The Representation of Arithmetic Facts in Memory: Results from Retraining a Brain-Damaged Patient

36, 290–309 (1998) BR980997

BRAIN AND COGNITION ARTICLE NO.

The Representation of Arithmetic Facts in Memory: Results from Retraining a Brain-Damaged Patient Tony Whetstone East Carolina University This single case study was designed to gather evidence regarding whether the mental representations mediating multiplication fact retrieval make use of single or multiple codes. MC is a brain-damaged volunteer whose numerical processing impairments were limited to multiplication fact retrieval. He relearned three sets of multiplication facts. Each set was relearned in one of three input formats: Arabic, written verbal, or spoken verbal. Following training all facts were tested in all input formats. MC’s posttraining performance was virtually error free and showed no effects of input format. However, reaction-time data showed fact retrieval was fastest when the training format matched the test format. Results are discussed in relation to single- and multiple-code models of multiplication fact retrieval.  1998 Academic Press

In recent years several models of numerical processing have been proposed. Some of these make conflicting claims about the nature of numerical mental representations (Campbell & Clark, 1988; Clark & Campbell, 1991; Dehaene, 1992; McCloskey, Caramazza, & Basili, 1985; McCloskey, Macaruso, & Whetstone, 1992). One issue of contention concerns the nature of the codes used to represent numerical information. The theoretical positions on this issue can be grouped into two broad classes—single-code and multiple-code models. The purpose of the current study was to investigate the issue of multiplicity of codes in the context of a specific numerical task: multiplication fact retrieval. Throughout this paper the word ‘‘format’’ will be used exclusively for referring to the manner in which numbers are presented (e.g., the written This study was conducted while the author was a postdoctoral fellow in Cognitive Neuropsychology at Johns Hopkins University. I thank patient MC for his unfailing good humor and hard work, Mike McCloskey for valuable comments on an earlier draft of this paper, and Grethe Lindemann for helping to design the computer interface used to present digitized voice. Correspondence and reprint requests should be addressed to Tony Whetstone, Department of Psychology, East Carolina University, Greenville, NC 27858. E-mail: pswhetst@ecuvm. ecu.edu. 290 0278-2626/98 $25.00

Copyright  1998 by Academic Press All rights of reproduction in any form reserved.

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word ‘‘five,’’ the spoken word ‘‘five,’’ or the digit ‘‘5’’). The word ‘‘code’’ will be used to refer to proposals about the nature of numerical mental representations. Given that numbers can be input and output in many different ways, multiple-code positions propose that the mental representations for multiplication facts also exist in different codes which preserve at least some of the information specific to input format. Single-code positions on the other hand propose that at some point following perception numbers are represented in a single common code, regardless of whether the numbers were originally input in differing formats. According to a single-code position, in order to access a multiplication fact a problem must first be translated to the code used to store multiplication facts. The following sections outline these theoretical positions and their implications for multiplication fact retrieval. MENTAL REPRESENTATIONS IN MULTIPLICATION FACT RETRIEVAL

This investigation concerns the nature of the representations used to access and store memorized multiplication facts. In order to narrow the scope of this discussion, theoretical positions and models are considered only as they relate to single-digit multiplication. However, findings should have implications for multidigit multiplication performance since the retrieval of singledigit facts from the memorized multiplication tables is a component process of multidigit multiplication. Furthermore, it seems reasonable to assume that findings from the current study may generalize to models of fact retrieval for other arithmetic operations such as addition and subtraction, at least to the extent that skilled performance of these operations relies on retrieval of memorized facts. Single-Code Positions Also called ‘‘translation’’ models (Noe¨l & Seron, 1993), single-code theoretical positions have in common the idea that only one code is used for storage and access of multiplication facts. In different models, this single code may or may not be the same code used to carry out other numerical processes such as numerical comparison (i.e., ‘‘Which is larger, 5 or 9?’’). These models will be discussed here only in relation to their claims for multiplication fact retrieval. According to single-code positions, when a person is queried with a multiplication problem such as ‘‘9 ⫻ 9 ⫽ ?,’’ the numerical processing system must translate the problem to the same code used to store the answers to problems. Variants of this position concern the possible candidates for the single code used to store multiplication facts and gain access to them. For example, the single code may be characterized as semantic representations that do not contain reference to sensory qualities of perceived numbers but

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only information necessary for understanding their numerical meaning. Other possibilities characterize the single code as more imagelike, being similar to one of the possible types of input format. Although any number of input types is possible, most theoretical positions have characterized the imaginal qualities of these representations as being either more or less visual or auditory. Visual representations are said to retain some of the spatial qualities of Arabic digits or written number words and auditory representations are said to retain some of the qualities of spoken number words. Although many types of input are possible and therefore many representational codes are possible, discussion in this paper will focus on three representational possibilities. These correspond to Arabic input (i.e., the visually presented digit ‘‘5’’), written verbal input (i.e., the visually presented number word ‘‘five’’), and spoken verbal input (i.e., the aurally presented number word ‘‘five’’). Several papers have proposed single-code models of calculation performance. The modular model (McCloskey, Caramazza, & Basili, 1985) proposes amodal semantic representations for most numerical processing tasks including multiplication fact retrieval. A recent variation of the modular model adds transcoding algorithms that can directly translate from one input format to another output format (Cipolotti & Butterworth, 1995). However, like the McCloskey et al. model it proposes that calculation requires accessing a single amodal semantic code. The Triple-Code model (Dehaene, 1992) suggests that numbers are represented in different codes depending on what numerical task is being performed. It further assumes that retrieval of memorized multiplication facts is accomplished by translating input to a phonological code, presumably similar to the spoken verbal input format. Therefore, the Triple-Code model is classified as a single-code model when considering multiplication fact retrieval. Other investigators have proposed the preferred entry code hypothesis (Noe¨l & Seron, 1993) which stresses individual differences in tasks and people. According to the preferred entry hypothesis, the nature of some tasks may make them more suitable for certain representational codes, or individuals may for idiosyncratic reasons favor the use of certain representational codes. For example, Weddell and Davidoff (1991) concluded that calculation was mediated by visual representations for a brain-damaged participant they studied. They speculated that these findings might be specific to their patient, however, and that other individuals might prefer to use phonological representations. The idea that some tasks may be carried out more efficiently by certain codes seems reasonable, but it is not clear how individuals may directly prefer a certain code—if ‘‘code’’ is meant to refer to representation. It seems clear that individuals may prefer certain input formats, and perhaps greater practice with the preferred format will indirectly lead to greater reliance upon the code reflecting that format. One implication of the preferred entry code hypothesis is that group stud-

