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The Representation Of Arithmetical Facts: Evidence from two Rehabilitation Studies
THE REPRESENTATION OF ARITHMETICAL FACTS: EVIDENCE FROM TWO REHABILITATION STUDIES* L. Girelli, M. Delazer, C. Semenza, G. Denes (Department of Psychology, University College London, U.K.; Department of Neurology, University Clinic Innsbruck, Austria; Department of Psychology, University of Padua; Clinica Neurologica I, University of Padua, Italy)
ABSTRACT
We report the improvement following rehabilitation of two patients with a selective deficit on multiplication facts. The remediation experiments were conducted in specific designs to test a series of hypotheses about both the process of reacquisition and organisation of arithmetical facts in memory. The results show a significant and stable recovery and the pattern of reacquisition reflects the different strategy used by the patients to relearn the multiplication facts. Quantitative and qualitative changes in error patterns in the course of remediation reveal the underlying (re)organization of the memory network representing multiplication facts.
INTRODUCTION
Arithmetical fact knowledge consists in calculation problems which are retrieved from memory and do not require any calculation procedures. It constitutes a part of the calculation system (McCloskey, Caramazza and Basili, 1985; McCloskey, 1992, for a review) and it is essential for efficient calculation (Hittmair-Delazer, Semenza and Denes, 1994). We repeatedly use arithmetical fact knowledge in everyday life, often without being conscious of doing calculations. We double the number of plates when unexpected guests arrive, we calculate the delay with which we will arrive at the lecture, we quickly check the change when we pay a bill etc. Some neuropsychological reports show that arithmetical facts can be lost selectively after brain damage (Warrington, 1982; McCloskey, Harley and Sokol, 1991; Sokol, McCloskey, Cohen and Aliminosa, 1991; McCloskey, 1992; Cohen and Dehaene, 1994; Hittmair-Delazer et aI., 1994). Patients who are not able anymore to do the basic, memory-based calculations are severely handicapped. Despite this, few studies have been concerned with the remediation of arithmetical facts (Miceli and Capasso, 1991; Sokol and McCloskey, 1991; McCloskey, Aliminosa and Sokol, 1991; HittmairDelazer et aI., 1994). McCloskey et aI. (1991) reported a selective training effect for repeated multiplication problems which extended to the problems' complements, but did not improve performance in general (e.g., the training of 4 X 5 would improve 5 X 4, but not 4 X 6). Also the present study will focus on * Some of this material was presented at a special symposium at the International Neuropsychological Society in Madeira, June, 1993. Cortex, (1996) 32, 49-66
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multiplication, the operation which is thought to depend most on retrieval from semantic memory. Indeed, multiplication is an ideal field to study the internal structure of a part of semantic memory: multiplications are exactly defined, they are limited in number, they are taught systematically, they are used in everyday day life and they represent a common knowledge of virtually all educated people. Various models of arithmetical fact representation have been proposed on the basis of systematic effects observed in children and adults (for example the problem size effect; Groen and Parkman, 1972; Ashcraft and Battaglia, 1978; Campbell and Graham, 1985). Although the models (Siegler and Shrager, 1984; Siegler, 1988; Campbell and Graham, 1985; Campbell, 1987; Ashcraft, 1987) differ in their internal structure, they share some general assumptions: all models assume a network of associations between operands (and/or entire problems) and stored answers, both, correct and false ones. The strength of the single links between operands (or problems) and answers would depend on the frequency of former activation. All models assume a spreading activation as underlying process of retrieval (for a discussion of the models with respect to the performance of acalculic patients, see McCloskey, Harley and Sokol, 1991). We present two acalculic patients who totally lost multiplication facts. The initial examination and the following remediation experiments, conducted in specific designs, offer the possibility to test a series of hypotheses, not only about the representation of arithmetical facts in semantic memory, but also about the process of building up those mental representations. We assume that in the course of remediation, at least partially successful, patients build up a new semantic representation of lost facts or, at least, reorganise preserved notions. This process should be reflected by an increasing accuracy in solving multiplication problems. Moreover, we expect an approximation of the patients' qualitative error patterns to the typical distribution observed in normal subjects. We will describe how patients learn multiplication tables and which consequences different back-up strategies may have. Finally we will evaluate whether the effect of training is specific for taught items (McCloskey et al. 1991; Hittmair-Delazer et aI., 1994) or whether a selective training on a subset of facts has a not item-specific activation effect (resulting in a general improvement). We will also assess divisions before and after the training periods, in order to examine the relation between the two operations. Studies on arithmetical facts have rarely focused on simple divisions and little evidence is available for an autonomous representation of division facts in semantic memory. If this is the case, the loss of division facts could be overcome by using backup strategies based on multiplications (e.g., 56: 7 = -+ 7 X n = 56). As pointed out by Hittmair et al. (1994) a further alternative has to be considered: division facts could routinely be solved via multiplication assuming they are not independently represented in semantic memory. Although multiple interpretations are available we still consider interesting whether the training on multiplications will affect performance on divisions. Before we start with the description of the remediation experiments we shortly summarise the data collected from a group of healthy Italian subjects. It is possible that retrieval of arithmetical facts could be related to the process of acquisition, hence it could differ according to teaching method (e.g., according
Reacquisition of arithmetical facts
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12 11 10
8
4
2
o
2x
3x
4x
5x 6x Times table
7x
8x
9x
Fig. 1 - Normal subjects error rate as a function of the time-tables.
to the Italian teaching system the table of 5 consists of 5 X 2, 5 X 3, etc., according to the English one the same table consists of 2 X 5, 3 X 5, etc.). Normal Subjects The aim of this part of the study was to assess error rate and error pattern of normal Italian subjects in simple multiplications. All subjects (n = 100; mean age = 22 years; formal education, mean = 12 years) answered hundred written multiplications (operands from 0 to 9) in random order. The task was performed under time pressure. M X N facts (from 2 X 2 to 9 X 9; n = 6400) and rule-based items (including 0 or 1; n = 3600) were separately analysed. The overall error rate for m X n problems was 5.56%. The error rate differed considerably among the single multiplication tables. The times tables showed an increasing difficulty, with the exception of the table of 5, which is easier than would be expected by number size (see Figure 1). We found a strong correlation between the magnitude of a problem's result (e.g., 20 for 4 X 5) and the number of errors observed (Problem size effect: Spearman rank correlation r=0.769; p
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100 90 80 70 (260 "~ 50 0 cf 40 30 20 10 0
66.4
(/)
operand
close-miss
table Error type
non table
operation
Fig. 2 - Normal subjects distribution of errors.
The errors were classified into the following types (as suggested by Campbell and Graham, 1985; Siegler, 1988; McCloskey et aI., 1991): an operand error is the product of an operand with another number different from the second factor (e.g., 5 X 4 = 24). A close miss error is a number near to the correct result (plus or minus ten percent of the correct result, such as 6 X 7 = 38). A table error belongs to a table different from either operands' (e.g., 3 X 4 = 25). A non-table error is a number not included in the times-tables (e.g., 3 X 4 = 17). Finally, in an operation error the answer would be correct for a different arithmetic operation between the same numbers (e.g., 5 X 2 = 7). Errors were assigned to one error class only. Certain errors could be assigned to more than one class, i.e. close miss versus non-table/table errors. We decided to separately classify errors close to the correct result since they have been argued to be of specific interest (e.g., Siegler, 1988). The distribution of error types is shown in Figure 2. Two thirds of all errors are operand errors, 76.7% of which consisted in correct answers for problems that not only share an operand with the stimulus problems but are also close in magnitUde with respect to the other operand (e.g., 7 X 9 = 56, that is the correct response for 7 X 8). This so-called operand distance effect has been already reported both in normal subjects (Campbell and Graham, 1085; Miller, Perlmutter and Keating, 1984) and in patients (Sokol et aI., 1991; Cohen and Dehaene, 1994). In a separate analysis of the problems including 0 or 1 as operands (which are thought to be solved by a stored rule; Sokol et aI., 1991) a lower error rate is found (3.22%). Out of 116 errors, 113 can be attributed to the application of a wrong rule (e.g., 5 X 0 = 5 from N X 0 = N). Most of the errors (91 %) occurred in O's problems. Summing up, normal subjects have a mean error rate of 5.56% in solving
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TABLE I
Background Data for TL and ZA
TL ZA
Age
Years of education
22 46
8 13
Lesion (eva) Wide perisylvian Wide perisylvian
Onset 6 years 2 years
Neurological examination Right hemiparesis Right hemiparesis
Raven
Digit span
Corsi
43/48 48/48
3 4
5 5
mUltiplications from 2 X 2 to 9 X 9 and of 3.22% in solving rule-based problems. In m X n problems, errors consist typically in a false product of one of the operands which is close to the correct result. In problems including 0 or 1, errors appear due to the application of a wrong rule. We found some of the effects already observed in normal controls (e.g., problem size-effect, operand distance effect). In comparison to McCloskey et al. (1991) we found a higher error rate in m X n problems and a relatively low error rate in rule-based problems (although McCloskey's data referred to only 15 subjects presented with very few problems).
