Chapter 14 In Defense Of The Encoding-Complex Approach: Reply To Mccloskey, Macaruso, & Whetstone

The Nature and Origins of Mathematical Skills J.I.D.Campbell (EdTtor)

0 1992 Elsevier Science Publishcrs B.V. All rights reserved.

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Chapter 14 IN DEFENSE OF THE ENCODING-COMPLEX APPROACH REPLY TO McCLOSKEY, MACARUSO, & WHETSTONE

Jamie I. D. Campbell University of Saskatchewan

Summary

In the preceding chapter, McCloskey, Macaruso, and Whetstone (henceforthMM& W) present a number of theoretical and empirical challenges to the encoding-complex view of number processing advanced by Campbell and Clark (this volume; henceforth C&C). In this reply I examine their explanationsfor the effects of numberformat on numberfact retrieval found by C&C. I argue that the main alternative accounts offered by MM& W -- the "internaldeadline" hypothesis and the "encodingeficiency" hypothesis -- are not supported by the data and do not provide convincing alternatives to G W s p r o p s a I that retrieval processes differ as a function offormat. I also argue that aspects of the qlanations offered by MM&W compromise the basic abstmctmodular theory and in fact, undermine MM&Ws claim that the modular view is likely to be more productive than the encoding-complexapproach. I propose further that the abstract-code theory of number meaning assumed within fhe modular framework is counterproductive, because it takes for granted compIa, fundamental aspects of cognitive number processing. In contrast, it is a primary goal of the encoding-complex approach to provide explanatory mechanisms for these basic elements of numerical cognition. Introduction A pervasive theme in the preceding comment by MM&W is that there are clear and important advantages that follow from starting with relatively simple assumptions about the cognitive architecture underlying numerical skills. One presumed advantage that MM&W emphasize is that the simple, modular architecture they propose provides straightforward predictions and interpretations with respect to experimental data, thus making their model testable and falsifiable.

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MM&W also argue that the encoding-complex approach is too underspecified to support the generation of testable hypotheses, and is therefore unfalsiiable. These considerations, they claim, indicate that the abstract-modular approach is likely to be more productive scientifically. In this reply I argue that the differences between the two approaches with respect to predictiveness and testability are not as clear-cut as MM&W claim. It is true that C&C's analyses of format effects in multiplication were exploratory, and MM&W correctly point out that the encoding-complex view did not predict a specific pattern of format effects (p. 510). This is not an embarrassment for the encoding-complex position, however. We have stated clearly (Clark & Campbell, 1991) that we are still early in the process of developing a precise encodingcomplex theory around our basic assumptions, and exploratory analyses are an essential part of that process. Furthermore, it is important to recognize that the abstract-modular theory similarly did not predict any specific format-related phenomena. This is not surprising, despite MM&W's claims of predictiveness and testability for the abstract-modular model. Although the model is specific in its assumptions about the skeletal structure of the proposed number-processing systems, very little has been specified about how the proposed comprehension, calculation, and production systems can interact in the context of different tasks. Indeed, in order to accommodate the various effects of number format demonstrated by C&C, MM&W introduced several new possible extensions and elaborations to the model. It is, of course, a natural part of scientific theory development to modify assumptions to accommodate new findings. But the question I raise here is, given the sorts of ad hoc modifications and possibilities introduced by MM&W, is the abstract-modular theory really more testable, constrained, or predictive than the encoding-complex view? Although MM&W characterize their modifications as "specific limited modifications" (p. 508), I will argue that the implications are more far reaching than this. To begin, it will be worthwhile to briefly review how the abstract-modular and encoding-complex views differ with respect to predictions about format-effects on number-fact retrieval. Number format and number-fact retrieval In the abstract-modular theory, numbers are represented by format and modality independent abstract-codes, which are characterized by a digit in brackets, followed by an exponential term specifying the appropriate order of magnitude (e.g., 67 is (6)lOEXPl (7)lOEXPO). Number facts are stored in abstract form in a "calculation system" and are activated by abstract encodings of problems that are input from a separate "comprehension system." In contrast,