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ies of mathematics performance may not give clear answers about whether multiplication performance is carried out by a single or multiple codes; if different participants have different preferred access codes, the results might mimic multiple codes in averaged data. Because the current study is a case study, this particular problem is avoided. Of course, if the preferred entry code hypothesis is correct then the current study will not be conclusive as to what single codes are possible because the evidence will reflect only one individual’s preferred code. However, if evidence is obtained for a preferred entry code in this study, then the evidence may support the general class of single-code models versus multiple-code models. Multiple-Code Positions According to multiple-code positions, multiplication facts are represented and accessed in more than one code. This implies representational redundancy as each multiplication fact has the potential to be represented in more than one code. For normal performance in native English speakers these mental representations would probably correspond to the Arabic, spoken verbal, and written verbal input formats. One question raised by multiple-code positions concerns the degree of interaction between codes. An extreme position is that multiple codes are noninteracting. If multiple codes are noninteracting, then facts represented in long-term memory in a visual-Arabic code will only be accessed when problems are presented in the Arabic format. The noninteracting multiplecode position predicts notable impairment in calculation performance if facts are learned in one format and tested in another. For example, suppose one has learned ‘‘five ⫻ five ⫽ twenty-five’’ through repeated exposure to the written verbal format and is subsequently tested in the Arabic format (i.e., ‘‘5 ⫻ 5 ⫽ ?’’). The prediction arising from noninteracting multiple-code models is that either the correct answer will not be accessed at all or that there will be great slowing of fact retrieval. The slowing occurs because one presumably does not know in what code a particular fact may be represented and therefore must actively translate the query from one input code to another in a trial and error fashion. There are probably no existing multiple-code models that make the claim for noninteracting representations. A less extreme form of multiple-code model involves interacting codes. Unlike single-code positions, the code used to access the stored representation need not be the same code in which the representation is stored. Therefore, when a fact stored in a code corresponding to the Arabic format is activated it will likely activate the corresponding fact in other codes. Although representations in differing codes may interact, representations in the same code should be more strongly connected than representations in differing codes. Therefore, a fact represented in a code corresponding to the Arabic format should be more quickly and more accurately accessed when the query is presented in the Arabic format.

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Implications for Multiplication Fact Retrieval Single- and multiple-code positions differ in their implications for multiplication performance because they make differing predictions about transfer of knowledge given different input formats. Consider a single-code model where all queries regardless of input format must be translated to a code corresponding to the Arabic format before the answer can be accessed. Once comprehension has taken place there should be no effect of input format on the observed patterns of reaction time (RT) and accuracy measures. This is because the same representations are activated whether the original input was in the Arabic, written verbal, or spoken verbal format. Individuals may be faster or more accurate in translating one or another of the input formats into the single internal representational code, so the prediction is not one of absolute equality among testing formats. However, there should be no differences in performance once the operands are comprehended. Therefore, the single-code positions, regardless of what the specific single code is, predict complete transfer of knowledge across differing input formats. In contrast to single-code models, multiple-code models predict only partial transfer. The noninteracting multiple-code position predicts no transfer at all because each input format activates exclusively its own stored representations. This creates the potential for patterns of reaction times and error rates to vary widely for the same problems when posed in different formats. The interacting multiple-code position predicts partial transfer. Responses will be faster and more accurate when input format and representational code match, but even when input format and representational code do not match, participants may show little impairment in accuracy. Interacting multiplecode positions predict reaction time and error differences between input formats, but due to the possibility of interaction among codes, these may be subtle and difficult to detect.

CURRENT STUDY

In order to explore the properties of the mental representations used to accomplish multiplication fact retrieval, this study examined the numerical processing competence of MC, who after recovering from surgery for a brain tumor found he was impaired in the retrieval of basic multiplication facts although he was largely unimpaired in several other areas of basic numerical processing. MC was retrained on the basic multiplication facts with each fact relearned in only one format. Following relearning, his performance was examined for patterns of transfer across formats. A study previously published by Kashiwagi, Kashiwagi, and Hasegawa (1987) also retrained brain-damaged patients on the basic multiplication facts. However, certain aspects of that study prevent it from providing clear answers about the representational codes used in multiplication performance.