CASE DESCRIPTION
Two subjects, TL (female, 22 years) and ZA (male, 46 years), participated in the study. Both, TL and ZA, had good arithmetical knowledge before their disease and were highly motivated to recover arithmetical skills. Both had been acalculic and aphasic for 6 and 2 years at the start of the study and a spontaneous remission had to be excluded (Table I). Both patients performed the AAT (Versione italiana: Luzzatti, Willmes and DeBleser, 1991) and a complete assessment of number processing and calculation (Miceli and Capasso, 1991). The results are shown in Appendix A. In the Raven matrices both patients performed in the average range (Spinnler and Tognoni, 1987). Their calculation deficit consists of a selective loss of arithmetical fact knowledge (procedures being intact) which was not secondary to other neuropsychological deficits. Transcoding processes were sufficiently preserved to produce reliable results in the calculation training: both were able to understand written and spoken numbers up to two digits and to produce an answer by writing the result or by pointing the number from a table. Although they had speech therapy since their trauma they never had any rehabilitation treatment for arithmetical problems.
DESIGN OF THE REMEDIATION EXPE RIMENTS
Using a cross-over design (as proposed by Coltheart, 1983; see also McCloskey et aI., 1991), we divided all multiplications in two sets, set A and Set B, which were presented in separate training periods (sets differed for the two patients). Performance was assessed before training, after the first and the second training period and in a follow-up examination. In these assessments, where overt strategies were not allowed, all multiplications from 2 X 2 through 9 X 9 were presented and answered in written form (by writing digits or pointing from a table).
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Patient TL First Assessment In a first examination (Tl) only 6 multiplications out of 64 (problems from 2 X 2 to 9 X 9) were answered correctly. Interestingly, 41% of the erroneous responses consisted in far-miss errors, defined as number over 100% of the product, e.g., 2 X 9 = 44; most of them were non-table numbers (50%). This type of errors had never been observed both in normal or acalculic patients. Following the previous classification we found 50% non-table errors (41 % of which were far-miss numbers), 22.4% table errors, 19% operand errors, 5.2% close miss errors and finally 3.4% operation errors. Consistent with the extremely low accuracy in production, TL scored at chance level in a verification task (e.g., 2 X 3 = 9?; 19/36 correct) and in a multiple choice task (e.g., 3 X 4 = 8, 12, 24?; 8/24 correct). In rule-based items TL failed to answer 50% (9/18) of the problems. Moreover, errors were only partially grounded in the application of the wrong rule (e.g., 1 X 6 = 1; 0 X 7 = 7 (2/9). Most, of the errors consisted in numbers different from either operands (517 were multiple of the larger number; e.g., 4 X 1 = 8). Remediation Procedure Multiplications (problems from 2 X 2 to 9 X 9) were divided in two sets of 32 items matched for difficulty - according to the classification of Campbell and Graham, 1985 (see Appendix B). Set A was repeated twice a week in the first month of training, Set B in the second month of training. Problems were presented in written form and read aloud by the experimenter; TL answered, as she preferred, pointing the number from a table - frequently speaking the number aloud on the same time - or writing the arabic digits. Errors were always corrected immediately and TL repeated a few times the whole problem aloud. Soon after the beginning of the training TL tried to get the solution by reciting one operand's table (e.g., 7 X 3 = 7, 14,21). She produced a large number of errors in reciting the tables which were again corrected. All multiplications were tested at the starting point (Tl), after training on set A (T2), and a second time, after the training on set B (T3). One further assessment was carried out one month after the remediation period (T4). Quantitative Results The overall error rate in Tl was 90.6% (Set A and B were equally compromised). A significant improvement was observed for both sets in the respective training periods. The error rate decreased from 87.5% (Tl) to 12.5% (T2) for set A (McNemar's test dJ. 11 = 22.04; p<.005) and from 37.5% (T2) to 3% (T3) for set B (McNemar's test d.f. 1 =9.09; p<005). In T2 (after the first training period) both trained and untrained problems were answered more
Reacquisition of arithmetical facts
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TABLE II
TL's Error Rate at Tl, T2, T3 and T4 Tl
Set A Set B
87,5% 93.7%
T2 --->
12.5% 37.5%
--->
T3
T4
12.5% 3.1%
9.4% 9.4%
The arrow indicates the period of training for the single set.
accurately. The performance in the trained set, however, improved more than the performance in the untrained one (see Table II). In order to clarify these results we separately analysed the improvement in difficult and easy items in each set. Interestingly, the improvement was mostly confined to easy items: difficult items, instead, needed specific training (see Figure 3). In contrast with what was observed in the first training session, training on set B did not affect performance on set A. For set A, error rates in T2, T3 and T4 were substantially the same. Indeed, errors occurred only on difficult problems which were less likely to benefit from general training. Error rate in problem complements (e.g., 3 X 4 vs 4 X 3) correlated significantly (Spearman rank correlation r = .871; p<.005). The problem size effect resulted significant only for problems in set B (Spearman correlation rank, Set B, r=.731, p<0.005; set A, r=.281, p=0.245).
Qualitative Results The qualitative analysis of the errors shows a remarkable modification across the testing sessions (Figure 4). The unrelated non-table errors disappeared totally while more plausible errors emerged. An additional analysis of the errors recorded across the single training sessions shows clearly the gradual change in error pattern. In the first half of the training sessions between Tl and T2, the most frequent errors were nontable (35.7%) which decreased drastically as the training went on. Moreover, the close-miss errors produced in the first session (7.9%) consisted mostly (80%) in non-table numbers, while in the subsequent presentation they involved mainly table numbers. The percentage of operand errors increased over the course of training and they constituted the most frequent error produced all over the second half of the training (Set B).
Divisions Divisions were presented before and after the training. In Tl, TL did not solve a single division problem (often she did not even try to work them out). In T4 she was still very slow, but answered 80% of the problems. Mostly she inverted divisions to the complement multiplications: 20 divided by 5 is 4, because 5 times 4 is 20. Thus, she demonstrated good understanding of divisions and the direct application of re-learnt facts.
L. Cirelli and Others
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100 90 80 70 (/J
e
60
"-
"Q)
'+-
0
50
cf(
40 30 20 10 o+---------~ ---------~~--------~----------.
T3 Times table
T1
- x - setA easy
~
setA diff.
~
T4 set 8 easy
____ set 8 diff.
Fig. 3 - TL's error rate in different problems sets from Tl to T4.
Rule-based Items
In T4 100% of the O's problems and 61 % of the 1's problems were solved correctly. Errors always resulted from the application of a wrong rule, such as Nx1=1. Discussion
The remediation was very efficient and showed long lasting effects. TL's performance improved clearly in the training periods, where the effect of training was specific for difficult problems and unspecific for easy problems. We explain
Reacquisition of arithmetical facts
57
this result with the back-up strategy adopted by TL, that is reciting the tables from N X 1 forward until she could get the solution. As a result of this procedure she got a much more intensive training with problems consisting of small operands (e.g., 3 X 4) than with problems consisting of big operands (e.g. 7 X 8). This is probably the reason why the problem size effect emerged only in the second part of the training. In the course of the training the error types changed drastically from unsystematic errors to errors clearly related to the correct answer. The range of
100 90 80
Q)
70
1\5 '-
e '-
'Q)
60
co
'Q)
>
0
-
50
Q)
..c +-' 0
+-'
:::l
40
0
?f2-
30 20 10 0
T1
T2 •
operand
D table
T3 •
close miss
T4 •
non table
operation
Fig. 4 - TL' s distribution of errors in different testing sessions (Close-miss errors consisting in non table numbers: T1: 2/3; T2: 2/3; T3: 2/1; T4: 3/3).
