What Makes Multiplication Facts Difficult - Hogrefe eContent

What Makes Multiplication Facts Difﬁcult Problem Size or Neighborhood Consistency? Frank Domahs,1,2 Margarete Delazer,3 and Hans-Christoph Nuerk4 1

Interdisziplina ¨res Zentrum fu ¨r Klinische Forschung (IZKF), Universita ¨tsklinikum der RWTH Aachen, Germany 2 Lehr- und Forschungsgebiet Neuropsychologie, Universita ¨tsklinikum der RWTH Aachen, Germany 3 Clinical Department of Neurology, Medical University, Innsbruck, Austria 4 Fachbereich Psychologie, Paris hodron Universita ¨t, Salzburg, Austria

Abstract. Two current models of arithmetic fact retrieval, the network interference theory (NIT; Campbell, 1995) and the interacting neighbors (IN) model (Verguts & Fias, 2005a), predict that errors in simple multiplication should be more probable, if they include the same digit as the correct result (i.e., if they are “consistent,” compared with “inconsistent” errors). In a reanalysis of error data originally reported by Campbell (1997), we provide first empirical evidence for this prediction. Furthermore, these results support the notion of different quantity representations for decades and units as proposed by Nuerk, Weger, and Willmes (2001). However, the NIT and IN-model differ in their explanations of the problem-size effect, a hallmark finding robustly observed in arithmetic fact retrieval. Only the IN-model predicts that a correct answer’s neighborhood consistency can fully account for the problem-size effect, which was confirmed in our analysis. Keywords: multiplication fact retrieval, consistency, operand errors, problem-size effect

Introduction Errors in Simple Multiplication What does a typical error in simple multiplication look like? The overwhelming majority of wrong responses consist of so-called operand errors, belonging to the table of one of the operands (e.g., 3 ⳯ 7 ⳱ 28; 3 ⳯ 7 ⳱ 18). In healthy educated adults, operand errors make up between about two thirds and four fifths of all errors (Campbell, 1997). Yet operand errors do not occur arbitrarily, and they are not equally distributed over all problems. First, one of the core findings in simple arithmetic is the problem-size effect: error rates and response latencies increase when the problems get larger (typically measured as the sum of the operands or size of the result; Campbell & Graham, 1985; LeFevre, Bisanz, Daley, Buffone, Greenham, & Sadesky, 1996; for an overview see Zbrodoff & Logan, 2004). However, the problem-size effect is modulated for two specific types of problems: problems with repeated operands (“ties”; e.g., 3 ⳯ 3) and problems involving 5 as an operand (“five problems”) are less errorprone and can be solved reliably faster than predicted by their problem size (Campbell & Graham, 1985; LeFevre et al., 1996; Siegler, 1988). Most of the operand errors are numerically close to the 1

correct result. In fact, about 73% of all operand errors reported by Campbell (1997) involved an error distance (“operand split”) of Ⳳ 1, about 22% involved operand splits of Ⳳ 2, and only the remaining 5% involved larger operand splits. Operand errors are also much more likely to be related to the smaller (min) than to the larger (max) operand (e.g., 4 ⳯ 9 ⳱ 32 is more likely than 4 ⳯ 9 ⳱ 27), regardless of operand order. Operand errors are particularly likely if both operands are related to the same incorrect answer, thus leading to convergent erroneous activation (Campbell, 1997; Campbell & Graham, 1985). Operand errors may co-occur together with “naming errors” (e.g., 4 ⳯ 8 ⳱ 4, 8, or 48) or “operand-intrusion errors” (e.g., 4 ⳯ 8 ⳱ 28), meaning that at least one of the operands intrudes into the result. Operand-intrusion errors1 can be interpreted as interference from reading the operands as a two-digit number at a certain level of encoding. This explains why the operand order in this case plays a role, the left operand being primarily associated with the decade position and the right operand being primarily associated with the unit position of a two-digit result (Campbell, 1995, 1997; Campbell & Clark, 1992). There is evidence that nonretrieval plausibility strategies can be used at least to check multiplication results. In a verification paradigm, Lochy, Seron, Delazer, and Butterworth (2000) found that subjects accepted multiplication

The term “operand-intrusion error” will be used to denote both classes of errors because naming errors can be regarded a subgroup of operand intrusions.