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Kashiwagi et al.’s study will be discussed next, followed by the specific methods of the present study. Kashiwagi, Kashiwagi, and Hasegawa (1987) studied seven brain-damaged individuals who were described as ‘‘aphasic with an absence of severe verbal apraxia, agraphia, agnosia, or dementia.’’ In addition to their language impairments, the patients were also severely impaired in recalling the answers to simple multiplication facts. The patients participated in a multiplication retraining program. The three patients of interest studied the problems with simultaneous Arabic and spoken verbal presentation. Practice tests also presented problems in both formats simultaneously and required responses in both formats. Training ranged between 30 and 60 sessions. Posttests were conducted in the following combinations: (1) spoken verbal test, spoken verbal answer, (2) spoken verbal test, Arabic answer, (3) Arabic test, spoken verbal answer, and (4) Arabic test, Arabic answer. The three patients who trained with simultaneous Arabic and spoken verbal presentation averaged less than 10% correct on the posttest combinations with spoken verbal input, 42% correct with Arabic input and spoken verbal output, and 89% correct with Arabic input and output. This study has been cited as positive evidence for the multiple-code position (Campbell & Clark, 1992) because the patients could not relearn the multiplication facts with spoken verbal presentation and practice but could relearn the facts with Arabic presentation and practice. This was taken as evidence that Arabic and spoken verbal input differ in their capacity to activate different representations of multiplication facts. However, it is not clear from Kashiwagi et al.’s description whether the aphasic patients were impaired in language comprehension, language production, or both. If one makes the assumption that the diagnosis of aphasia refers to an impairment in spoken verbal comprehension, the results may be explained as easily by a single-code account as by a multiple-code account. A single-code interpretation of the data follows. This account assumes (1) participants were unimpaired in comprehending Arabic input and producing Arabic output, (2) participants were profoundly impaired in comprehension of spoken verbal input and somewhat less impaired in producing spoken verbal output, and (3) because of (1) and (2), participants were able to relearn the facts with Arabic presentation but not spoken verbal presentation. These assumptions about the comprehension abilities of the patients predict Kashiwagi et al.’s results while saying nothing about the long-term memory code used to store the facts. During training, participants were able to relearn the facts presented in Arabic because Arabic comprehension was unimpaired. Once comprehended the facts could be translated to some single internal code for storage. During posttesting, problems presented in Arabic were successfully comprehended and then translated to the single code used to store facts. This code was used to cue the memory system. When queried with Arabic input and asked to produce spo-

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ken verbal output, patients were only partially successful due to impaired spoken verbal production, not because they were unable to retrieve answers in an internal code corresponding to the input format. The grossly impaired performance when test input was in the spoken verbal format is due to impaired spoken verbal comprehension, not necessarily an inability to retrieve answers in a specific code. The patients cannot be expected to produce the answers to questions they cannot understand. The Kashiwagi et al. findings do not provide evidence that clearly supports either single- or multiple-code accounts. The current study concerned the performance of an individual, MC, whose general processing of numbers was unimpaired except for impairment in the retrieval of multiplication facts. MC relearned three groups of multiplication facts with each group relearned in only one format. All three groups of facts were then tested in all formats. Because MC was unimpaired in numerical comprehension and production, any differences in performance should be attributable to interactions between input format and representational code for the multiplication facts.

ASSESSMENT Case History

MC is right-handed and was 42 years old at the time of testing. In August of 1990 he underwent surgery for a brain tumor. Postsurgical examination revealed a left parietal lesion, described in medical records as ‘‘extending to the splenium of the corpus callosum, without crossing midline and afferent of the cortical sulci.’’ The medical records also described MC as having mild diffuse brain atrophy and further stated that he retained right-sided weakness following surgery. No other neurological information was available. During the year he was tested, MC was treated with medications for the control of seizures and edema. Prior to discovery of the brain tumor MC was employed as a computer programmer. He and his wife reported that before surgery he had no difficulties with number processing or multiplication. MC holds an Associate of Arts degree and stated he did well in a course in business mathematics while at college. MC was tested at his home during weekly one-hour sessions from February 1991 through May 1993. Initial screenings for aphasia revealed no obvious problems in language comprehension or production. Formal testing of language abilities was confined to the production and comprehension of numbers, as specified below. Initial screening tests of numerical processing skills suggested MC’s comprehension and production of numbers in all tested formats was unimpaired. His only numerical processing deficit appeared to be difficulty in the retrieval of basic mathematical facts. These impressions were confirmed with the more thorough assessments presented below.

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Comprehension and Production Assessment