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possible answers declined and included an increasing proportion of table numbers (table errors close to the correct solution), particularly multiples of the problem's operands (operand errors). A similar developmental trend has been reported by Campbell and Graham (1985) and Graham (1987) in the acquisition of multiplication facts. They describe the change of error types from Grade 2 through Grade 5 as follows: "As skill progress, the set of candidate responses becomes increasingly constrained to the answers familiar as correct responses to some multiplication problems ... The pattern of specific errors made increasingly approximates the adult pattern" (Campbell and Graham, 1985, page 351). The qualitative analysis of errors is a powerful source of information about the underlying cognitive representations. As assumed in arithmetical facts models (for example, Siegler and Shrager, 1984) errors might reflect associative links between operands (or entire problems) and candidate answers. Accordingly, decreasing error rates and changing error types reflect the reorganisation of a network, the associative links of which get more and more specific. This suggestion is supported by the finding that the typical effects found in the control group (namely (a) problem size effect, (b) frequent operand errors, (c) operand distance effect and (d) close miss errors), appeared only (a, b) or mostly (c, d) during the second training period, when the associative network reached a certain stage of specificity. The assumption that the network is shaped by frequently used associations is supported by another finding: during the training 87% of table and operand errors in list A were correct results of other problems in list A, while only 30% of the errors in list B were answers of list B (and 70% of list A). This error distribution suggests that earlier established associations may function as a source of proactive interference (Campbell and Graham, 1985; Campbell, 1987). Finally, a considerable improvement in divisions was noted where TL efficiently used the relearned multiplication facts (this finding will be discussed later). Patient ZA
The first assessment showed a clear dissociation between preserved addition and subtraction skills and severely compromised multiplications. Out of 64 onedigit problems (from 2 X 2 to 9 X 9) only 12 were answered correctly. Errors consisted in omissions (52%), operand errors (25%), table errors (7%), nontable errors (6%) and close miss errors (10%). In contrast, nearly all rule-based items were solved correctly (38/40). ZA answered only 5 out of 64 divisions. Rule-based items were performed without problems (18/18). Moreover, ZA was able to solve multi-digit additions and subtractions; the poor performance in written multiplications was entirely due to failure in retrieving arithmetical facts. In conclusion, we found a selective loss of multiplication tables with intact knowledge of rules and procedures.
Reacquisition of arithmetical fa cts
59
Remediation Procedure One of the major points to be tested was the selective acquisition of facts. We therefore tried to reduce the possible interference or overlapping of learning effects between the two sets of problems. To prevent these effects induced, for example, by backup strategies as TL's procedure, we chose the table of 3 and 5 and their complement problems as a first set to be trained (set A: 28 onedigit multiplications including 3 or 5 as operands). These tables do not implicate each other (apart from problem 3 X 5) and include a minimum of problems in other tables. Another reason why we chose the tables of 3 and 5 as starting point, was their minor difficulty with respect to other tables (as found in our control data). We therefore assumed them to be easier to learn. However, it is important to note that ZA's performance in Tl was equally compromised across the tables. The second set (set B) comprised the remaining 36 problems (where 3 or 5 never appeared as operands). All multiplications were tested in September (Tl), October (T2), November (T3) and December (T4). From T1 to T2 set A was trained, from T2 to T3 set B. From T3 to T4 no specific training was performed. Divisions were tested in Tl and T4. All material was presented in written form. The answer was written down or pointed from a table. All items of the training sets were repeated at least 20 times. ZA did not learn the answers as "labels" but worked them out by repeated additions (and sometimes subtractions). He was very slow in the beginning, but automatised multiplications in the course of the training. During the training periods errors were always immediately corrected, and all corrections which seemed not convincing to ZA were discussed and worked out together. The preferred strategy to reach the correct result was the repeated addition of the second operand (e.g. 4 X 5 = 5 + 5 + 5 + 5 = 20). The way how ZA relearned multiplications possibly influenced the emerging error pattern, as will be later seen. Quantitative Results The overall error rate for m X n multiplications in Tl was 81.2%, errors being equally distributed in set A and set B. The trained set A improved from T1 to T2 significantly (McNemar'S test d.f. 1 = 16.4; p< .005) and a lower, but significant improvement could be observed for set B (McNemar'S test dJ. 1 = 4.92; p<.05). However, in T2 the performance on set A was significantly more accurate than on set B (Chi Square dJ. 1 = 13.75; p< .OOI). Only set B improved from T2 to T3 (McNemar's test d.f. 1 = 8.64; p< .005). A follow up TABLEllI
Z4 's Error Rate at Tl , 12, T3 and T4
Set A Set B
TI
T2
T3
82% 81 %
11% 56%
22%
The arrow indicates the period of training for the single set.
14%
T4 7%
14%
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100 90 80 70 60 (/)
e "-
"Q)
0
50
?f2,
40 30 20 10 0
T1
T2 ~
T3
Times table setA ___ set B
T4
Fig. 5 - Z4. 's error rate in different problems sets from Tl to T4.
test in T4 demonstrated a slight improvement for both sets (see Table III). From T3 to T4 no specific training for multiplication tables has been performed, but some written calculation, transcoding and problem solving tasks were given. Although both sets improved slightly from T3 to T4, the error rate remained considerably higher than the controls' one (Figure 5). Across the four presentations of all problems we found a significant problem size effect (Spearman rank correlation r=0.692; p
Reacquisition of arithmetical fact s
61
Qualitative Analysis With the decreasing error rate the distribution of error types changed considerably. As expected, omissions and non-table errors (not being close to the correct result) disappeared. A high percentage of close miss errors emerged (see Figure 6). Operand errors resulted mostly from the substitution of the smaller operand.
100 90 80
--co
70
ill l...
e l... l...
60
(j)
a3 l...
(j)
> 0
50
--o --
40
(j)
..c Y-
::::J
0
cf
30 20 10 0
T2
T1
operand
T3
o table
•
.. _
close miss
T4 •
non table
II omission
-- - - _.
Fig. 6 - ZA's distribution (!l errors ill dif/£'rent testing sessions (Close-miss errors consistillg in non table numbers: Tl: 5/5; T2: 10/11; T3: 10/11; T4: 5/6).
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Divisions While in Tl only 13/64 divisions were answered correctly, in T4 62/64 problems were performed successfully. Most of them were answered by inverting the complement multiplication. Seven divisions were answered by repeated addition (adding n times the divisor till the result matches the dividend).
Discussion What we clearly found in this remediation experiment was a specific improvement of performance. The training effect concerned the single items and did not improve the general performance to the same extent. A second result was the change of error pattern from omissions and non table errors to close miss errors and operand errors. This qualitative change in the type of errors is thought to reflect the functioning of a recovering semantic network, as argued also for TL. The striking difference between ZA' s errors and the errors observed in the control group is the predominance of close miss errors which mainly were non-table numbers (e.g., 5x6=31). ZA's preferred back-up strategy was the repeated addition of the second factor. Interestingly, he seldom failed the number of operands to be added, but made calculation errors in the addition itself (additional tests, in fact, showed problems in successive additions). With this interpretation we do not suggest that all these errors (close miss and operand errors) occurred due to actual strategy failures. On the contrary, overt strategy use was not allowed in the control lists. We have two possible explanations: First, ZA used internal, fast strategies that occasionally failed. Second, the errors were erroneously retrieved numbers, which were connected to the problem in the semantic network of multiplication facts. The constancy of certain errors (e.g., 9X 9 was answered with 82 in T1, T2 and T3) argues for wrong associations between problems and candidate answers. With the assumption that repeated answers, correct and false ones, shape the network connections, backup strategies and their results become a crucial factor in the building up of the semantic representation (Siegler, 1988, makes the same argument for children). Both explanations, first, the use of internal fast strategies and second, failed fact retrieval, are possible and do not exclude each other. In fact, procedural and automatic retrieval might take place in parallel, the faster process providing the answer (as argued by Ashcraft, 1992, in his "horse race model"). An additional point should be mentioned concerning the occurrence of nontable errors. Such errors, as 5 X 6 = 31, are very unusual for normal subjects. When presented verbally, the problems "5 X 6 = 30" and "5 X 6 = 31" sounded "equally good" to ZA (the same was valid for the other errors). He had totally lost the capacity to check the correctness of a calculation by the verbal rhyme. As children in different cultures learn the tables in verbal sequences (e.g., Kashiwagi A. , Kashiwagi T. and Hasegawa, 1991), we assume that the verbal form provides a considerable help in generating or checking an answer. The lack of auditory feedback certainly favoured the occurrence of non-table errors, which would be rejected by healthy people.
Reacquisition of arithmetical faers
63
Summing up, in this experiment we found good recovery of arithmetical facts and selective training effects for single sets. Furthermore, we found support for the claim that repeated back-up strategies can shape one's semantic network.