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according to the encoding-complexview (e.g., Clark & Campbell, 1991), numberfact retrieval is based on modality-specificrepresentations, such as visual codes for digits, and visual and phonological codes for number words. Given these different assumptions, a natural approach to testing the abstract- versus specific-code views is to examine effects of number format (e.g., digits vs. words) on number-fact retrieval. The abstract-code view implies that the calculation process for a given task (e.g., simple multiplication or addition) is the same regardless of the format in which problems are presented. Therefore, the abstract-code view seems to predict that there should be no substantial effect of format on performance, and, indeed, Sokol, McCloskey, Cohen and Aliminosa (1991) concluded that their failure to find format effects in the multiplication errors of an acalculic patient is consistent with the abstract-code hypothesis. The contrasting view, that memory for number-facts is modality specific, implies that retrieval processes can vary as a function of format. For example, it seems very likely that simple arithmetic problems are encountered more frequently in digit format (e.g., 2 t 6 = 8 and 4 x 6 = 24) than in written number-word format (two t six = eight and four x six = twenty-four). As a consequence, a digit problem would be more likely to activate a visual representation than a numberword problem. Retrieval via number words must depend more on auditoryphonological representations of problems. Retrieval via digits, therefore, should be easier because it is mediated both by well-established visual and phonological "routes,"whereas retrieval via number-word format will not provide a strong, direct visual basis for retrieval. Although the encoding-complexview leads to the general prediction that digitbased retrieval should normally be easier than word-based retrieval, more precise predictions at this early stage of theory development are not possible. Predictions are complicated by the possibility of indirect paths of activation due to strong associations between visual codes and the corresponding phonological codes. For example, retrieval via written number-words presumably can include activation of visual-codes for digit-based representations, because visual codes for number words potentially activate digits codes via a common phonological association. Furthermore, phonological representations of problems likely will be strongly activated whether problems are presented as digits or as number words. These considerations imply that there will be substantial overlap in the retrieval structures activated by problems presented in digit format or in number-word format. Consequently, effects of format are likely to be quite subtle, and there are bound to be many parallels in retrieval given number-word or digit formats. Nonetheless, in the C&C experiment, careful analysis uncovered several systematic effects of format.

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Comments on MM&W's explanations for various effects of format

Although MM&W state that the experiment reported by C&C "did not reveal gross differences in performance between the word and digit problems" (p. 510), there were, in fact, very substantial differences in accuracy and speed for simple multiplication problems as a function of format. The mean rate of errors was 51% higher for word-format problems than for the digit format (12.5% vs. 8.3% of trials), and mean time for a correct response was 36% slower with words than digits (1107 ms vs. 816 ms)'. C&C proposed that these and other phenomena suggest that retrieval processes can differ as a function of format. In contrast, MM&W argue that these overall differences could be due, directly or indirectly, to longer encoding times for the word stimuli relative to digits.

The "internal deadline" hypothesis MM&W propose that longer encoding times could have contributed to the high error rate for word problems in the following way: Subjects may have been motivated to generate an answer prior to some "internal processing deadline" (p. 522). This deadline corresponds to the maximum amount of time a subject permits to elapse before generating a response. Total multiplication time includes encoding and calculation (i.e., retrieval) stages, and both stages run to completion unless the deadline is exceeded. Because of longer encoding times for words, subjects were more likely to encounter the deadline for word-format problems than digit-format problems. As a consequence, word problems were afforded less retrieval-stage processing on average relative to digit problems, making retrieval errors more likely for the word stimuli. Although the internal deadline theory is plausible, it is almost certainly not the correct explanation for the overall higher error rate for word-format multiplication in C&C's experiment. Pervasive use of a deadline in the manner proposed by MM&W would impose an upper bound or ceiling on word-format RTs. This effect would appear as a clustering of RTs for the more difficult word-format problems at the cutoff defined by the average deadline. The absence of such a ceiling for word-format RTs would indicate that subjects' responses generally were not constrained by a deadline as suggested by MM&W. Figure 1 presents the mean RTs from C&C's study for each problem in each format, with word-format

'Campbell (1990) also observed that word-based addition and multiplication were both about 300 ms slower compared to digit format.

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RT on the y-axis plotted against digit-format RT for the corresponding problem on the x-axis. The deadline hypothesis predicts a rightward bend at the top of the function corresponding to the ceiling imposed on word-format RTs by the putative deadline. As the Figure shows, there was no evidence that word-format RTs were constrained by a deadline; instead, the Figure shows that the relation between word and digit RTs across the 64 problems was linear (r = .913) across the entire range of word-format RTs.

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Figure 1. Mean RT for M x N multiplication problems presented in word or digit format (Campbell & Clark, this volume).