Dyscalculia Battery MC was administered the Johns Hopkins Dyscalculia Battery, which is comprised of tasks designed to assess numerical processing. Each test taps one or more numerical processing component skills to varying degrees (see McCloskey, Aliminosa, & Macaruso, 1991, for details). MC was 95 to 100% correct on all subtests of the battery that assess comprehension or production. The magnitude comparison test (e.g., Which number is larger, 406 or 503?) probes numerical comprehension without involving production, as the patient merely indicates the larger number without producing it. MC did very well in all three input formats, scoring 100% (20/20) correct in the spoken verbal and Arabic formats, and 95% correct (19/20) in the written verbal format. The numerical transcoding portion of the battery assesses comprehension and production by observing all input and output combinations of Arabic, written verbal, and spoken verbal numbers. For example, in an Arabic to spoken verbal transcoding test, MC might be presented with the numeral ‘‘55’’ and be expected to say ‘‘fifty-five.’’ MC was 100% (20/20 in each combination) correct transcoding, if one scores as correct a few written verbal output spelling errors (e.g., ‘‘ninty’’ for ninety), where the intended number was obvious. Follow-Up Assessment Comprehension was first tested with a transcoding task that presented all 100 numbers from 0 to 99 in Arabic, written verbal, and spoken verbal input formats. All numbers were converted to spoken verbal output (when presented with spoken verbal input MC was instructed to repeat the numbers spoken by the experimenter). MC was tested in this way because MC’s multiplication retraining was later to be assessed by examination of his spoken verbal reaction time for multiplication fact retrieval. MC was 100% correct with all input formats. Numerical competence, free of conventional input and output formats, was also tested with a ‘‘dots’’ apparatus. The dots apparatus consists of a casing containing strips that are pulled out to reveal from one to nine filled circles (dots). The rightmost strip indicates ones, and proceeding from right to left, each succeeding strip represents tens, hundreds, and so on. For example, the number 55 can be signified by pulling out five dots on the ones strip and five dots on the tens strip. During the period the dots stimuli were being tested MC usually performed flawlessly. However, on one day he incorrectly answered 10% of the stimulus presentations. He appeared less alert than usual that day and stated his medication had been changed. He was retested the following week with the same stimuli and made no errors. Excluding

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the ‘‘off’’ day, MC was 100% correct in transcoding numbers from 0 to 99 from Arabic, written verbal, and spoken verbal input to dots output. He was also 100% correct transcoding from dots input to spoken verbal output. Discussion The magnitude comparison and transcoding results from initial and follow-up assessments indicated preserved comprehension and production abilities for all three input formats and spoken verbal output. Further evidence that MC’s numerical processing was impaired only in the recall of multiplication facts, and not in comprehension or production of numbers, comes from analysis of MC’s errors on multiplication problems (see next section under results of follow-up assessment). Multiplication Assessment

Dyscalculia Battery In contrast to the excellent performance on magnitude comparison and transcoding tasks, MC incorrectly answered 3/10 simple single-digit multiplication facts when asked to produce the answers without using pencil or paper. Further, he correctly answered only 1/10 multidigit paper and pencil multiplication problems. Examination of MC’s performance on the multidigit problems showed that all but two of his errors involved fact retrieval errors (for examples of MC’s fact retrieval errors, see Table 1). His knowledge of multiplication procedures (e.g., when to carry, when to start a new row, spatial aligning of succeeding rows) appeared unimpaired. MC had some knowledge of when he produced incorrect answers. On one multidigit problem there was evidence he used a strategy of breaking down a larger multiplication fact to a smaller fact for which he could correctly retrieve an answer. For example, when he could not recall the answer to 6 ⫻ 7, he said aloud that he correctly recalled the fact that 6 ⫻ 6 ⫽ 36. He then added 6 to get 42. TABLE 1 Examples of MC’s Pretraining Multiplication Errors 3 4 4 4 7 7 8 9

⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻ ⫻

8 3 3 9 7 9 8 4

⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽ ⫽

27 16 18 28 47 54 53 20

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Follow-up Assessment For 10 consecutive sessions MC was tested with the 64 single-digit multiplication facts in the range of 2 ⫻ 2 through 9 ⫻ 9. Problems were presented in three input formats: Arabic (e.g., 2 ⫻ 2), written verbal (e.g., two ⫻ two), and spoken verbal (e.g., ‘‘two times two’’ spoken by the experimenter). In all cases MC gave spoken verbal responses. Written verbal and Arabic problems were presented in random order one at a time centered on paper in a 24-point font, with 13 problems to a page. Input format was alternated every page and the order of input formats was changed every session. MC was instructed to try to recall problem answers from memory and to avoid ‘‘figuring out’’ the answers to any problems. MC left no problems unanswered. He stated that in all cases, he reported answers that came to mind, and did not try to use adding strategies to arrive at them. Comprehension and production. Further evidence that MC is unimpaired in the comprehension and production of numbers comes from MC’s patterns of performance on multiplication problems. This shows that the dissociations observed in MC’s performance are not specific to certain tasks. In other words, MC does not simply understand numbers better on one task labeled comprehension and understand numbers more poorly on a task labeled fact retrieval (Sokol, McCloskey, Cohen, & Aliminosa, 1991). MC erred on only 5 (1.1%) of the problems involving a two in one or both operands, yet he erred on 26.5% of the problems involving 3 thorough 9. Because the numbers 3 through 9 appear in the twos times table problems, MC’s very low error rate on the twos times table problems indicates intact comprehension of the numbers involved as operands in the simple multiplication problems. Evidence for intact production comes from the fact that 380 of MC’s 394 errors (96.4%) are correct answers to other problems in the single digit multiplication table, and 73.4% of MC’s errors were correct answers to another problem in the same times table (e.g., 3 ⫻ 4 ⫽ 18). One would not expect such a systematic pattern of errors given a production deficit rather than fact retrieval deficit (Sokol, McCloskey, Cohen, & Aliminosa, 1991). Multiplication. MC’s fact retrieval performance improved over the course of testing. His average percent error rate per session declined from 35.94 to 11.46%. If problems and their complements are counted as the same problem (3 ⫻ 4 is counted the same as 4 ⫻ 3), MC showed marked impairment in 18 of the 36 unique multiplication problems. Impairment rates for a given problem ranged from 8 to 100% (error rates are presented in Table 2). Average error rate for these 18 problems over the nine weeks of testing was 42%. MC’s fact retrieval performance is similar to that of the patient PS as reported by Sokol et al. (1991). Like PS, most of MC’s errors are correct answers to other numbers in the same times table, and although he made more errors on larger problems, he also erred on smaller problems. MC’s error patterns like those of PS, show a lack of support for the hypothesis of