CONCLUSIONS
We investigated the possibilities and limits of relearning arithmetical facts in two severely acalculic patients. Both patients improved considerably during the training periods and showed a good performance in the follow-up examination. Decreasing elTor rates and a systematic change of elTor types were argued to indicate the growing or reorganisation of a semantic network with associations between operands (or problems) and answers (not only correct ones, but also false ones). We argued that backup strategies play an important role in this process. Training effects mainly concerned the respective training set. This finding supports the idea of independent fact representations for the single problems (since complement problems were always trained contemporaneously, we cannot draw any conclusion on their representation). However, our patients showed also a general, not specific improvement which may be due to an activation of preserved notions, but may also arise from higher motivation and self-confidence. After the training both patients tried to work out problems they could not retrieve from memory (as divisions), answered correctly rule-based items and also solved easy text problems. The remediation experiments demonstrate that even in severely acalculic patients a re-organisation of fact knowledge can be induced by specific training. Since arithmetical facts constitute knowledge not only required in explicit calculation problems, but continuously used in every day life, such a training can considerably improve the patients' quality of life. Acknowledgements. The research reported was supported by Grant SCI*-CT91-0730 from the European Community. We would like to thank Brian Butterworth for helpful comments on the first draft of this paper.
REFERENCES ASHCRAFT, M.H. Cognitive arithmetic: a review of data and theory. Cognition, 44: 75-106, 1992. ASHCRAFT, M.H. Children's knowledge of simple arithmetic: developmental model and simulation. In 1. Bisanz, CJ. Brainerd and R. Kail (Eds), Formal Methods in Developmental Psychology: Progress in Cognitive Developmental Research. New York: Springer-Verlag, 1987, Ch. 9, pp. 302-338. ASHCRAFT, M.H., and BATTAGLIA, J. Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition. journal of Experimental Psychology: Human Learning and Memo ry, 4: 527538, 1978. CAMPBELL, J.l.0. Network interference in mental multiplication . journal of Experimental Psychology: Learning, Memory and Cognition, 13: 109-113, 1987. CAMPBELL, 1.1.0. , and GR AHAM, OJ. Mental multiplication skill: structure, process and acquisition. Canadian journal of Psychology, 39: 338-366, 1985. COHEN, L., and DEHAENE, S. Amnesia for arithmetical facts: A sin gle case study. Brain and Language, 47: 214-232, 1994. COLTHEART, M. Aphasia therapy research: a single case study approac h. In C. Code and OJ. Mull er (Eds.), Aphasia Therapy. London: Edward Arnold, 1983, Ch. 18, pp.193-202. GRAHAM, O.F. An associative retrieval model: How children learn to multiply. In 1. Sloboda and D.
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Rogers (Eds.), Cognitive Processes in Mathematics. Oxford, England: Oxford University Press, 1987, Ch. 7, pp. 123-141. GROEN, G.J. , and PARKMAN, 1.M . A ch ronometric analysis of simple addition. Psychological Review, 93: 411-428, 1972. HITTMAIR-DELAZER, M., SEMENZA, c., and DENES, G. Concepts and facts in calculation. Brain , 117: 715-728, 1994. KASHIWAGI, A., KASHIWAGI, T. , and HASEGAWA, T. Improvement of deficits in mnemonic rhyme for multiplication in 1apanese aphasics. Neuropsychologia , 25: 443-447, 1987. L UZZATTI, C., WILLMES, K., and DE BLESER, R. Aachener Aphasia Test; Versione 1taliana. Verlag fur Psychologie, Dr. C.J. Gottingen: Hogrefe, 1991. M CCLOSKEY, M. Cognitive mechanisms in numerical processing: E vidence from acquired dyscalculia. Cognition, 44: 107-157,1992. MCCLOSKEY, M., ALIMINOSA, D., and MACARUSO, P. Theory-based assessment of acquired dyscalculia Brain and Cognition, 17: 285-308, 1991. MCCLOS KEY, M. , ALIMI NOSA, 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 , 199 1. MCCLOS KEY, M., HARLEY, W ., and SOKOL, S.M. Models of arithmetical fact retrieval: An evaluation in light of findings from normal and brain-damaged subjects. Journal of Experimental Psychology: Learning, Memory and Cognition, 17: 377-397, 1991. MCCLOS KEY, M., CARAMAZZA, A. , and BASILI, A. Cognitive mechani sms in number processing and calculation: Evidence from dyscalculia. Brain and Cognition, 4: 117-196, 1985. MICELI, G. , and CAPASSO, R. I disturbi del calcolo. Diagnosi e riabilitazione. Milano: Masson, 1991. MILLER, K. , PERLMUTTER, M. , and KEATING, D. Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory and Cognition, 10: 46-60, 1984. SI EGLER, R Strategy choice procedures and development of multiplication skill. Journal of Experimental Psychology: General, 11 7: 258-275 , 1988. SIEGLER, R , and SHRAGER, 1. A modelof strategy choice. In C. Sophian (Ed.), Origin of Cognitive Skills. Hill sdale, N l : Erlbaum, 1984, Ch. 3, pp. 229-293. SOKOL, S.M., and MCCLOSKEY, M . Cognitive mechanism s in calculation. In R Sternberger and P.A. Frensch (Eds.), Complex Problem Solving: Principles and Mechanisms. Hill sdale, N1: Erlbaum . 1991, Ch. 3, pp. 85-116. SOKOL, S., MCCLOSKEY, M. , COHEN, N.J., and AUMINOSA, 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, 199 1. SPINNLER, H., and TOGNONI, G. Standardizzazione e taratura ita1iana di test neuropsicologici. The Italian Journal of Neurological Sciences, Supplemento al n. 6, 1987. WARRINGTON, E.K. The fractionation of arithmetical skills: A single case study. Quarterly Journal of Experimental Psychology, 34a: 3 1-51, 1982. Luisa Girelli. Department of Psychology, University Collcge London. Gower Street, London WC IE 6BT,
U.K.
Reacquisition of arithmetical facts
65
ApPEN DIX A
Aachener Aphasie Test
TL Token Repeati tion Wri tten Language Nam ing Comprehens ion Alloc. class.
ZA
Row score
(deficit)
Row score
(deficit)
35/50 1151150 65/90 1161120 801120 Broca aphasia
(slight) (medium-slight) (med ium) (slight) (medium-slight)
41150 11 211 50 52/90 631120 971120 Not class.
(severe) (medium) (medium) (medium) (slight)
Number processing and calculation (Miceli and Capasso, 1991)
Non numerical tasks Magni tude comparison Dots pattern seriation
TL
ZA
10/ 10 5/5
10/10 5/ 5 1-2 d *
Magnitude comparison Visual-digits Vi sual n umberwords Verbal Transcodification tasks Repeti tion Reading numbers Read ing numberwords Writing from dictation Writing (words to digits) Recognition of arit. signs Visual Verbal Verbal ca lculation Addition Subtraction Multipli cation
* Percentage
1-2 d*
34/34 29/34 32/34
100% 100% 100%
33/34 33/34 32/34
100% 100% 100%
30/55 35/55 20/55 25/55 35/55
100% 100% 80% 100% 100%
30/55 18/55 17/55 23/55 31155
100% 68% 68% 92% 96%
8110 8110
10110 8/ 10
6120 6120 2120
20/20 20/20 5115
of correct a nswers with numbers up to 2-digits.
66
L. Girelli and Others ApPENDIX
B
Problems Included in Set A and Set B for the Patients' Training
TL
ZA
SET A
SET B
SET A
SET B
2x2 2x3 2x6 2x7 3x2 3x3 3x5 3X6 3x9 4x5 4x7 4x8 5X3 5X4 5X8 5x9 6x2 6x3 6X9 7X2 7X4 7X7 7X8 8X4 8x5 8x7 8x8 8X9 9X3 9X5 9x6 9x8
2x4 2X5 2x8 2x9 3x4 3x7 3x8 4x2 4x3 4x4 4x6 4x9 5X2 5X5 5x6 5x7 6X4 6X5 6x6 6x7 6x8 7x3 7x5 7x6 7x9 8X2 8X3 8x6 9x2 9x4 9x7 9X9
3x2 3x3 3x4 3x5 3x6 3x7 3x8 3x9 5X2 5x3 5X4 5x5 5x6 5X7 5x8 5x9 2x3 4X3 6x3 7x3 8X3 9X3 2x5 4x5 6x5 7x5 8x5 9x5
2x4 2X6 2X7 2X8 2X9 4X2 4X4 4x6 4X7 4X8 4x9 6X2 6X4 6X5 6X6 6X7 6X8 6X9 7x2 7x4 7x6 7X7 7X8 8X2 8X4 8x6 8X7 8X8 9x2 9X4 9X6 9X7 9X8 9X9
ABSTRACT
We report the improvement following rehabilitation of two patients with a selective deficit on multiplication facts. The remediation experiments were conducted in specific designs to test a series of hypotheses about both the process of reacquisition and organisation of arithmetical facts in memory. The results show a significant and stable recovery and the pattern of reacquisition reflects the different strategy used by the patients to relearn the multiplication facts. Quantitative and qualitative changes in error patterns in the course of remediation reveal the underlying (re)organization of the memory network representing multiplication facts.