Despite the apparent linearity, a quadratic component in the relation between digit and word RTs might signal the curvilinear trend predicted by the deadline hypothesis. To test for evidence of a quadratic trend in Figure 1, a multiple regression on the 64 word-problem RTs was performed with the corresponding digit RTs and their squares used as predictors. The squared-RT variable provides a test for the quadratic component. Once the linear relation with digit RTs was factored out, however, the quadratic component did not enter the equation (a to enter = .05; partial r = -.22). When only word-format problems with RTs over loo0 ms were included in the analysis, the partial correlation for the quadratic component was -.16 (p > .25). This affirms that there was no curvilinear trend

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associated with the upper end of the function. Thus, far from there being clear visible evidence of a ceiling on word-format RTs, there was no statistical evidence of a curvilinear trend in Figure 1 as would be expected given the internal deadline idea. The data appear to directly disconfirm the deadline hypothesis as a plausible explanation for the 50% higher rate of errors in the word-format condition. The unfamiliarity of word-based multiplication Apart from no evidence to support a RT ceiling for word problems, MM&W's deadline proposal encounters another major difficulty. The deadline proposal implies that differences in encoding times for the word and digit problems must have been substantially in excess of 300 ms (i.e., because the deadline proposal assumes that time in the retrieval stage was less for words than for digits, the overall 300 ms longer RTs for words must be underestimating the differences in encoding time). A difference of this magnitude is difficult to reconcile, however, with evidence that naming times for individual number-words and digits in the range from two through nine (i,e., the numbers used as multipliers in the C&C experiment) differ by as little as 11 ms on average (Campbell, 1990). The naming data suggest that words and digits do not differ greatly with respect to the amount of time required for encoding. MM&W argue (p. 512), however, that the 300 ms differences can be explained in terms of differences in encoding times. They propose that such factors as stimulus width and frequency (discussed below), and the "unfamiliar task (p. 512) of word multiplication would have contributed to longer encoding times for words. The proposal that encoding times were long because of low familiarity with word-based multiplication seems to be at odds with the basic abstract-modular viewpoint, which places great emphasis on the functional independence of the comprehension and calculation processes (e.g., MM&W, pp. 496-497). Functional independence implies that the process of converting stimuli into abstract codes (i.e., the comprehension process) should be the same irrespective of the subsequent processing that will take place. Consequently, practice and experience at "comprehending"number words in any task that requires the putative abstract codes (e.g., reading numbers; cf,, McCloskey, Sokol & Goodman, 1986), should transfer to the multiplication task. As people do have extensive experience with number words in non-calculation contexts, the claim that low-familiarity would contribute to longer encoding times for word-based multiplication, per se, seems to contradict the assumption of functional independence. Indeed, later in their comment, MM&W use number-word and digit frequencies that were estimated from book and magazine counts to predict differences in

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multiplication RTs as a function of format (pp. 515-521). For the p u r p o s of this analysis, which was to discount a retrieval-based explanation of the diffelences in favor of an encoding-based account, MM&W appear to assume that general experience with encoding number-words should transfer directly to the multiplication task. Transfer seems to be predicted when it yields results potentially consistent with the abstract-modular position, but is not predickd when it would yield results that potentially contradict the abstract-modular view. MM&Ws claim that the abstract-modular model affords simplicity, prediaiveness, and testability ultimately is based on the assumption that there is a generic comprehension process that is common to most number-processing tasks; however, the actual criteria for determining when the model predicts generic comprahension processes and when they could be format or task specific, are elusive. There is also a more general point to be made about MM&Ws suggesrion that lack of familiarity with word-based multiplication accounts for the longer wordformat RTs: This suggestion amounts to asserting that encoding-fbrmat x calculation-task interactions are to be expected within the abstract-modular framework. I will argue that by admitting such interactions, MM&W substantially weaken their claim that the assumption of functionally independent modules provides powerful predictive and interpretive constraints. Frequency, encoding eflciency, and problem diflculry C&C reported multiple regression analyses of speed and accuracy across problems in the digit and word formats (pp. 466-468).These analyses showed that the physical width of word-format problems was positively related to word-format RTs, but not to digit-format RTs, indicating that time to read or encode (he word stimuli contributed to the pattern of differences between word and digit RTs. Once these effects were partialled out, however, variables theoretically related to retrieval difficulty (problem size and "fan"), accounted for significant residual variance in the pattern of format-related differences in RTs and errors. In general, the word-format deficit tended to increase with problem difficrlty, and C&C suggested that this finding was consistent with the conclusion that retrieval processes differed between the word and digit formats. As an alternative account, MM&W proposed that the apparent interaction of format with problem difficulty might be due to differences in encoding difficulty between digits and number words (pp. 515-521). MM&W report data (rables 2 and 3, pp. 515-516) that indicate that number-word and digit frequencies are negatively correlated with numerical value over the range from two to nne, and that the slope of the relation is steeper for number words than digits. 'I%us, if it