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TABLE 2 Pretraining Error Frequency and Retraining Format Problem

% Error

Recency

3⫻ 8⫻ 3⫻ 6⫻ 4⫻ 7⫻ M

9 8 6 7 6 8

Arabic 12 13 15 52 60 77 38

1.57 2.75 6.22 6.45 4.58 5.74 4.55

3⫻ 8⫻ 5⫻ 7⫻ 4⫻ 6⫻ M

7 9 7 9 4 8

4⫻ 7⫻ 4⫻ 3⫻ 3⫻ 6⫻ M

7 7 8 8 4 9

Written verbal 8 10 22 32 57 100 38 Spoken verbal 8 23 27 27 38 100 37

5.20 4.67 2.08 2.95 5.12 5.50 4.25 1.60 2.29 3.13 4.06 3.22 5.50 3.30

Note. Problems are grouped according to format in which they were subsequently retrained. Recency is the average session number in which errors were committed.

noninteracting multiple representational codes. If multiplication problems were stored in noninteracting multiple format specific codes, overall error rates should differ when problems are tested in different formats. This was not the case. The mean per session percentage error rates for problems tested in the Arabic, written verbal, and spoken verbal input formats were 20, 20, and 22%, respectively. The differences among error rates are not statistically significant (F(18, 2) ⫽ .77). However, because we do not know MC’s lifetime exposure to the problems in the varying formats, we have no way of knowing what postdamage levels of performance should be expected. For example, initially weak representations of problems in a code corresponding to written verbal input might be damaged only slightly, while initially stronger representations of problems in a code corresponding to Arabic might be extensively damaged, producing very similar overall error rates. A critical

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test of the nature of mental representations used in multiplication fact retrieval requires control over training exposure to problems in the different formats. If multiplication problems and answers were represented in multiple codes one might also expect that brain damage could lead to differing patterns of errors on the problems when tested in different formats. For example, the code corresponding to Arabic input might be damaged on the fact 3 ⫻ 6, whereas the code corresponding to spoken verbal input might be damaged for the problem 7 ⫻ 6. However, percent correct correlations among the 64 problems in the three formats were very high, ranging from .94 to .96. These findings are in line with predictions made by single-code models, and in fact they contradict the assumption of noninteracting multiple representational codes. However, an interacting version of the multiple-code hypothesis can account for the results in the following way. If the answer to a particular problem is no longer stored in a particular code, the input representation of that code may still activate another code that is able to activate the answer to the problem. Differences between formats may be masked by this process when the dependent measure is commission of errors. A more sensitive analysis might be had by examining reaction time differences among the problems when tested in the different formats, as one might expect that the circuitous route outlined above (input code activating another code which then activates the answer) should take more time than direct activation (input code activating answer). The problems in the preceding paragraphs can be solved through examination of MC’s performance after multiplication fact retraining. First, when problems are retrained, pretesting exposure to problems in various formats can be precisely controlled. Second, once problems are relearned in a particular format, reaction time can be examined to determine performance costs when problems are trained in one format and tested in another. MULTIPLICATION RETRAINING

Each problem was retrained in only one of three different formats: Arabic, written verbal, or spoken verbal. Predictions are based on the amount of transfer between formats. Consider problems tested in Arabic. This test included problems trained in all three formats. If total transfer has taken place (single-code prediction), performance should be the same in a given test format regardless of training format. However, if transfer is partial (multiplecode prediction), then some performance advantages should be seen in accuracy and/or reaction time when training format matches test format. Materials MC was impaired in answering 18 of the 36 unique multiplication problems in the range of 2 ⫻ 2 through 9 ⫻ 9. For purposes of retraining, the

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18 problems were divided into the three groups shown above in Table 2. Problem groups were equated on several measures. First, problem groups were equated on difficulty as measured by error rate. Second, problem groups were equated on difficulty as measured by average recency of error. Recency of error was assessed by assigning a value from one to ten to an error, based on the week in which the error occurred (e.g., 1 for an error in week one, 10 for an error in week ten). For example, if MC incorrectly answered a problem in a particular format once in week five and once in week two, the average recency of error score would be 3.5. The third consideration for equating problem groups was the number of times a particular operand appeared in each problem group. When possible, operands appeared in the same positions equally often in the three groups. This was not always possible as some operands occurred less often than required for equal distribution among the three groups. For example, examination of Table 2 reveals that the operand 3 appears twice in the first problem group and the third problem group but only once in the second problem group. To compensate, filler problems were drawn from the ones and twos times tables. These are problems that MC never missed in the 10-week multiplication evaluation stage. For example, to have the operand three appear as often in the second problem group as in the first and third problem group, the filler problem 3 ⫻ 1 was added to the second problem group. When all problem groups were equated, each problem group contained six critical problems and five fillers from the ones and twos times tables. Equating operand frequency ensured that MC had equal practice comprehending individual operands in all formats. Therefore, any performance differences found between formats on posttraining multiplication tests cannot be attributed to operand encoding processes rather than the nature of numerical representations. The fourth measure taken to equate the three problem groups was an even distribution of the three tie problems (e.g., 8 ⫻ 8), thus ensuring an equal number of observations in each group. Once the three problem groups were equated, each was assigned randomly to one of the three formats for retraining. Training Procedure and Results In weekly sessions, MC studied the problem groups in their assigned retraining formats. A session consisted of three blocks, each block consisting of study–test opportunities for one of the three problem groups. Block order was changed weekly so that problem groups appeared in each position in the sequence an equal number of times. A block consisted of the following events. First, a pretest without feedback was given in which a problem was presented and MC produced the answer into a microphone (output format was spoken verbal for all problem groups). Second, MC studied the problems