INTRODUCTION
Arithmetical fact knowledge consists in calculation problems which are retrieved from memory and do not require any calculation procedures. It constitutes a part of the calculation system (McCloskey, Caramazza and Basili, 1985; McCloskey, 1992, for a review) and it is essential for efficient calculation (Hittmair-Delazer, Semenza and Denes, 1994). We repeatedly use arithmetical fact knowledge in everyday life, often without being conscious of doing calculations. We double the number of plates when unexpected guests arrive, we calculate the delay with which we will arrive at the lecture, we quickly check the change when we pay a bill etc. Some neuropsychological reports show that arithmetical facts can be lost selectively after brain damage (Warrington, 1982; McCloskey, Harley and Sokol, 1991; Sokol, McCloskey, Cohen and Aliminosa, 1991; McCloskey, 1992; Cohen and Dehaene, 1994; Hittmair-Delazer et aI., 1994). Patients who are not able anymore to do the basic, memory-based calculations are severely handicapped. Despite this, few studies have been concerned with the remediation of arithmetical facts (Miceli and Capasso, 1991; Sokol and McCloskey, 1991; McCloskey, Aliminosa and Sokol, 1991; HittmairDelazer et aI., 1994). McCloskey et aI. (1991) reported a selective training effect for repeated multiplication problems which extended to the problems' complements, but did not improve performance in general (e.g., the training of 4 X 5 would improve 5 X 4, but not 4 X 6). Also the present study will focus on * Some of this material was presented at a special symposium at the International Neuropsychological Society in Madeira, June, 1993. Cortex, (1996) 32, 49-66
50
L. Girelli and Others
multiplication, the operation which is thought to depend most on retrieval from semantic memory. Indeed, multiplication is an ideal field to study the internal structure of a part of semantic memory: multiplications are exactly defined, they are limited in number, they are taught systematically, they are used in everyday day life and they represent a common knowledge of virtually all educated people. Various models of arithmetical fact representation have been proposed on the basis of systematic effects observed in children and adults (for example the problem size effect; Groen and Parkman, 1972; Ashcraft and Battaglia, 1978; Campbell and Graham, 1985). Although the models (Siegler and Shrager, 1984; Siegler, 1988; Campbell and Graham, 1985; Campbell, 1987; Ashcraft, 1987) differ in their internal structure, they share some general assumptions: all models assume a network of associations between operands (and/or entire problems) and stored answers, both, correct and false ones. The strength of the single links between operands (or problems) and answers would depend on the frequency of former activation. All models assume a spreading activation as underlying process of retrieval (for a discussion of the models with respect to the performance of acalculic patients, see McCloskey, Harley and Sokol, 1991). We present two acalculic patients who totally lost multiplication facts. The initial examination and the following remediation experiments, conducted in specific designs, offer the possibility to test a series of hypotheses, not only about the representation of arithmetical facts in semantic memory, but also about the process of building up those mental representations. We assume that in the course of remediation, at least partially successful, patients build up a new semantic representation of lost facts or, at least, reorganise preserved notions. This process should be reflected by an increasing accuracy in solving multiplication problems. Moreover, we expect an approximation of the patients' qualitative error patterns to the typical distribution observed in normal subjects. We will describe how patients learn multiplication tables and which consequences different back-up strategies may have. Finally we will evaluate whether the effect of training is specific for taught items (McCloskey et al. 1991; Hittmair-Delazer et aI., 1994) or whether a selective training on a subset of facts has a not item-specific activation effect (resulting in a general improvement). We will also assess divisions before and after the training periods, in order to examine the relation between the two operations. Studies on arithmetical facts have rarely focused on simple divisions and little evidence is available for an autonomous representation of division facts in semantic memory. If this is the case, the loss of division facts could be overcome by using backup strategies based on multiplications (e.g., 56: 7 = -+ 7 X n = 56). As pointed out by Hittmair et al. (1994) a further alternative has to be considered: division facts could routinely be solved via multiplication assuming they are not independently represented in semantic memory. Although multiple interpretations are available we still consider interesting whether the training on multiplications will affect performance on divisions. Before we start with the description of the remediation experiments we shortly summarise the data collected from a group of healthy Italian subjects. It is possible that retrieval of arithmetical facts could be related to the process of acquisition, hence it could differ according to teaching method (e.g., according
Reacquisition of arithmetical facts
51
12 11 10
8
4
2
o
2x
3x
4x
5x 6x Times table
7x
8x
9x
Fig. 1 - Normal subjects error rate as a function of the time-tables.
to the Italian teaching system the table of 5 consists of 5 X 2, 5 X 3, etc., according to the English one the same table consists of 2 X 5, 3 X 5, etc.). Normal Subjects The aim of this part of the study was to assess error rate and error pattern of normal Italian subjects in simple multiplications. All subjects (n = 100; mean age = 22 years; formal education, mean = 12 years) answered hundred written multiplications (operands from 0 to 9) in random order. The task was performed under time pressure. M X N facts (from 2 X 2 to 9 X 9; n = 6400) and rule-based items (including 0 or 1; n = 3600) were separately analysed. The overall error rate for m X n problems was 5.56%. The error rate differed considerably among the single multiplication tables. The times tables showed an increasing difficulty, with the exception of the table of 5, which is easier than would be expected by number size (see Figure 1). We found a strong correlation between the magnitude of a problem's result (e.g., 20 for 4 X 5) and the number of errors observed (Problem size effect: Spearman rank correlation r=0.769; p
52
L. Girelli and Others
100 90 80 70 (260 "~ 50 0 cf 40 30 20 10 0
66.4
(/)
operand
close-miss
table Error type
non table
operation
Fig. 2 - Normal subjects distribution of errors.
The errors were classified into the following types (as suggested by Campbell and Graham, 1985; Siegler, 1988; McCloskey et aI., 1991): an operand error is the product of an operand with another number different from the second factor (e.g., 5 X 4 = 24). A close miss error is a number near to the correct result (plus or minus ten percent of the correct result, such as 6 X 7 = 38). A table error belongs to a table different from either operands' (e.g., 3 X 4 = 25). A non-table error is a number not included in the times-tables (e.g., 3 X 4 = 17). Finally, in an operation error the answer would be correct for a different arithmetic operation between the same numbers (e.g., 5 X 2 = 7). Errors were assigned to one error class only. Certain errors could be assigned to more than one class, i.e. close miss versus non-table/table errors. We decided to separately classify errors close to the correct result since they have been argued to be of specific interest (e.g., Siegler, 1988). The distribution of error types is shown in Figure 2. Two thirds of all errors are operand errors, 76.7% of which consisted in correct answers for problems that not only share an operand with the stimulus problems but are also close in magnitUde with respect to the other operand (e.g., 7 X 9 = 56, that is the correct response for 7 X 8). This so-called operand distance effect has been already reported both in normal subjects (Campbell and Graham, 1085; Miller, Perlmutter and Keating, 1984) and in patients (Sokol et aI., 1991; Cohen and Dehaene, 1994). In a separate analysis of the problems including 0 or 1 as operands (which are thought to be solved by a stored rule; Sokol et aI., 1991) a lower error rate is found (3.22%). Out of 116 errors, 113 can be attributed to the application of a wrong rule (e.g., 5 X 0 = 5 from N X 0 = N). Most of the errors (91 %) occurred in O's problems. Summing up, normal subjects have a mean error rate of 5.56% in solving
Reacquisition of arithmetical facts
53
TABLE I
Background Data for TL and ZA
TL ZA
Age
Years of education
22 46
8 13
Lesion (eva) Wide perisylvian Wide perisylvian
Onset 6 years 2 years
Neurological examination Right hemiparesis Right hemiparesis
Raven
Digit span
Corsi
43/48 48/48
3 4
5 5
mUltiplications from 2 X 2 to 9 X 9 and of 3.22% in solving rule-based problems. In m X n problems, errors consist typically in a false product of one of the operands which is close to the correct result. In problems including 0 or 1, errors appear due to the application of a wrong rule. We found some of the effects already observed in normal controls (e.g., problem size-effect, operand distance effect). In comparison to McCloskey et al. (1991) we found a higher error rate in m X n problems and a relatively low error rate in rule-based problems (although McCloskey's data referred to only 15 subjects presented with very few problems).