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is assumed that encoding efficiency will be directly related to frequency, then it follows that frequency could account for the interaction between format and problem size that appeared in C&C's multiple regression analyses. MM&W report new regression analyses showing that when frequency differences are taken into account the problem size and fan variables no longer account for unique variabillty. There is good reason, however, to doubt the frequency-based account. The plausibility of the frequency-based explanation proposed by MM&W is weakened, if not contradicted, by the results of the number encoding-time experiment they report (see Tables 4 and 5 in MM&W, pp. 519-520). The results showed that the estimated encoding times for digits (Table 5 ) were unrelated to frequency differences for digits (Table 3, p. 516), despite substantial variability in digit frequencies. If anything, digit frequency and digit-encodingtime estimates tended to be negatively related (r = -.20,p. 520). Although number-word encoding times were positively correlated with number-word frequencies, estimates of digit encoding times were unaccountably not predictable from frequency differences. The data indicate, therefore, that frequency counts are not a good basis for estimating number-encoding times. Thus, the statisticallysignificant prediction of RT differences by the frequency variable in MM&Ws multiple regression analyses might have nothing to do with encoding processes. Instead, for example, number frequency may simply be correlated with problem frequency and, hence, with retrieval difficulty.

Effects of format on specific features of errors The analyses performed by C&C also uncovered a number of format effects on the characteristics of specific errors. The most important format-related effects on specific errors concerned 1) the tendency for word-format errors to be more distant from the correct answer, and 2) for operand-intrusionerrors to occur more frequently with the word format. Error distance C&C found that errors that were two or more operand-units distant from the correct answer (e.g., 4 x 6 = 32) were about 40% more common for words than for digits. C&C also noted that, relative to digit problems, a higher percentage of word errors were multiples of the larger operand than the smaller operand. C&C emphasized that the specific causes of these effects were uncertain (p. 472), but suggested that both effects might reflect weaker involvement of magnitude factors for word relative to digit multiplication. This interpretation is contrary to the

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abstract-modular view because it implies that the involvement of mgnitude representations can be format specific, whereas magnitude representation is associated with the abstract-codes in the abstract-modular theory, and should presumably be a constant factor across different formats. In their comment, MM&W correctly point out that the higher rate of maxrelated errors for words is actually an artifact of the higher rate of intrusim errors for words (see MM&W, p. 525) and probably has nothing to do with mgnitude per se. Nonetheless, format effects on intrusion errors present a major Qlallenge to the abstract-modular view (see below). Furthermore, whereas the higher incidence of intrusions can account for differences in the frequency of ma-related errors, it does not account for the operand-distance effect. MM&W propose that the tendency for tabled-related errors to be more distant with word stimul is more plausibly interpreted in terms of the processing-deadline hypothesis discussed above, rather than to differential involvement of magnitude factors for digits versus words. They argue that, although the word-format yielded much lorger RTs overall, less time was spent on retrieval processes relative to digit Woblems. Consequently, processing in the retrieval system was terminated earlier on average for word problems than digit problems. In this case, digit errors would tend to be less remote than word errors, MM&W suggest, if it is assumed that t k "set of candidates is gradually narrowed during retrieval" (p. 524). A similar explanation (p. 525) is offered to explain the higher rate of miscellaneous errors (he, errors involving non-products) in the word format. Although this explanation is not implausible, it takes for granted the varacity of the deadline hypothesis, which, as demonstrated previously, is not supporttd by the data: Contrary to the deadline hypothesis, there was no evidence of a ceiling for word-format RTs. Furthermore, there is other experimental evidence consistent with C&C's proposal (see Clark & Campbell, 1991, for a more extensive discussion). For example, Foltz, Poltrock, and Potts (1984) found t h a format (words vs. digits) interacted with numerical distance in a magnitude jdgement task; specifically, there was a strong magnitude-congruity effect with digt stimuli but not with number-word stimuli. These results suggest that magnitub: factors were not as important in numerical processing of number words as digits. Similarly, the effects of format on the distance of multiplication errors may be plausibly interpreted as evidence that number-words and digits can diflbr in the extent to which they involve the processing of magnitude information. Tks, given the apparent inadequacy of MM&Ws internal-deadline explanation, the finding that retrieval-error distances vary as a function of format remains a swstantial challenge to the view that number-fact retrieval is based only on formatindependent, abstract codes.

