased additions (48% errors), and N-] problems more impaired (17% errors) than factbased subtraction (12.5% errors). Moreover, the dissociation among the three operations was still present when only the errors for the fact-based problems were considered (subtraction: 12.5%, addition: 48%, multiplication: 59%). Paired comparisons indicated that addition and multiplication did not differ from one another (t3 = - 1.2; p> .3), and, for the 28 fact-based problems sharing the same operands, that subtraction differed significantly from addition (t3 = - 4.9; p<.02) and multiplication (t3 = - 12.9; p<.OOI). TABLE III
Percentage of Errors by Operation and Kind of Problem Problems Problems with "0" Problems with "I" N-N IO-N Fact-based problems Fact-based problems size constant (28 items)
Operations Subtraction 0 16.7 2.5 4.5 12.5 12.5
(0/40) (6/36) (1/40) (2/44) (141112) (141112)
Addition 2.6 (2/76) 8.8 (6/68) 48 (123/256) 48.2 (541112)
Multiplication 88.2 (67/76) 11.8 (8/68) 59.4 (152/256) 60.7 (681112)
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M. Pcscntiand .others
Discussion A strong dissociation among operations was thus observed, with a selective preservation of subtraction. Usually, dissociations among operations take the form of consistently worse performance for multiplication than for .addition or subtraction, .and sometimes worse for subtraction than for addition (McCloskey et aI., 1991a). However, such patterns of results ~ould reflect differences among operations in premorbid strength of the stored facts rather than ~segregation of the representations of arithmetical facts by operation in memory. The preservation of subtraction reported by D&M and the one presented here rule out this interpretation since a presumably easier or better learned operation (e.g., addition) showed greater impairment than a more difficult or less well-learned operation (e.g., subtraction). Nevertheless, a critical point has to be discussed before certifying BB' s preservation for subtraction. D&M used 100 subtraction problems such that for each m + n ;= p problem, .a complementary p-n;=m problem was tested (e.g., 15-7;=8 for 8+7;=15). In this study, however, subtraction never presented operands and consequently responses greater than 10, and the comparison analysis was carried out on what we considered to be real arithmetical faet, namely, holding constant the size of the operands across operations, implying that subtraction never presented operands and responses greater than 9. From this, one could object that subtraction and addition are not comparable in the way this was done, and even that subtraction appeared less impaired because addition involved greater or more difficult problems. Several considerations argue against this objection. From a theoretical point of view, we decided not to use D&M's complementary subtraction set because of the somewhat unclear cognitive status of this set. Let us recall that we are not considering addition or subtraction in general, but rather addition and subtraction arithmetical facts. In this sense, it is not clear whether 15 - 7 =; 8 belongs to any stored subtractive network. Such subtraction problems are generally not included in the "simple subtraction set" (Geary, Frensch and Wiley, ]993; Siegler and Schrager, 1984; Woods, Resnick and Groen, 1975) and are more likely solved using procedllml (counting, decomposition ... ) than direct retrieval strategies (Siegler, 1989; Svenson and Hedenborg, 1979). And even if it were to be considered as a fact-based problem, it is not certain that it is really comparable to 8 + 7;= J5 which D&M supposed to be its .counterpart. Does the mathematical complementarity between addition and subtraction mean such a cognitive correspondence? We do not tbink that problems with answers greater than 10 arec.omparable to problems startin,g from operands greater than 10. Thus, rather than matching addition and subtraction problems in the way used by D&M, we JIllltched them with respect to the size of their .operands. This is, in many respects, the way everywhere used t.o compare addition and mUltiplication. It remains that, matching them this way, subtraction and addition differ in the range of numbers to be produced as answers Cnumbersgreater than 10 for addition, below 10 for subtraction), and that the dissocation m~ght reflect some numeral production deficit. This interpretation, however, does not hold for BB. First, her performance in numeralcornprehensionand production tasks was nearly perfect a.nd she could easily produce numbers in the range of addition as well as subtraction responses4. Furthermore, restricted to the 21 fact-based addition and subtraction problems with answers in the range 2-9, her rate of errors was lower for subtraction (15.5%) than for addition (32%), and for the 10 addition and subtraction problems with 10 as answer or operand, it was still lower for subtraction (5%) than for addition (35%). It thus seems legitimate for us to conclude that subtraction is relatively preserved. Concerning arithmetic rules, the results confirm the distinction between rule-based and fact-based problems. Except for the NX a rule, problems potentially solvable by rules were better performed than problems requiring retrieval of specific facts. Such a dissociation supports the distinction between the mental representation of arithmetical rules and arithmetical facts (see for convergent evidence McCloskey et aL, 1991a; McCloskey et aI., 1991b; Sokol et aL., 1991). Moreover, the pattern of results for N+] and N -] problems suggests that
BB'~~rrors in the transcoding tasks concerned numbers greater than 100; all her errors in spoken verbal numerals were of the form described at note 2, Moreover. at the I.as! session, BBcompletea lwo time5 :Ii yerbaJ fluency task for the numbers (she was asked to' onllly produce numbers duting I minute). Produced numbers ranged from! to 41 the lirs! time, and from I to 100 .the second lime, with numbers under 10 and between 10 .and 20 equally frequent. 4
Selective impairment oj arithmetical facts
661
these sets of problems, without appealing to formal rules, could be treated differently than fact-based problems. This, of course, needs further investigations.