䉷 2006 Hogrefe & Huber Publishers

Experimental Psychology 2006; Vol. 53(4):275–282 DOI 10.1027/1618-3169.53.4.275

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Domahs et al.: Neighborhood Consistency in Multiplication

results faster when they were even than when they were odd. The reverse was true when wrong results were to be rejected. This finding may be linked to a “high familiarity” with even results, based on the fact that three quarters of all multiplication results are even (see Lemaire & Reder, 1999, for a different interpretation). Finally, research by Campbell (Campbell, 1991; Campbell & Arbuthnott, 1996) has shown that errors can be primed by answers given to preceding problems in the sequence of trials. This repetition priming can be negative (inhibiting the result just stated before) or positive (facilitating the results stated about 2 to 10 trials before). In sum, it is well known that the typical multiplication error occurs more often with large than with small problems. Furthermore, it is operand-related and numerically close to the correct result and may be influenced by some additional variables, such as parity, the surface form of operand presentation, and the sequence of trials presented. We provide in this article first empirical evidence that the erroneous retrieval of multiplication facts is strongly influenced by another variable, decade consistency, which has not been considered by previous empirical research. The influence of decade consistency is, however, predicted by two current models of fact retrieval in multiplication (Campbell, 1995; Verguts & Fias, 2005a). We test the predictions of these models and evaluate their account for the data. Most importantly, we show in the data at hand that the hallmark effect in simple multiplication, the problemsize effect, can be completely drawn back to decade consistency, as predicted by Verguts and Fias (2005a). This finding has important theoretical implications in that it seems to suggest that the problem-size effect in multiplication is confounded with decade consistency, and it is therefore conceptually different from magnitude effects in other numerical tasks.

Decade-Consistency Effects and the Representation of Multiplication Facts Because operand errors are so prominent, they form one of the core aspects of multiplication performance to be explained in cognitive models. Indeed, virtually all current models can account for them quite naturally, given that they agree on the assumption of associative relationships between multiplication problems and their solutions (Ashcraft, 1987; Campbell, 1995; McCloskey & Lindemann, 1992; Rickard, 2005; Siegler, 1988; Verguts & Fias, 2005a). However, most of the models mentioned are silent about the probability of an operand error to be produced above instead of below the correct result (e.g., 3 ⳯ 7 ⳱ 24 instead of 3 ⳯ 7 ⳱ 18) or vice versa. Yet it is possible to derive some predictions about the direction of operand errors (matched for all the factors mentioned earlier) from two current models on multiplication fact retrieval (Campbell, 1995; Verguts & Fias, 2005a). The interacting neighbors (IN) model of Verguts & Fias (2005a; see also Verguts & Fias, 2005b) adopts the notion that two-digit numbers (as most simple multiplication products) are represented in separate decade–unit compoExperimental Psychology 2006; Vol. 53(4):275–282

nents (Nuerk, Weger, & Willmes, 2001; Nuerk & Willmes, 2005). Thus, at a certain stage of production, information about the answer to simple multiplication problems is thought to be split into decade-related and unit-related information. Similar to assumptions in language processing (Segui & Grainger, 1990), the IN-model assumes that the success of producing the correct answer to a simple multiplication problem depends on the consistency of its decade and unit digits with the decade and unit digits of closely related operand errors (i.e., whether product and operand errors activate the same decade or unit position). Products that are highly consistent with neighboring answers can be retrieved faster and more accurately than products that are less consistent. According to Verguts and Fias (2005a), the relative advantage with respect to error rates and response times for problems with repeated operands and problems including 5 as one of the operands (“tie effect” and “five effect”) can be explained with a more consistent neighborhood for those problems: ties have less neighbors in general, and because neighbors tend to be inconsistent rather than consistent, less competition is predicted for tie problems. For all five-problems there are related false answers with an operand split of 2 including the same unit digit as the correct result (unit-consistent answers; e.g., for the problem 7 ⳯ 5, the correct result would be 35, and the unit-consistent operand errors would be 25 and 45). Although the influence of these “far” neighbors is assumed to be small, it could account for the five effect. Extending the consistency account to operand errors, the following prediction can be derived: production of incorrect answers should be more probable if one of the digits is consistent with the correct product than if both are inconsistent with it. Similar ideas have been put forward in the modified network interference theory (NIT) by Campbell (1995). Crucially, the NIT includes the assumption of decade and unit positions being separately activated by multiple nodes competing within a network for control of production mechanisms. At least a subset of operand errors is explained by competing activation of the decade position from the correct result itself and from neighboring answers. Although both the NIT and the IN-model predict an influence of a correct answer’s neighborhood on the probability of specific operand errors, the models differ in their explanation of the problem-size effect typically found in simple multiplication. Within the IN-model, the problemsize effect can be explained by different consistency characteristics in the neighborhood of small and large problems: products of small problems tend to be more consistent with their neighboring answers than products of large problems, and thus small problems’ solutions are retrieved faster and more successfully than large problems’ solutions. Thus, problem size may cease to be a predictive variable if neighborhood consistency is taken into account. Within the NIT, the problem-size effect is traced back to a compressed quantity representation (“magnitude code”), similar to the mental number line proposed by Dehaene (2003). Accordingly, a large magnitude code would activate more problems than a small magnitude code, leading to more interference and inferior performance for large 䉷 2006 Hogrefe & Huber Publishers