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Operand-priming as evidence of encoding-format x retrieval-process interactions The operand-priming effects reported by C&C provide some of the most challenging phenomena for the abstract-modular theory, and therefore it is worth reviewing the phenomena and C&C's characterization of them in some detail. Many of the multiplication errors observed by C&C appeared to involve "intrusions" of the problem's operands (e.g., 8 x 4 = twentyfour). Cases where the entire response corresponded to one or both of the operands were called naming errors (e.g., 2 x 9 = nine; 2 x 7 = twenty seven). Many other cases were simple intrusions, however, in which only one operand appeared in the error (e.g.,9 x 6 = thirty sir, 6 x 9 = sir& three). Format had a very strong effect on intrusions: Naming errors were about 4.5 times more common with words than with digits (90 vs. 16 errors), and simple intrusions similarly were much more common in the word format than the digit format (597 vs. 326 errors), accounting on average for 51% and 38% of word and digit errors, respectively. Thus, the data showed that operand intrusions were a substantial factor in performance, and that the influence of the processes producing intrusions was much greater for word stimuli. C&C argued that operand intrusions involve number-fact retrieval processes. If intrusions arose by priming of post-retrieval lexical codes, for example, intrusions ought to be associated frequently with miscellaneous answers (e.g., 7 x 4 = 34). Instead, about 80% of operand intrusions co-occurred with arithmetically associated products (7 x 4 = 24), suggesting that operand intrusions resulted from priming or activation of number-fact representations. Consistent with this interpretation, C&C also found that when the position of an operand matched the corresponding number in the correct answer (e.g., 6 x 4 = 24 vs. 4 x 6 = 24, 6 x 8 = 48 vs. 8 x 6 = 48, etc.), there were significantly fewer errors and correct RTs were faster relative to when there was not a positional match. These findings paralleled the observation that intrusion errors preserve the position of the matching operand more often that expected by chance (e.g., 8 x 4 = 24 tends to be a more common error than 4 x 8 = 24). C&C proposed that intrusion errors and related effects arise because of direct priming of number-fact representations that contain features or components that match the operands. More specifically, they proposed that the effects result from an interaction of fact-retrieval processes and the number-reading processes engaged when the problem is encoded. Number-reading processes could activate number-word codes that match corresponding verbal-code representations of multiplication problems and answers. The effect of this activation would be facilitative when the correct problem-answer representation is primed, but produce interference when related, but irrelevant, representations are primed (cf. Campbell

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& Oliphant, this volume). With respect to format effects on intrusions, C&C proposed that the higher rate of intrusions for the word format would occur if number words evoked stronger activation of verbal codes relative to digit stimuli, perhaps because general reading experience makes the reading response more automatic with number words than digits. Another possibility is that greater experience calculating with digits than with words allows individuals to develop digit-specific inhibitory strategies that reduce interference from irrelevant numberreading processes (see Clark, this volume, for an extensive discussion of inhibitory mechanisms in mental arithmetic). C&c)s emphasis on number-reading mechanisms is supported by the observation that intrusions tend to preserve the position of the intruding operand (e.g. 6 x 9 = sixty three; 9 x 6 = thirty six). This finding suggests that the pair of operands is encoded as if it were a two-digit number, with the left operand encoded as a tens words (sixty) and the right operand as a units word (six). Although other explanations may be possible, the most obvious explanation for why the pair of operands in a horizontally-oriented multiplication problem (e.g., 3 x 6) would be encoded as a single numeral is that there is a strong tendency to read the stimulus as if it was a pair of numbers without a multiplication sign (i.e., 3 x 6 activates the verbal response thirty six, among others). One implication of this account is that number-fact retrieval and numberreading processes cannot truly be said to be functionally independent; rather, this view of operand-priming effects implies that number-reading and number-fact retrieval are integrated processes that are activated simultaneously and compete for common representational structures (e.g., verbal-code representations). Indeed, successful multiplication may normally require inhibition of the relatively automated number-reading response. Based on such considerations, C&C suggested (p. 474) that operand-priming effects provide evidence of format-specific interactions between number-reading and fact-retrieval mechanisms that at least complicate, and perhaps contradict, the simple modular view espoused by MM&W. According to the latter, number reading implicates the comprehension and production modules (e.g., MM&W, p. 499), but should have no direct involvement with processes in the calculation module (cf. MM&W, p. 497).

MM&W's alternative account of operand-priming phenomena Surprisingly, MM&Ws account of operand-priming effects very closely resembles C&C's account. They apparently accept the evidence that operandintrusion errors "often arise from arithmetic fact retrieval processes" (p. 526), and that the same factors "may have contributed to the higher relative frequency of