STUDY
2:
MULTIPLICATiON VERIFICATION
Experiment 1: Simple Verification Task Method In October 1992, BE was proposed a verification task for multiplication in which she had to judge problems presented with correct or' incorrect answers. Each of the 100 table multiplication problems was presented twice, once with the correct answer (true items) and once with an incorrect answer (false items). Five kinds of incorrect answers: were randomly assigned5 : operand distractor (an answer that would result from randomly increasing or decreasing one of the problem operands by 1 of 2, "4 X 9 = 28"), table distractor (a correct response for a problem that shares no common operand with the problem, "7 X 8 = 45"), operation distractor (the correct response for the same operands in addition, "2X 8 = 10"), nontable distractor (an impossible multiplication response, "5X4=29"), and frequent error distractor (an erroneous response frequently given by BB in the multiplication production task6 , "8 X 4 = 2j.'');. Incorrect answers outside the range of correct answers for multiplication were exclUded. Both, problems and answers were written on sheets of papltc (10 problems, per sheet); BE responded bY' ti<>:king off the correct or incorrect boxes:.
Results The percentage and pattern of errors were similar'to those observed in the multipl'i'cati'on production task. Overall, BB was 46% wrong (901197 7 errors, 27.5% for the true items and 63% for false'items). As in the production task, NX 0 problems were more impaired (89% errors, true items: 83%, false items: 64%; production: 89.5%), and NX 1 problems less impaired (20.5% correet, true items: 12%, false items: 29%; production: 12%) than factbased problems (52% correct, true'items: 16%, false items: 87.5%; production: 59%). For false items, the reverse of the classical split effect (Ashcraft and Battaglia, 1978) was observed: BB rejected more easily false items with small splits than with large splits (split of I and 2: 62.5% correct, split of 3 to 10: 36.5% correct, split larger than 10: 15.6% correct; X2 (3, N = 198) = 4 1.2; p< .000 I}. Finally, BB took very little advantage of the parity rule (Krueger, 1986; Krueger and Hallford, 1984): she performed just a little better with respect to false items violating the parity rule than to non-violating ones (violating: 43% correct; non-violating: 26.5% correct; X2 (I, N = 100) = 3. I.; p<.08).
Discussion Although these results strikingly resemble those observed in the production task and thus suggest no dissociation between verification and production in multiplication, they are harMy interpretable due to the fact that chance performance is 50% correct in a verification. task. The results, mainly for the facts-based problems, do not say much about BB's real level of impairment. In order to reduce this "chance bias", a multiple choice verification task was designed for mUltiplication.
'(Except for the NxO and Nx I prOblems that. were all operation distractors),. 6 Frequent error distractors were also table or nontables distractors. 7 Due to errors in preparing' the' item~, Ox 0;. both for tme and false items, artd 9X 7 for tme ones- were nor inducted in' the analysis,
M. Pesenti and Others
668
TABLE IV
Percentages of Errors for the Multiplications in Production First operand
o I
2
3 4 5 6 7
8 9 Total
Second operand
0 0 25 100 100 100 75 100 100 100 100 80
75 0 75 0 25 25 25 0 0 0 22.5
2
3
4
5
6
7
8
9
Total
100 0 25 0 0 0 25 0 0 50 20
100 25 0 0 25 0 75 75 100 100 50
100 0 50 75 0 25 0 100 100 75 55
100 0 0 0 0 0 100 100 100 75 47.5
100 0 50 75 75 75 50 100 100 100 72.5
100 25 0 0 100 100 100 100 75 100 70
100 0 0 100 100 100 25 75 100 100 70
100 0 50 100 75 100 100 100 100 100 82.5
87.5 7.5 35 45 50 50 60 75 77.5 80 57
Experiment 2: Multiple Choice Verification Task Method
In November 1992, BB was proposed a multtple choice verification task for multiplication. For each problem (from 0 X 0 through to 9 X 9), she had to select the correct answer among 6 different responses (see above for definitions; examples are given for "6 X 5"): the correct answer ("30"), an operand distractor ("20"), a table distractor ("16"), an operation distractor ("11"), a nontable distractor ("43"), and a frequent error distractor ("41"). Both problems and answers were written on sheets of paper (8 problems per sheet); for each problem, the positions of the correct response and distractors were randomly assigned. TABLE V
Errors in the Multiple Choice Verification Task for the Multiplications First operand
0
1
2 3 4 5 6 7 8 9
Total
Second operand
0 E E E E E E E E E
9
E E E E
2
3
4
5
6
7
8
9
E
E
E
E
E
E
E
E
9 2
E E
E E E
E E E
E E E
5 5 2 5 4 8 8 49
E
E
E E
5
E E
5
E E E
4
E
3
E E E E
5
E E E
4
6
7
Total
1
Results