problems. This is a crucial theoretical difference between the two models. The NIT assumes that the problem-size effect in multiplication is due to the same magnitude representations employed in other numerical tasks (e.g., number comparison). By contrast, the IN-model assumes that a specific attribute of multiplication tables, neighborhoodconsistency characteristics, results in the problem-size effect, because these consistency characteristics are confounded with problem size. Consequently, the IN-model assumes that the problem-size effect in multiplication has little in common with the magnitude effects in other tasks (such as number comparison) although it bears a similar name. A number of other accounts have also been proposed to explain the problem-size effect in simple multiplication, including frequency of occurrence (Ashcraft, 1987; Ashcraft & Christy, 1995), age and order of acquisition (Campbell & Graham, 1985), procedural errors during acquisition (Siegler, 1988), and others. Because of space limitations, they cannot be discussed here in any detail.2 Crucially, however, the IN-model of Verguts & Fias (2005a) is the only one predicting that problem size may disappear as an independent factor once consistency is taken into consideration. The present study aims to investigate two main predictions of the IN-model: (1) Does consistency affect the erroneous choice of an operand-related answer in a way that decade-consistent operand errors are more likely to be produced than otherwise comparable decade-inconsistent operand errors (also compatible with the NIT account), and (2) can the consistency characteristics of a correct answer’s neighborhood explain error probability without the need of problem size as an independent variable (uniquely predicted by the IN-model)? To answer these questions, we performed a reanalysis on error data reported by Campbell (1997). Question 1 is tackled by a categorical analysis; question 2 is addressed by a regression analysis.

Categorical Error Analysis: Are There Decade-Consistency Effects? Method Our analysis is based on data published by Campbell (1997). He examined 44 psychology students who solved the standard set of 36 simple multiplication problems (operands from 2 to 9) six times (three times in each operand

2 3

4

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order). Altogether, this procedure resulted in a database of 132 trials per problem, 9,504 trials altogether. Overall, 737 (7.8%) wrong responses were recorded, 645 (87.5%) of them being operand-related. In our analysis only problems leading to at least one operand error with a maximum operand split of Ⳳ 2 from the correct result were included because this split was considered to define neighborhood by Verguts and Fias (2005a). Furthermore, problems leading to a one-digit correct result were excluded because it was unclear how to calculate decade consistency in these problems. For the remaining problems, decade consistency was classified for all their operand errors with a maximum operand split of Ⳳ 2 operands from the correct result. An operand error was classified to be consistent if its decade matched the decade of the correct result. It was classified as inconsistent if there was no match in the decade position.3 For every problem m ⳯ n this gives four pairs of decade-consistency classifications (m Ⳳ 1, m Ⳳ 2, n Ⳳ 1, n Ⳳ 2). Only pairs of operand errors with differing decade-consistency classifications entered our analysis. For this reason not every pair of operand errors was analyzed, but for some problems more than one pair of operand errors was included4 (see Table 1). For example, the following three pairs of operand errors of the problem 7 ⳯ 4 (m ⳱ 7; n ⳱ 4) could have entered the analysis: m Ⳳ 1: 32 (inconsistent) versus 24 (consistent); m Ⳳ 2: 36 (inconsistent) versus 20 (consistent); n Ⳳ 1: 35 (inconsistent) versus 21 (consistent), whereas the pair n Ⳳ 2 was not considered because both operand errors (42 vs. 14) are inconsistent with the correct product. The prediction for the three “critical” pairs of this specific problem would be that operand errors above the correct product are much less probable than operand errors below the correct product because all above errors are inconsistent and all below errors are consistent with the target answer. Furthermore, error pairs of ties (e.g., 4 ⳯ 4) were only considered once. Altogether, this procedure led to a set of 31 “critical” pairs of operand errors with differing decade-consistency classifications, representing a total of 141 wrong answers (see Table 1).