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with their answers for 5 seconds each. Once all 11 problems had been studied, they were studied again for 5 seconds each in a different random order. Following the second study opportunity, a posttest without feedback was given. Arabic and written verbal problems were studied and tested in a large font centered on a computer screen. Spoken verbal problems were presented by the computer via the experimenter’s digitized voice. During study, spoken verbal problems were followed by 3 seconds of silence to equate for the 5 seconds study time allotted to Arabic and written verbal problems. Throughout the retraining program, problem groups were studied and tested only in their assigned formats. MC was instructed to learn the answers to the problems and to avoid repeating problems out loud. No specific mention was made of presentation format. Multiplication retraining took 12 sessions. MC averaged less than one error per problem group on the last four weeks’ pretests. By the end of training, all problems had been answered correctly on the pretest for at least two consecutive sessions. An indication of MC’s improvement over the course of training is found by comparing pretraining error rate with posttraining error rate. In the 9 weeks prior to training, MC averaged 42% errors for the 18 problems which he was to relearn. Following training, MC averaged 3% errors for these problems. Posttraining Testing Procedure and Results Posttraining testing was designed to assess the costs and benefits to multiplication performance of a match or nonmatch between training and testing input format. The testing consisted of nine weekly sessions. All 64 problems were tested three times each session, once in each format. Input format was blocked, and block order was changed weekly. Arabic, written verbal, and spoken verbal problems were presented with the same fonts and digitized voice apparatus used in the retraining phase. MC was instructed to produce the answers into a microphone and was asked to proceed as fast as was comfortable for him, but to avoid mistakes. MC’s RTs averaged 4873 ms (SD 2825 ms), and about 3% were over 10 s. MC described long RTs as occasions when ‘‘no answer would come quickly into my mind.’’ In other words, these are occasions when he was paying attention and was apparently actively engaged in retrieval. Therefore, all analyses were performed on untransformed RTs and only errors were excluded from analyses. Analysis of log RTs and RTs trimmed for extreme scores three standard deviations from the mean were also performed for all the effects reported. These analyses obtained the same patterns of findings, and therefore are not reported here. Sessions were treated as subjects in all analyses to follow. For each of the nine testing sessions, the mean RT for correct answers to match and nonmatch problems in each testing format was calculated. All statistics reported are significant at the .05 level of confidence, unless otherwise noted. Before

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FIG. 1. effects.

TONY WHETSTONE

MC’s RTs for the multiplication times tables, demonstrating the size and fives

examining the match effect, MC’s performance will be examined for the presence of RT effects commonly found in multiplication fact retrieval studies with non-brain-damaged participants. Size, fives, and ties effects. It might be argued that MC’s multiplication performance is somehow different from the performance of other people. Therefore, MC’s fact retrieval RTs were assessed for three RT effects commonly found in the multiplication fact retrieval performance of non-braindamaged participants. These effects are the size effect (problems with smaller operands, e.g., 2 ⫻ 3, are answered more quickly than problems with larger operands, e.g., 8 ⫻ 9), the fives effect (responses to the fives times table are given more quickly than would be expected from the numerical size of the operands, and the ties effect (‘‘tie’’ problems, where the first operand is identical to the second operand, e.g., 6 ⫻ 6, are answered more quickly than nontie problems where the first operand differs from the second operand, e.g., 6 ⫻ 7). In the analyses that follow, the means for each input format in each weekly session were treated as a subject and the size and ties effects were treated as within-subjects variables. Figure 1 depicts the size effect. The figure groups RTs by times tables (i.e., all problems with a two as an operand, all problems with a three as an operand, etc.). Inspection of Fig. 1 shows patterns generally found in multiplication reaction times for normal subjects. For example, problems with smaller operands are in general answered more quickly than problems with larger operands. This effect is commonly seen in experiments with nonbrain-damaged populations. Also seen with non-brain-damaged populations is an advantage for the fives times table, for which facts are answered more quickly than would be expected by size alone. For statistical comparison of the size effect, problems were categorized as small (product of operands less

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TABLE 3 Mean Millisecond Reaction Time for Match and Nonmatch Problems Input format Arabic M SD Written verbal M SD Spoken verbal M SD

Match

Nonmatch

4250 512

4496 727

5188 866

6060 628

3568 742

4109 743

than 27) or large (product of operands 27 or greater). Tie problems were not included in this analysis because tie problems with larger operands are known to be answered more quickly than would be expected given operand size. MC’s RT for large operand problems was significantly slower than his RT for small operand problems (difference ⫽ 884 ms.), t(26) ⫽ 8.39, SE ⫽ 105.27, p ⬍ .001. The mean ties effects (nonties RT minus ties RT) was 1381 ms. This difference was significant t(26) ⫽ 11.98, SE ⫽ 115.34, p ⬍ .001. To summarize the above findings, examination of effects commonly found in the multiplication literature suggest that MC carries out multiplication fact retrieval in the same manner as non-brain-damaged individuals. Match effect. The data of most interest for current purposes concern the match effect. Both error and RT data were examined. All statistics reported are for critical nonfiller problems. As expected, MC’s error rate following 12 weeks of training was low. MC was 97% correct both when training format matched test input format and when training format did not match test input format. A two (match) by three (training format) ANOVA with match as a within subjects effect was performed on the error data. Neither the main effects nor the interaction were significant. However, because MC’s fact retrieval performance was near ceiling following training, the RT data are of more interest. Analysis of RTs revealed input format differences. Inspection of Table 3 shows overall effects of format differences in RT. Problems presented in the spoken verbal format were answered most quickly, followed by problems presented in the Arabic format and problems presented in the written verbal format. The main effect of format was statistically significant for all RT analyses reported in this study, but will not be dealt with further, as it probably reflects encoding time differences among the formats and therefore is of little utility in answering questions about representation. The match RT effect is displayed in Table 3. For each test input format, RTs were fastest for problems studied in that format as compared to problems