CASE DESCRIPTION
Two subjects, TL (female, 22 years) and ZA (male, 46 years), participated in the study. Both, TL and ZA, had good arithmetical knowledge before their disease and were highly motivated to recover arithmetical skills. Both had been acalculic and aphasic for 6 and 2 years at the start of the study and a spontaneous remission had to be excluded (Table I). Both patients performed the AAT (Versione italiana: Luzzatti, Willmes and DeBleser, 1991) and a complete assessment of number processing and calculation (Miceli and Capasso, 1991). The results are shown in Appendix A. In the Raven matrices both patients performed in the average range (Spinnler and Tognoni, 1987). Their calculation deficit consists of a selective loss of arithmetical fact knowledge (procedures being intact) which was not secondary to other neuropsychological deficits. Transcoding processes were sufficiently preserved to produce reliable results in the calculation training: both were able to understand written and spoken numbers up to two digits and to produce an answer by writing the result or by pointing the number from a table. Although they had speech therapy since their trauma they never had any rehabilitation treatment for arithmetical problems.
DESIGN OF THE REMEDIATION EXPE RIMENTS
Using a cross-over design (as proposed by Coltheart, 1983; see also McCloskey et aI., 1991), we divided all multiplications in two sets, set A and Set B, which were presented in separate training periods (sets differed for the two patients). Performance was assessed before training, after the first and the second training period and in a follow-up examination. In these assessments, where overt strategies were not allowed, all multiplications from 2 X 2 through 9 X 9 were presented and answered in written form (by writing digits or pointing from a table).
54
L. Girelli and Others
Patient TL First Assessment In a first examination (Tl) only 6 multiplications out of 64 (problems from 2 X 2 to 9 X 9) were answered correctly. Interestingly, 41% of the erroneous responses consisted in far-miss errors, defined as number over 100% of the product, e.g., 2 X 9 = 44; most of them were non-table numbers (50%). This type of errors had never been observed both in normal or acalculic patients. Following the previous classification we found 50% non-table errors (41 % of which were far-miss numbers), 22.4% table errors, 19% operand errors, 5.2% close miss errors and finally 3.4% operation errors. Consistent with the extremely low accuracy in production, TL scored at chance level in a verification task (e.g., 2 X 3 = 9?; 19/36 correct) and in a multiple choice task (e.g., 3 X 4 = 8, 12, 24?; 8/24 correct). In rule-based items TL failed to answer 50% (9/18) of the problems. Moreover, errors were only partially grounded in the application of the wrong rule (e.g., 1 X 6 = 1; 0 X 7 = 7 (2/9). Most, of the errors consisted in numbers different from either operands (517 were multiple of the larger number; e.g., 4 X 1 = 8). Remediation Procedure Multiplications (problems from 2 X 2 to 9 X 9) were divided in two sets of 32 items matched for difficulty - according to the classification of Campbell and Graham, 1985 (see Appendix B). Set A was repeated twice a week in the first month of training, Set B in the second month of training. Problems were presented in written form and read aloud by the experimenter; TL answered, as she preferred, pointing the number from a table - frequently speaking the number aloud on the same time - or writing the arabic digits. Errors were always corrected immediately and TL repeated a few times the whole problem aloud. Soon after the beginning of the training TL tried to get the solution by reciting one operand's table (e.g., 7 X 3 = 7, 14,21). She produced a large number of errors in reciting the tables which were again corrected. All multiplications were tested at the starting point (Tl), after training on set A (T2), and a second time, after the training on set B (T3). One further assessment was carried out one month after the remediation period (T4). Quantitative Results The overall error rate in Tl was 90.6% (Set A and B were equally compromised). A significant improvement was observed for both sets in the respective training periods. The error rate decreased from 87.5% (Tl) to 12.5% (T2) for set A (McNemar's test dJ. 11 = 22.04; p<.005) and from 37.5% (T2) to 3% (T3) for set B (McNemar's test d.f. 1 =9.09; p<005). In T2 (after the first training period) both trained and untrained problems were answered more
Reacquisition of arithmetical facts
55
TABLE II
TL's Error Rate at Tl, T2, T3 and T4 Tl
Set A Set B
87,5% 93.7%
T2 --->
12.5% 37.5%
--->
T3
T4
12.5% 3.1%
9.4% 9.4%
The arrow indicates the period of training for the single set.
accurately. The performance in the trained set, however, improved more than the performance in the untrained one (see Table II). In order to clarify these results we separately analysed the improvement in difficult and easy items in each set. Interestingly, the improvement was mostly confined to easy items: difficult items, instead, needed specific training (see Figure 3). In contrast with what was observed in the first training session, training on set B did not affect performance on set A. For set A, error rates in T2, T3 and T4 were substantially the same. Indeed, errors occurred only on difficult problems which were less likely to benefit from general training. Error rate in problem complements (e.g., 3 X 4 vs 4 X 3) correlated significantly (Spearman rank correlation r = .871; p<.005). The problem size effect resulted significant only for problems in set B (Spearman correlation rank, Set B, r=.731, p<0.005; set A, r=.281, p=0.245).
Qualitative Results The qualitative analysis of the errors shows a remarkable modification across the testing sessions (Figure 4). The unrelated non-table errors disappeared totally while more plausible errors emerged. An additional analysis of the errors recorded across the single training sessions shows clearly the gradual change in error pattern. In the first half of the training sessions between Tl and T2, the most frequent errors were nontable (35.7%) which decreased drastically as the training went on. Moreover, the close-miss errors produced in the first session (7.9%) consisted mostly (80%) in non-table numbers, while in the subsequent presentation they involved mainly table numbers. The percentage of operand errors increased over the course of training and they constituted the most frequent error produced all over the second half of the training (Set B).
Divisions Divisions were presented before and after the training. In Tl, TL did not solve a single division problem (often she did not even try to work them out). In T4 she was still very slow, but answered 80% of the problems. Mostly she inverted divisions to the complement multiplications: 20 divided by 5 is 4, because 5 times 4 is 20. Thus, she demonstrated good understanding of divisions and the direct application of re-learnt facts.
L. Cirelli and Others
56
100 90 80 70 (/J
e
60
"-
"Q)
'+-
0
50
cf(
40 30 20 10 o+---------~ ---------~~--------~----------.
T3 Times table
T1
- x - setA easy
~
setA diff.
~
T4 set 8 easy
____ set 8 diff.
Fig. 3 - TL's error rate in different problems sets from Tl to T4.
Rule-based Items
In T4 100% of the O's problems and 61 % of the 1's problems were solved correctly. Errors always resulted from the application of a wrong rule, such as Nx1=1. Discussion
The remediation was very efficient and showed long lasting effects. TL's performance improved clearly in the training periods, where the effect of training was specific for difficult problems and unspecific for easy problems. We explain
Reacquisition of arithmetical facts
57
this result with the back-up strategy adopted by TL, that is reciting the tables from N X 1 forward until she could get the solution. As a result of this procedure she got a much more intensive training with problems consisting of small operands (e.g., 3 X 4) than with problems consisting of big operands (e.g. 7 X 8). This is probably the reason why the problem size effect emerged only in the second part of the training. In the course of the training the error types changed drastically from unsystematic errors to errors clearly related to the correct answer. The range of
100 90 80
Q)
70
1\5 '-
e '-
'Q)
60
co
'Q)
>
0
-
50
Q)
..c +-' 0
+-'
:::l
40
0
?f2-
30 20 10 0
T1
T2 •
operand
D table
T3 •
close miss
T4 •
non table
operation
Fig. 4 - TL' s distribution of errors in different testing sessions (Close-miss errors consisting in non table numbers: T1: 2/3; T2: 2/3; T3: 2/1; T4: 3/3).