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intrusions in the word condition" (p. 528). Having acknowledged these points, however, MM&W argue that making such allowances does not necessarily contradict the abstract-modular theory. To account for the various operand-priming phenomena, MM&W argue that the basic theory outlined by C&C can be construed to be consistent with the abstractmodular model. Specifically, MM&W propose (pp. 526-528) that the comprehension system not only encodes the operands individually, but also treats the entire problem as a single numeral. For example, 7 x 6 would give rise to abstract-codes representing 7, 6, and 76 (e.g., (7)lOEXPl (6)lOEXPO). This produces intrusions because "(t)he numeral representation may then interfere with the arithmetic fact retrieval process, leading to activation of answer representations with which it shares one or both quantities" (p. 527). For example, in the case of 7 x 6, abstract representations of answers in the calculation system that contain either (7)lOEXPl or (6)lOEXPO may be directly activated and promoted as intrusions errors. The tendency for intrusions to preserve position (e.g., 7 x 6 = 56) occurs because the exponents in MM&Ws abstract codes correspond directly to position (i.e., lOEXF'l maps on to the tens position and lOEXPO on to the units position). The higher rate of intrusions in the word-format, MM&W suggest, might be due to a stronger tendency for the problem to be treated as a single numeral with word stimuli, because, "for example, seven x sir arguably resembles seventy-sir more closely than 7 x 6 resembles 7 6 (p. 527-528). One important difference between MM&Ws and C&C's explanations of operand-priming effects is that MM&Ws account assumes that intrusions reflect the activation of abstract codes. In contrast, C&C's account proposes that intrusions arise from reading processes that activate verbal codes for number words (e.g., 7 x 6 activates phonological codes for seven, sir, and seventy sir). MM&W claim, however, that their abstract-code account is superior because "the intruding response may occur as a tens word or a teens word, and therefore, "intrusions are apparently best characterized as operand quantity intrusions and not operand name intrusions" (p. 527). This claim, however, is based on an oversimplification of C&C's proposal. According to C&C's account, operand position is preserved in intrusions because position influences whether a tens or a units word is activated by number-reading processes. In other words, the intruding number names may be tens or units words depending on the position of the corresponding operand. A second difference between the explanations is that MM&W attribute the tendency for the pair of operands to be encoded as a single numeral to generic "comprehension processes" (p. 527), rather than to "reading processes," per se. Nonetheless, it seems reasonable to propose that it is long experience with reading

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horizontally presented pairs of digits and number words that tends to promote encoding of the left operand as a tens item and the right operand as a units item. In the absence of the reading hypothesis there seems to be no basis within the abstract-modular theory to predict or explain why the pair of operands would be encoded as a single numeral. As pointed out previously, however, if it is allowed that intrusions are retrieval phenomena, and also allowed that intrusionphenomena arise because of format-specific interference from number-reading processes, then the modularity assumption that calculation relations are functionally separated from basic number reading processes (ix., in the presumed comprehension and production systems) is weakened or contradicted. The following section examines this issue more closely.

Fonnat-specific calculation effects and the merits of modular architecture MM&W's proposed explanation of the operand-priming phenomena unambiguously entails format-specific interactions of comprehension and calculation processes. That is, MM&W allow that operand intrusions are a number-fact retrieval phenomenon (i.e., reflect priming of number-fact representations), and also that format differentially influences this retrieval phenomenon (is., some features of the priming effect are more probable with word stimuli; pp. 527-528). Therefore, within the abstract-modular theory, basic calculation phenomena can differ as a function of number format. Allowing such interactions, however, sharply reduces the ostensible scientific value of assuming a simple architecture with functionally independent comprehension and calculation modules. The boundaries between MM&W's proposed modules are, in effect, defined by the abstract codes through which they communicate (pp. 496-497), and the abstract codes, in turn, are defined as being independent of surface form (p. 497). Allowing format-specific effects of comprehension processes that directly effect calculation processes, obscures, both empirically and theoretically, the boundaries between the modules. The boundaries are obscured empirically because the assumption of abstract codes no longer places constraints on the sorts of effects that can be transmitted between modules. Allowing format-specific interactions with calculation mechanisms (and presumably, therefore, with the proposed production mechanisms as well; cf. p. 508), renders the model capable of accommodating practically any imaginable pattern of format-specific phenomena. Furthermore, allowing formatspecific calculation effects comes close to violating the theoretical principles of functionally independent modules: If format-specific encoding processes in the proposed Comprehension system (whose presumed function is to abstract over

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formats) can have unique, task-specific consequences on retrieval processes in the calculation system (cf. MM&Ws explanation for operand-priming effects, p. 526528), it is not clear what is gained by calling comprehension and calculation "functionallyindependent modules,'' Practically speaking, format-specific influences of comprehension processes on calculation processes means that their functions are not independent. Instead, as the encoding-complex view espouses, such interactions strongly suggest a continuity of encoding and retrieval processes in which features of retrieval processes vary systematically with features of the encoding process. The questionable necessity of the assumption of abstract-quantity codes