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studied in other formats. A two (match) by three (training format) ANOVA on correct RTs to critical problems revealed significant main effects of both match, F(1, 24) ⫽ 13.38, MSError ⫽ 308845, p ⬍ .05 and input format, F(2, 24) ⫽ 21.49, MSError ⫽ 703752, p ⬍ .05. The interaction of match with training format was not significant, F(2, 24) ⫽ 1.43, MSError ⫽ 308845. Operand order. MC relearned the problems with unequal operands (nontie problems) in only one order. For example, the problem 3 ⫻ 7, was always studied and tested during training as 3 ⫻ 7, and never as 7 ⫻ 3 (Table 2 shows the problems in the operand order used for relearning). Posttraining RTs for both match and nonmatch problems were analyzed for operand order effects, with each of the retrained problems contributing both a studied order and nonstudied order. There were no significant effects of operand order, and in fact, both match problems (4336 vs 4479 ms) and nonmatch problems (4892 vs 5171 ms) were answered nonsignificantly faster in the nonstudied operand order than in the studied order (both t’s ⬍ 1). Discussion of Posttraining Test Results After retraining, MC had near flawless performance on retrained problems. No effects of format were found for error data. However, MC did have faster response times to problems that were tested in the same format in which they were trained. The finding of a study–test match effect for all formats shows that simple multiplication can be affected by the format in which problems are trained and tested. GENERAL DISCUSSION

Before retraining, MC was impaired in the production of answers to simple multiplication facts although his ability to correctly comprehend and produce numbers in different formats was unimpaired. MC’s selective impairment provided an opportunity to test whether single or multiple representational codes are used in the retrieval of multiplication facts. Analysis of MC’s pretraining error data replicated earlier findings of no differences in multiplication performance between input formats (Sokol et al., 1991), which contradicts predictions from the more extreme noninteracting version of multiple code models. However, multiple-code models, which allow for interaction between codes might handle MC’s error data by supposing that problems presented in formats that do not correspond to the code in which the answer is represented, can still access that code. A test of the interacting version of the multiple-code models is better investigated by retraining and observing reaction time and error data. MC was retrained on the multiplication facts for which his performance was most impaired. Examination of MC’s RT data for effects commonly found in the multiplication fact retrieval literature suggest that MC carries out fact retrieval no differently than non-brain-damaged participants. MC’s error data

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revealed no format effects. However, analysis of MC’s reaction time data by input format revealed a match effect; MC answered problems more quickly when they were tested in the same format in which they had been trained. What do the two major posttraining findings indicate for models of multiplication fact retrieval? First, the absence of format effects in the error data is in line with the predictions arising from single-code models but becomes somewhat more difficult for interacting multiple code models. For example consider MC’s performance on the problem ‘‘six times eight’’ which was retrained in the spoken verbal format. Prior to retraining MC failed this problem regardless of input format. Assuming that the multiple codes representing this problem were lost prior to retraining, after retraining MC should have an internal spoken verbal representation. MC’s performance on this problem after retraining was highly accurate, regardless of format. An interacting version of the multiple-code model will need to explain more precisely how this problem—which should be represented in only one code—can be activated by other codes without practice. On the other hand, the reaction time data revealed a match effect which indicates that models of multiplication fact retrieval do need to include multiple representational codes at some level in the numerical processing system. At least three codes should be considered. Facts are not represented in only a verbal and a visual code, for example, because the match effect was found for two types of verbal representations: spoken verbal and written verbal. The reaction time evidence from MC’s reaction time performance is consistent with an interacting view of multiple representational codes. Although the single-code class of models predicts MC’s error data, the reaction-time data requires some elaboration in order to accommodate the evidence. One possibility is the activation of two memory systems or two different stages involved in memory retrieval. Multiple codes may be used in comprehension while a single code may underlie fact retrieval. Schacter (1990) has hypothesized that a perceptual representation store may be responsible for many findings of repetition priming. This system is hypothesized to store the general perceptual characteristics of specific types of material without reference to semantic features. A system such as that proposed by Schacter could influence numerical processing by storing the visual features or auditory characteristics of multiplication problems. This system could have its effect during the comprehension process in single-code models. These representations would then be translated to a single code for fact retrieval. For example, McCloskey et al.’s modular model includes comprehension modules for visual and verbal materials, and these modules are assumed to store some knowledge about sensory input. Comprehension processes are hypothesized to retain some information about the individual operands of a problem, making it possible that a given module can become more efficient encoding a frequently encountered op-