58
L. Girelli and Others
possible answers declined and included an increasing proportion of table numbers (table errors close to the correct solution), particularly multiples of the problem's operands (operand errors). A similar developmental trend has been reported by Campbell and Graham (1985) and Graham (1987) in the acquisition of multiplication facts. They describe the change of error types from Grade 2 through Grade 5 as follows: "As skill progress, the set of candidate responses becomes increasingly constrained to the answers familiar as correct responses to some multiplication problems ... The pattern of specific errors made increasingly approximates the adult pattern" (Campbell and Graham, 1985, page 351). The qualitative analysis of errors is a powerful source of information about the underlying cognitive representations. As assumed in arithmetical facts models (for example, Siegler and Shrager, 1984) errors might reflect associative links between operands (or entire problems) and candidate answers. Accordingly, decreasing error rates and changing error types reflect the reorganisation of a network, the associative links of which get more and more specific. This suggestion is supported by the finding that the typical effects found in the control group (namely (a) problem size effect, (b) frequent operand errors, (c) operand distance effect and (d) close miss errors), appeared only (a, b) or mostly (c, d) during the second training period, when the associative network reached a certain stage of specificity. The assumption that the network is shaped by frequently used associations is supported by another finding: during the training 87% of table and operand errors in list A were correct results of other problems in list A, while only 30% of the errors in list B were answers of list B (and 70% of list A). This error distribution suggests that earlier established associations may function as a source of proactive interference (Campbell and Graham, 1985; Campbell, 1987). Finally, a considerable improvement in divisions was noted where TL efficiently used the relearned multiplication facts (this finding will be discussed later). Patient ZA
The first assessment showed a clear dissociation between preserved addition and subtraction skills and severely compromised multiplications. Out of 64 onedigit problems (from 2 X 2 to 9 X 9) only 12 were answered correctly. Errors consisted in omissions (52%), operand errors (25%), table errors (7%), nontable errors (6%) and close miss errors (10%). In contrast, nearly all rule-based items were solved correctly (38/40). ZA answered only 5 out of 64 divisions. Rule-based items were performed without problems (18/18). Moreover, ZA was able to solve multi-digit additions and subtractions; the poor performance in written multiplications was entirely due to failure in retrieving arithmetical facts. In conclusion, we found a selective loss of multiplication tables with intact knowledge of rules and procedures.
Reacquisition of arithmetical fa cts
59
Remediation Procedure One of the major points to be tested was the selective acquisition of facts. We therefore tried to reduce the possible interference or overlapping of learning effects between the two sets of problems. To prevent these effects induced, for example, by backup strategies as TL's procedure, we chose the table of 3 and 5 and their complement problems as a first set to be trained (set A: 28 onedigit multiplications including 3 or 5 as operands). These tables do not implicate each other (apart from problem 3 X 5) and include a minimum of problems in other tables. Another reason why we chose the tables of 3 and 5 as starting point, was their minor difficulty with respect to other tables (as found in our control data). We therefore assumed them to be easier to learn. However, it is important to note that ZA's performance in Tl was equally compromised across the tables. The second set (set B) comprised the remaining 36 problems (where 3 or 5 never appeared as operands). All multiplications were tested in September (Tl), October (T2), November (T3) and December (T4). From T1 to T2 set A was trained, from T2 to T3 set B. From T3 to T4 no specific training was performed. Divisions were tested in Tl and T4. All material was presented in written form. The answer was written down or pointed from a table. All items of the training sets were repeated at least 20 times. ZA did not learn the answers as "labels" but worked them out by repeated additions (and sometimes subtractions). He was very slow in the beginning, but automatised multiplications in the course of the training. During the training periods errors were always immediately corrected, and all corrections which seemed not convincing to ZA were discussed and worked out together. The preferred strategy to reach the correct result was the repeated addition of the second operand (e.g. 4 X 5 = 5 + 5 + 5 + 5 = 20). The way how ZA relearned multiplications possibly influenced the emerging error pattern, as will be later seen. Quantitative Results The overall error rate for m X n multiplications in Tl was 81.2%, errors being equally distributed in set A and set B. The trained set A improved from T1 to T2 significantly (McNemar'S test d.f. 1 = 16.4; p< .005) and a lower, but significant improvement could be observed for set B (McNemar'S test dJ. 1 = 4.92; p<.05). However, in T2 the performance on set A was significantly more accurate than on set B (Chi Square dJ. 1 = 13.75; p< .OOI). Only set B improved from T2 to T3 (McNemar's test d.f. 1 = 8.64; p< .005). A follow up TABLEllI
Z4 's Error Rate at Tl , 12, T3 and T4
Set A Set B
TI
T2
T3
82% 81 %
11% 56%
22%
The arrow indicates the period of training for the single set.
14%
T4 7%
14%
60
L. Girelli and Others
100 90 80 70 60 (/)
e "-
"Q)
0
50
?f2,
40 30 20 10 0
T1
T2 ~
T3
Times table setA ___ set B
T4
Fig. 5 - Z4. 's error rate in different problems sets from Tl to T4.
test in T4 demonstrated a slight improvement for both sets (see Table III). From T3 to T4 no specific training for multiplication tables has been performed, but some written calculation, transcoding and problem solving tasks were given. Although both sets improved slightly from T3 to T4, the error rate remained considerably higher than the controls' one (Figure 5). Across the four presentations of all problems we found a significant problem size effect (Spearman rank correlation r=0.692; p
Reacquisition of arithmetical fact s
61
Qualitative Analysis With the decreasing error rate the distribution of error types changed considerably. As expected, omissions and non-table errors (not being close to the correct result) disappeared. A high percentage of close miss errors emerged (see Figure 6). Operand errors resulted mostly from the substitution of the smaller operand.
100 90 80
--co
70
ill l...
e l... l...
60
(j)
a3 l...
(j)
> 0
50
--o --
40
(j)
..c Y-
::::J
0
cf
30 20 10 0
T2
T1
operand
T3
o table
•
.. _
close miss
T4 •
non table
II omission
-- - - _.
Fig. 6 - ZA's distribution (!l errors ill dif/£'rent testing sessions (Close-miss errors consistillg in non table numbers: Tl: 5/5; T2: 10/11; T3: 10/11; T4: 5/6).
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L. Girelli and Others
Divisions While in Tl only 13/64 divisions were answered correctly, in T4 62/64 problems were performed successfully. Most of them were answered by inverting the complement multiplication. Seven divisions were answered by repeated addition (adding n times the divisor till the result matches the dividend).
Discussion What we clearly found in this remediation experiment was a specific improvement of performance. The training effect concerned the single items and did not improve the general performance to the same extent. A second result was the change of error pattern from omissions and non table errors to close miss errors and operand errors. This qualitative change in the type of errors is thought to reflect the functioning of a recovering semantic network, as argued also for TL. The striking difference between ZA' s errors and the errors observed in the control group is the predominance of close miss errors which mainly were non-table numbers (e.g., 5x6=31). ZA's preferred back-up strategy was the repeated addition of the second factor. Interestingly, he seldom failed the number of operands to be added, but made calculation errors in the addition itself (additional tests, in fact, showed problems in successive additions). With this interpretation we do not suggest that all these errors (close miss and operand errors) occurred due to actual strategy failures. On the contrary, overt strategy use was not allowed in the control lists. We have two possible explanations: First, ZA used internal, fast strategies that occasionally failed. Second, the errors were erroneously retrieved numbers, which were connected to the problem in the semantic network of multiplication facts. The constancy of certain errors (e.g., 9X 9 was answered with 82 in T1, T2 and T3) argues for wrong associations between problems and candidate answers. With the assumption that repeated answers, correct and false ones, shape the network connections, backup strategies and their results become a crucial factor in the building up of the semantic representation (Siegler, 1988, makes the same argument for children). Both explanations, first, the use of internal fast strategies and second, failed fact retrieval, are possible and do not exclude each other. In fact, procedural and automatic retrieval might take place in parallel, the faster process providing the answer (as argued by Ashcraft, 1992, in his "horse race model"). An additional point should be mentioned concerning the occurrence of nontable errors. Such errors, as 5 X 6 = 31, are very unusual for normal subjects. When presented verbally, the problems "5 X 6 = 30" and "5 X 6 = 31" sounded "equally good" to ZA (the same was valid for the other errors). He had totally lost the capacity to check the correctness of a calculation by the verbal rhyme. As children in different cultures learn the tables in verbal sequences (e.g., Kashiwagi A. , Kashiwagi T. and Hasegawa, 1991), we assume that the verbal form provides a considerable help in generating or checking an answer. The lack of auditory feedback certainly favoured the occurrence of non-table errors, which would be rejected by healthy people.
Reacquisition of arithmetical faers
63
Summing up, in this experiment we found good recovery of arithmetical facts and selective training effects for single sets. Furthermore, we found support for the claim that repeated back-up strategies can shape one's semantic network.