Given the evidence of format-specific retrieval effects and other phenomena demonstrating modality-specific representations, C&C raised the question of whether the assumption of abstract codes is necessary (p. 481). Among other points, C&C pointed out that similarity of performance across different number formats does not provide direct evidence for the existence of abstract codes. MM&W recast this observation to make the point that "even a finding of no differences between formats in retrieval processes would apparently not be an embarrassment to the encoding complex position" (footnote p. 510). Although MM&W think that this constitutes an indictment of the encoding-complex view, it is a simple, logical fact that similarity does not demonstrate either the existence of abstract codes or the absence of modality-specific codes. Similarity across formats could mean that performance is based on a common (but not necessarily abstract) code, or that the different formats activate different internal codes that are processed similarly because of common functional or experiential factors. Thus, similarity of performance across number formats, in and of itself, does not necessitate the assumption of abstract codes. MM&W also express puzzlement in their commentary (p. 497) over the view that it might not be necessary to posit any internal representations that are not based on modality-specific codes. The puzzlement, apparently, is over how C&C could not see the self-evident need to hypothesize abstract codes. MM&W state that they assume that "the semantic representations [of numbers] abstract away from... surface details to represent quantity or magnitude" and that it is "uncontroversial that any theory of numerical processing will need to posit some form of internal quantity or magnitude representation" (p. 497). Based on this pair of statements, MM&W conclude that "the answer is clearly 'yes' to [C&C's] question, 'Is the hypothesis of abstract number codes necessary?"' (p. 497).

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This confident conclusion, however, seems to be based on a confusion of the issue of abstraction and the issue of magnitude representation. It certainly is necessary to posit internal processes that represent magnitude, but it is decidedly unclear that it is necessary to assume that they are abstracted processes. Under the encoding-complex view it is assumed that the representation of magnitude can take a variety of forms. For example, a visuo-spatial representation of distance or position (e.g., a number-line) is one possible form of magnitude representation; but associative connections between successive number-words and digits also can support relative magnitude judgements. Under this view, magnitude is not a unitary psychological construct, as assumed or implied by MM&W's abstract-code view; rather, the representation of magnitude corresponds to a set of specific learned relations and processes (e.g., labelling of perceptual groups or intensities, uses of counting and other basic arithmetic relations to represent changes in quantity), Furthermore, under the encoding-complex view, these magnitude skills generally are based on modality-specific codes (e.g., verbal or visuo-spatial representations). Viewed from this perspective, MM&Ws conclusion that the hypothesis of abstract codes is necessary because of the need to posit a representation for magnitude, seems to be only a dogmatic assertion of the abstract-code hypothesis. Magnitude representation does not require or imply the assumption of abstract (i.e., modality independent) codes.

The dubious explanatorypower of the abstract-quantitycodes In the abstract-modular theory, comprehension of a number is practically equated with activation of the appropriate abstract-quantity code. This special code is assumed to represent the "basic quantities in a number, and power of ten associated with each" (MM&W, p. 495). The abstract codes are represented using a notation in which quantity is specified by a digit in brackets, followed by an exponential term specifying the appropriate order of magnitude. For example, the abstract-code notation for 50 is (5)lOEXPl. The information contained in the proposed abstract-codes potentially contributes to many basic number processing skills. For example, the information specified potentially accounts for the ability to encode or produce syntactically and semantically correct strings of numbers. The abstract codes also provide a basis for judgements of absolute and relative magnitude, as well as processing of relations associated with order of magnitude. Sokol, Goodman-Schulman and McCloskey (1989) hinted at the possibility that the abstract-code might also specify the odd-even status of a number (p. 108), providing the explanatory basis for a whole variety of other basic, numerical tasks and judgements.

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However, a fundamental problem with the abstract-code hypothesis is that it is these "built in" assumptions about the properties of abstract codes that are doing much of the important explanatory work. MM&W state that "the internal base-10 representational scheme is built up from more fundamental concepts of quantity, through experience with base-10 number systems" (p. 497), but the nature of these more fundamentalconcepts is left unspecified. Furthermore, such statements leave completely unexplained how the abstract codes represent quantity, and therefore provide no concrete explanation of how they provide the capacity to decide, for example, that 5 is less than 6. In other words, the abstract-code hypothesis does not explain how people comprehend this relation, rather it simply takes for granted that the capacity to process this relation is entailed in the abstract specification. The abstract-code theory of number representation is, in effect, only a reification of the abstract-code notation. In this sense the abstract-code theory seems to evade, rather than explain, how quantity and order of magnitude is represented and processed. Number concepts and the encoding-compler hypothesis Under the encoding-complexview, the "semanticrepresentation of quantity" and the "comprehension of numbers" refer to a variety of specific number skills, rather than to a unitary, symbolic code2. Indeed, the representation of quantity and order of magnitude are complex components of number processing that need to be explained, rather than taken for granted. By focusing on how people use specific types of mental codes to process and represent numerical relations, the encoding-complex approach seeks to provide genuine explanation for these elementary features of numerical skill. For example, positing a visuo-spatial medium for magnitude (e.g., an imagistic line or area) entails positing representational structure that directly mediates the processing or understanding of magnitude (e.g., relative position on the imaginary line). Similarly, the series of verbal associationsunderlyingwell-learned counting strings represent magnitude explicitly in terms of the temporal order of elements in the verbal string. Individual number words and digits gain meaning in terms of their location within counting series, and by referring to specific perceptual representations of quantity

'MM&W now allow that "the model does not contend that all numerical processing is mediated by abstract semantic representations. For example, internal spatial representations may well be

implicated in solving geometry problems" @. 498). It is still unclear, however, what principle allows spatial representations to be functional in geometry, but excludes them from more basic types of number skills.