Again, the percentage and pattern of errors were very similar to those observed in the multiplication production task. Dissociation among production and verification tasks? Tables IV and V clearly show that, both for production and multiple choice verification, (I) NX 0 problems were severely impaired (respectively, 89.5% and 95% errors in the production and verification tasks), with Ox 0 being the only preserved product; (2) NX 1 problems were relatively preserved
Selective impairment of arithmetical facts
669
(respectively, 12% and 18%), with the vast majority of errors being the correct answer for addition; and (3) for the fact-based problems, the rate of errors was average (respectively 59% and 42%), but higher for large problems (respectively 89% and 81 %). Finally, the kind of distractors erroneously chosen by BB in verification was similar to the kind of errors she made in production (see Table VI), Discussion The multiple choice verification task confirmed the pattern of performance observed in the simple verification and production tasks: rule-based and fact-based problems presented nearly the same level of impairment in the three tasks and the kinds of errors in verification were highly comparable with production errors. In contrast, the patient RG (D&M) presented different patterns in multiplication production and verification: he was, respectively, 8% but 24% correct for the N X O problems, 81% but 39% for the NX! problems, and 22% but 65 % for the fact-based problems. Moreover, the dissociation among operations observed in production was not found in verification, leading D&M to conclude that production and verification must involve different retrieval processes. For BB, addition and subtraction verification tasks were not administered because her level of performance began to change for these operations at that time. Nonetheless, when comparing verification and production tasks for multiplication, one cannot disregard the strong similarity of error patterns which was not present in RG, and it seems highly implausible to us that this similarity could be TABLE VI
Percentage of Kind of Errors in the Production and Multiple Choice Verification Tasks Errors/distractors Operands Tables Non-tables Operation Frequent error Non-responses
Tasks Production 57 18 15 10
(87/152) (27/152) (23/152) (15/152)
o
Verification 63 (17127) 15 (4127) 11 (3/27) 7 (2127) 4 (1127)
o
just fortuitous . On the contrary, it suggests that the deficit responsible for errors in both tasks is the same: either the same process (or processes) is (are) involved in production and verification, and this process is impaired in BB, or production and verification rely on different processes leading to the same mental representations, and these later are altered in BB. This alternative clearly has to be further investigated. It must be noted that BB's performance did not improve in the two verification tasks, as is classically observed (Campbell, 1987b; Zbrodoff and Logan, 1990), suggesting that the proposed answers did not act as primes, maybe because BB did not take advantage of item structure guiding judgement in normal subjects (split, parity rule).
CONCLUSIONS
A patient suffering from precocious evolving dementia with impaired arithmetic performances is described here. This patient showed during a long period of time specific di ssociations in basic mental arithmetic. First, in arithmetic production tasks, a strong dissociation among operations was observed: BB performed much better in subtraction than in addition and multiplication. From what we know, D&M's case and this one are the unique systematic reports of such a subtraction preservation. This selective preservation suggests the idea of a segregation by operation of the arithmetical facts mental representations.
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M. Pesenti and Others.
S'econd, except for the NXO' rule, rule-based problems were better performed than factbased problems sustaining the distinction of arithmetical rules and arithmetical facts. mental representations. Moreover, the results for N+ 1 and N~ 1 problems suggest that these sets of problems might be solved using other strategies than fact retrieval. Finally, BB's pattern of performance in the production and veri.fication tasks presented a strong similarity, both for the severity of the disorder and the pattern of errors. Seemingly, BB did not use item structure (split, parity rule) to guide her judgements .. This may explain why production and verification lead to the same pattern of results. ABSTRACT
We· report hen: th€· {;ase· ofa patient (BB). suffering from 31 preca.:ious evolving d¢mentiw with impaired arithmetic' performance-, who, showed specilk and theoretical pertinent dissociations: in basic mental arithmetic. First, in a task invoLving production of answers to simple arithmetic. problems, a strong di"s50ciation was, found among operations:. while multiplication was severely, impaired, addition was moderately' and subtra
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