CONCLUSIONS
We investigated the possibilities and limits of relearning arithmetical facts in two severely acalculic patients. Both patients improved considerably during the training periods and showed a good performance in the follow-up examination. Decreasing elTor rates and a systematic change of elTor types were argued to indicate the growing or reorganisation of a semantic network with associations between operands (or problems) and answers (not only correct ones, but also false ones). We argued that backup strategies play an important role in this process. Training effects mainly concerned the respective training set. This finding supports the idea of independent fact representations for the single problems (since complement problems were always trained contemporaneously, we cannot draw any conclusion on their representation). However, our patients showed also a general, not specific improvement which may be due to an activation of preserved notions, but may also arise from higher motivation and self-confidence. After the training both patients tried to work out problems they could not retrieve from memory (as divisions), answered correctly rule-based items and also solved easy text problems. The remediation experiments demonstrate that even in severely acalculic patients a re-organisation of fact knowledge can be induced by specific training. Since arithmetical facts constitute knowledge not only required in explicit calculation problems, but continuously used in every day life, such a training can considerably improve the patients' quality of life. Acknowledgements. The research reported was supported by Grant SCI*-CT91-0730 from the European Community. We would like to thank Brian Butterworth for helpful comments on the first draft of this paper.
REFERENCES ASHCRAFT, M.H. Cognitive arithmetic: a review of data and theory. Cognition, 44: 75-106, 1992. ASHCRAFT, M.H. Children's knowledge of simple arithmetic: developmental model and simulation. In 1. Bisanz, CJ. Brainerd and R. Kail (Eds), Formal Methods in Developmental Psychology: Progress in Cognitive Developmental Research. New York: Springer-Verlag, 1987, Ch. 9, pp. 302-338. ASHCRAFT, M.H., and BATTAGLIA, J. Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition. journal of Experimental Psychology: Human Learning and Memo ry, 4: 527538, 1978. CAMPBELL, J.l.0. Network interference in mental multiplication . journal of Experimental Psychology: Learning, Memory and Cognition, 13: 109-113, 1987. CAMPBELL, 1.1.0. , and GR AHAM, OJ. Mental multiplication skill: structure, process and acquisition. Canadian journal of Psychology, 39: 338-366, 1985. COHEN, L., and DEHAENE, S. Amnesia for arithmetical facts: A sin gle case study. Brain and Language, 47: 214-232, 1994. COLTHEART, M. Aphasia therapy research: a single case study approac h. In C. Code and OJ. Mull er (Eds.), Aphasia Therapy. London: Edward Arnold, 1983, Ch. 18, pp.193-202. GRAHAM, O.F. An associative retrieval model: How children learn to multiply. In 1. Sloboda and D.
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Rogers (Eds.), Cognitive Processes in Mathematics. Oxford, England: Oxford University Press, 1987, Ch. 7, pp. 123-141. GROEN, G.J. , and PARKMAN, 1.M . A ch ronometric analysis of simple addition. Psychological Review, 93: 411-428, 1972. HITTMAIR-DELAZER, M., SEMENZA, c., and DENES, G. Concepts and facts in calculation. Brain , 117: 715-728, 1994. KASHIWAGI, A., KASHIWAGI, T. , and HASEGAWA, T. Improvement of deficits in mnemonic rhyme for multiplication in 1apanese aphasics. Neuropsychologia , 25: 443-447, 1987. L UZZATTI, C., WILLMES, K., and DE BLESER, R. Aachener Aphasia Test; Versione 1taliana. Verlag fur Psychologie, Dr. C.J. Gottingen: Hogrefe, 1991. M CCLOSKEY, M. Cognitive mechanisms in numerical processing: E vidence from acquired dyscalculia. Cognition, 44: 107-157,1992. MCCLOSKEY, M., ALIMINOSA, D., and MACARUSO, P. Theory-based assessment of acquired dyscalculia Brain and Cognition, 17: 285-308, 1991. MCCLOS KEY, M. , ALIMI NOSA, 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 , 199 1. MCCLOS KEY, M., HARLEY, W ., and SOKOL, S.M. Models of arithmetical fact retrieval: An evaluation in light of findings from normal and brain-damaged subjects. Journal of Experimental Psychology: Learning, Memory and Cognition, 17: 377-397, 1991. MCCLOS KEY, M., CARAMAZZA, A. , and BASILI, A. Cognitive mechani sms in number processing and calculation: Evidence from dyscalculia. Brain and Cognition, 4: 117-196, 1985. MICELI, G. , and CAPASSO, R. I disturbi del calcolo. Diagnosi e riabilitazione. Milano: Masson, 1991. MILLER, K. , PERLMUTTER, M. , and KEATING, D. Cognitive arithmetic: Comparison of operations. Journal of Experimental Psychology: Learning, Memory and Cognition, 10: 46-60, 1984. SI EGLER, R Strategy choice procedures and development of multiplication skill. Journal of Experimental Psychology: General, 11 7: 258-275 , 1988. SIEGLER, R , and SHRAGER, 1. A modelof strategy choice. In C. Sophian (Ed.), Origin of Cognitive Skills. Hill sdale, N l : Erlbaum, 1984, Ch. 3, pp. 229-293. SOKOL, S.M., and MCCLOSKEY, M . Cognitive mechanism s in calculation. In R Sternberger and P.A. Frensch (Eds.), Complex Problem Solving: Principles and Mechanisms. Hill sdale, N1: Erlbaum . 1991, Ch. 3, pp. 85-116. SOKOL, S., MCCLOSKEY, M. , COHEN, N.J., and AUMINOSA, 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, 199 1. SPINNLER, H., and TOGNONI, G. Standardizzazione e taratura ita1iana di test neuropsicologici. The Italian Journal of Neurological Sciences, Supplemento al n. 6, 1987. WARRINGTON, E.K. The fractionation of arithmetical skills: A single case study. Quarterly Journal of Experimental Psychology, 34a: 3 1-51, 1982. Luisa Girelli. Department of Psychology, University Collcge London. Gower Street, London WC IE 6BT,
U.K.
Reacquisition of arithmetical facts
65
ApPEN DIX A
Aachener Aphasie Test
TL Token Repeati tion Wri tten Language Nam ing Comprehens ion Alloc. class.
ZA
Row score
(deficit)
Row score
(deficit)
35/50 1151150 65/90 1161120 801120 Broca aphasia
(slight) (medium-slight) (med ium) (slight) (medium-slight)
41150 11 211 50 52/90 631120 971120 Not class.
(severe) (medium) (medium) (medium) (slight)
Number processing and calculation (Miceli and Capasso, 1991)
Non numerical tasks Magni tude comparison Dots pattern seriation
TL
ZA
10/ 10 5/5
10/10 5/ 5 1-2 d *
Magnitude comparison Visual-digits Vi sual n umberwords Verbal Transcodification tasks Repeti tion Reading numbers Read ing numberwords Writing from dictation Writing (words to digits) Recognition of arit. signs Visual Verbal Verbal ca lculation Addition Subtraction Multipli cation
* Percentage
1-2 d*
34/34 29/34 32/34
100% 100% 100%
33/34 33/34 32/34
100% 100% 100%
30/55 35/55 20/55 25/55 35/55
100% 100% 80% 100% 100%
30/55 18/55 17/55 23/55 31155
100% 68% 68% 92% 96%
8110 8110
10110 8/ 10
6120 6120 2120
20/20 20/20 5115
of correct a nswers with numbers up to 2-digits.
66
L. Girelli and Others ApPENDIX
B
Problems Included in Set A and Set B for the Patients' Training
TL
ZA
SET A
SET B
SET A
SET B
2x2 2x3 2x6 2x7 3x2 3x3 3x5 3X6 3x9 4x5 4x7 4x8 5X3 5X4 5X8 5x9 6x2 6x3 6X9 7X2 7X4 7X7 7X8 8X4 8x5 8x7 8x8 8X9 9X3 9X5 9x6 9x8
2x4 2X5 2x8 2x9 3x4 3x7 3x8 4x2 4x3 4x4 4x6 4x9 5X2 5X5 5x6 5x7 6X4 6X5 6x6 6x7 6x8 7x3 7x5 7x6 7x9 8X2 8X3 8x6 9x2 9x4 9x7 9X9
3x2 3x3 3x4 3x5 3x6 3x7 3x8 3x9 5X2 5x3 5X4 5x5 5x6 5X7 5x8 5x9 2x3 4X3 6x3 7x3 8X3 9X3 2x5 4x5 6x5 7x5 8x5 9x5
2x4 2X6 2X7 2X8 2X9 4X2 4X4 4x6 4X7 4X8 4x9 6X2 6X4 6X5 6X6 6X7 6X8 6X9 7x2 7x4 7x6 7X7 7X8 8X2 8X4 8x6 8X7 8X8 9x2 9X4 9X6 9X7 9X8 9X9