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erand. In the current study, operands were balanced among problem groups so that participants had equal exposure to operands in all formats. However, it might be that with practice comprehension modules form a unitized problem representation. At least one current computational model of multiplication has assumed problem level representations (Campbell & Oliphant, 1992). Because problems were studied in only one format, the match effect could be interpreted as the result of faster encoding processes for a problem that is tested in its studied format. For example, if 8 ⫻ 7 was studied in Arabic and 8 ⫻ 8 was studied in the written verbal format, when tested in Arabic 8 ⫻ 7 may be faster than 8 ⫻ 8, not because problem and answer are stored in Arabic form, but because the Arabic comprehension module has been more practiced with encoding 8 ⫻ 9 than 8 ⫻ 8. In this account multiple representational formats may aid the comprehension process but they are converted to a single format in order to retrieve answers. There is a further puzzle to solve for a model that contains problem level representations at a perceptual (asemantic) level. A problem level representation that preserves features of the original perceptual experience without reference to the semantic nature of the problem suggests that if a problem with unequal operands is studied in a certain order (i.e., 7 ⫻ 8) then performance should be faster when the problem is tested in that order, rather than the reverse order (8 ⫻ 7). However, examination of MC’s fact retrieval reaction time performance for problems in their studied order versus nonstudied order found no differences. Of course the lack of an operand order effect will also need to be explained by multiple-code models. Why do representations retain specific information about input format but not operand order? The findings of this experiment are somewhat paradoxical. First, the preand posttraining error data were notable mainly for how similar MC’s performance was across formats. On the other hand, the posttraining reaction time results found clear evidence that format can influence performance. Any explanation of the representations involved in multiplication fact retrieval will need to be able to predict when similar or different patterns of findings should be obtained between input formats. One area that appears promising for future research is the type of test. Implicit memory measures often find effects of format on performance, whereas explicit memory tests often find no effects of format on performance (Schacter, 1990). Indeed, MC’s pretraining testing focused on explicit answers to multiplication facts and revealed no evidence of format effects. On the other hand, MC’s posttraining testing used a more implicit memory measure, reaction time, and format effects were revealed. A last issue concerns the preferred entry hypothesis. The current study is a case study, and therefore potentially limited in its generalizability. It may be possible that although MC uses multiple codes to achieve fact retrieval, others who prefer certain codes may represent facts in a single code. Further studies will be necessary to estimate the universality of multiple codes, but

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experience with MC does lead to an interesting observation concerning the preferred entry hypothesis. MC spontaneously volunteered more than once during testing that he subjectively disliked the written verbal input format for presenting multiplication problems and that he preferred the Arabic input format. From these statements it is obvious that MC had a preferred entry format. Yet MC performed most efficiently when the testing format of a multiplication problem involved the format in which it was originally studied, regardless of whether MC preferred encountering that format. This suggests that subjective preference may have little influence on the codes used to store multiplication facts. REFERENCES Campbell, J. I. D., & Clark, J. M. 1988. An encoding-complex view of cognitive number processing: Comment on McCloskey, Sokol, & Goodman (1986). Journal of Experimental Psychology: General, 117, 204–214. Campbell, J. I. D., & Clark, J. M., 1992. Cognitive number processing: An encoding-complex perspective. In J. I. D. Campbell (Ed.), The nature and origin of mathematical skills (pp. 457–491). North-Holland: Elsevier. Campbell, J. I. D., & Oliphant, M. 1992. Representation and retrieval of arithmetic facts: A network-interference model and simulation. In J. I. D. Campbell (Ed.), The nature and origin of mathematical skills (pp. 457– 491). North-Holland: Elsevier. Cipolotti, L., & Butterworth, B. 1995. Toward a multiroute model of number processing: Impaired number transcoding with preserved calculation skills. Journal of Experimental Psychology: General, 124, 375–390. Clark, J. M., & Campbell, J. I. D. 1991. Integrated versus modular theories of number skills and acalculia. Brain and Cognition, 17, 204–239. Dehaene S. 1992. Varieties of numerical abilities. Cognition, 44, 1–42. Kashiwagi, A., Kashiwagi, T., & Hasegawa, T. 1987. Improvement of deficits in mnemonic rhyme for multiplication in Japanese aphasics. Neuropsychologia, 25, 443–447. McCloskey, M., Aliminosa, D., & Macaruso, P. 1991. Theory-based assessment of acquired dyscalculia. Brain and Cognition, 17, 285–308. McCloskey, M., Caramazza, A., & Basili, A. 1985. Cognitive mechanisms in number processing and calculation: Evidence from dyscalculia. Brain and Cognition, 4, 171–196. McCloskey, M., Macaruso, P., & Whetstone, T. 1992. The functional architecture of numerical processing mechanisms: Defending the modular model. In J. I. D. Campbell (Ed.), The nature and origin of mathematical skills (pp. 493–537). North-Holland: Elsevier. Noe¨l, M. P., & Seron, X. 1993. Arabic number reading: A single case study or when 236 is read (2306) and judged superior to 1258. Cognitive Neuropsychology, 10, 317–339. Schacter, D. L. 1990. Perceptual representation systems and implicit memory: Toward a resolution of the multiple memory systems debate. Annals of the New York Academy of Sciences, 608, 543–571. Sokol, S. M., McCloskey, M., Cohen, N. J., & Aliminosa, D. 1991. Cognitive representations and processes in arithmetic: Inferences from the performance of brain-damaged patients. Journal of Experimental Psychology: Learning, Memory, and Cognition, 8, 320–335. Weddell, R. A., & Davidoff, J. B. 1991. A dyscalculic patient with selectively impaired processing of the numbers 7, 9, and 0. Brain and Cognition, 17, 240–271.