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or frequency. Processing of base-ten structure may be realized in terms of direct associative mapping between position in a visual string of digits, and related associative structures, such as counting by tens and ones, and other concrete representations of order of magnitude (cf. Fuson, Fraivillig & Burghardt, this volume). Approaching the difficult theoretical problems of quantity representation and knowledge of base-ten structure in such terms potentially provides concrete explanations of how these basic numerical skills are cognitively r e d i d . In contrast, to simply assert that there is an abstract-quantity code that provides these abilities, does not appear to explain these abilities in any substantial way.

Conclusions Whereas the diagram in Figure 1 of MM&Ws chapter does present a simple picture, the structures and processes outlined in that diagram obviously do not constitute the detailed models of specific tasks required to explain the complex and subtle phenomena that occur in number processing. When faced with these sorts of detailed experimental phenomena, the abstract-modular model requires the additional specification of a variety of ad hoc mechanisms and factors. At this level, the abstract-modular mode! appears to be as open to the criticism of underspecification as the encoding-complex approach, and also to be extremely flexible in its capacity to accommodate unpredicted phenomena. Although the abstract-modular theory provides a common-sense taxonomy of numerical skills organized in a plausible arrangement, this global level of architectural detail seems to provide few genuine constraints on the interpretation of data. Indeed, given the weak form of functional independence implied by MM&W’s various elaborations of the model, it appears that the general assumptions of abstract codes and modularity are not open to disconfirmation. Based on these considerations, I conclude, contrary to MM&Ws claims, that the abstract-modular approach is not inherently superior to the encoding-complex approach on the grounds of testability and predictiveness. At present, the abstract-modular approach, like the encoding-complex approach, is perhaps best thought of as a metatheoretical framework, providing specific directions and guidelines for experimental research and for the development of precise theoretical models of components of numerical cognition. Whereas both approaches point to worthwhile theoretical and empirical goals, it remains true that they differ fundamentally with respect to basic assumptions, and the two approaches cannot both be correct with respect to these assumptions: Cognitive number processing is either primarily representationally abstract and modular or it is primarily specific and integrated. It seems very unlikely, however, that a

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single, clear test of these alternative positions is possible; but this does not mean that there will never be a satisfactory resolution of these issues. As the two positions are developed and specified further, and faced with more empirical constraints, gradually the theoretical and practical benefits of one approach versus the other will become clear. Ultimately, theoretical coherence, plausibility, and utility in accounting for the accumulated weight of empirical evidence will settle the question. ACKNOWLEDGEMENTS I express my thanks to Paul Meagher, James Sawchyn, and especially to Valerie Thompson for very useful feedback on a previous draft of this chapter. This research was supported by Natural Sciences and Engineering Research Council of Canada grant OPG0001980 to Jamie Campbell. REFERENCES Campbell, J.I.D. (1990). Error priming in cognitive arithmetic: Effects of number format, Poster presented at the meetings of the Psychonomic Society, New Orleans. Campbell, J.I.D. & Clark, J.M. (1988). An encoding complex view of cognitive number processing: Comment on McCloskey, Sokol, and Goodman (1986). Journal of Experimental Psychology: General, 117, 204-214. Clark, J.M., & Campbell, J.I.D. (1991). Integrated versus modular theories of number skills and acalculia. Brain and Cognition, 17, 204-239. Foltz, G.S., Poltrock, S.E., & Potts, G.R. (1984). Mental comparisons of size and magnitude: Size congruity effects. Journal of Experimental Psychology: Learning, Memoty, and Cognition, 10, 442-453. McCloskey, M., Sokol, S.M., & Goodman, R.A. (1986). Cognitive processes in verbal-number production: Inferences from the performance of braindamaged subjects. Journal of Experimental Psychology: General, 115, 307330. Sokol, S.M., Goodman-Schulman, R., & McCloskey, M. (1989). In defense of a modular architecture for the number processing system: Reply to Campbell Sr Clark. Jounial of Experimental Psychology: General, 118, 105-110. 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 ExperimentalPsychology: Leaming, Memoy, and Cognition, 17,355-376.