Results Whereas inconsistent operand errors accounted for 48 of the 141 (34.0%) “critical” wrong responses in the subjects’ performance, consistent operand errors made up 93 of the 141 wrong responses (66.0%). This difference reached statistical significance (Wilcoxon exact p ⳱ .0263). Results

For more detailed discussions, see Verguts and Fias (2005a) or Zbrodoff and Logan (2004) and references cited therein. In principle, the IN-model and the NIT also predict consistency effects for the unit position. However, only very few neighboring operand errors show a unit consistency with the correct result. Therefore, we focus on decade consistency in the present analyses. There is one important exception worth mentioning: Five-related problems’ errors Ⳳ 2 are unit-consistent with the correct result, containing a 0 or 5 unit. Indeed, this unit consistency for “far” neighbors was made responsible for the five-effect in multiplication fact retrieval (Verguts & Fias, 2005a). For the problem 6 ⳯ 3, one pair of operand errors (24 and 12) can ambiguously be classified as n Ⳳ 1 or m Ⳳ 2. This specific pair of operand errors was only included once.


Experimental Psychology 2006; Vol. 53(4):275–282

Operand

n m m n m m n n m n m n n m n m n m m n m m n m n m n m m m n

m⳯n

6⳯2 2⳯6 6⳯3 6⳯3 3⳯6 7⳯3 3⳯7 4⳯5 7⳯4 4⳯7 8⳯4 4⳯8 6⳯5 5⳯6 5⳯6 7⳯5 5⳯7 8⳯5 5⳯8 5⳯8 7⳯6 6⳯7 6⳯7 8⳯6 8⳯6 6⳯8 6⳯8 4⳯4 7⳯7 6⳯2 2⳯6 Sum

Error Characteristics

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2

Dist. 18 18 21 24 24 24 24 24 32 32 36 36 36 36 35 40 40 45 48 45 48 49 48 54 56 56 54 20 56 16 16

Above 6 6 15 12 12 18 18 16 24 24 28 28 24 24 25 30 30 35 32 35 36 35 36 42 40 40 42 12 42 8 8

Below

Error Pair

0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1 5

Above 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 0 0 6

Below

Operand Intrus.

8 2 2 2 1 1 0 1 3 2 4 3 1 2 0 2 2 6 3 4 1 3 1 2 5 2 2 1 0 1 3 70

Above 1 1 0 2 1 0 1 0 2 6 0 1 0 0 1 5 3 0 0 0 6 1 7 7 2 1 12 1 7 1 2 71

Below

Error Freq.

8 2 0 2 1 1 0 1 2 6 4 3 1 2 0 5 3 6 3 4 1 3 1 7 2 1 12 1 7 1 3 93

Cons. 1 1 2 2 1 0 1 0 3 2 0 1 0 0 1 2 2 0 0 0 6 1 7 2 5 2 2 1 0 1 2 48

Incons.

Error Freq.


3 0

0 4 1

2

1 2

3 0 6

5

2 17

3

2

3 23

2

No Intrus.

1

Intrus.

Error Freq.

Table 1. Decade consistency and error frequency in the “critical” pairs of operand errors. Highlighted errors are decade-consistent with the correct result. For the operand-intrusion columns, 1 ⳱ intrusion error possible; 0 ⳱ intrusion error not possible. Intrus. ⳱ intrusion; Freq. ⳱ frequency; Dist. ⳱ distance (operand split); Cons. ⳱ consistent; Incons. ⳱ inconsistent

278 Domahs et al.: Neighborhood Consistency in Multiplication



remained principally the same when one-digit answers and teens were excluded from analysis.5 No general systematic preference for a specific error direction could be observed: within the set of critical pairs, operand errors above the correct result were produced equally often (70/141) as below the correct result (71/141). An analysis was performed analogous to the one described for decade consistency to examine possible effects of operand-intrusion errors. In 81 out of 136 (59.6%) cases, participants produced an error that included one of the operands in the “corresponding” position (i.e., the first operand intruded into the decade position and the second operand into the unit position), and in the remaining 55 out of 136 (40.4%) cases, the answer did not contain such a potential operand-intrusion error. This difference failed to reach statistical significance, which was also true for the comparison of possible operand-intrusion versus nonintrusion errors within the set of trials “critical” for decade consistency. The results remained principally the same in an analysis excluding answers in the teens range. However, for both intrusion and nonintrusion answers there were more decade-consistent than inconsistent errors per trial (see Figure 1).

Regression Analysis: The Contribution of Neighborhood Consistency and Problem Size Method In this regression analysis, we included all items and all errors of the Campbell (1997) data without any a priori

279

selection of items. The dependent variable used was number of errors per item. The independent variables included in the stepwise regression were problem size (in terms of the correct result),6 number of consistent neighbors, number of inconsistent neighbors, number of all neighbors (all neighborhood measures were in the range of Ⳳ 2 operands), parity of the correct result (even ⳱ Ⳳ2, odd ⳱ Ⳳ1), tie status (tie ⳱ 1, no tie ⳱ 0), five table problems (five table ⳱ 1, no five table ⳱ 0), and relative amount of possible intrusion errors (neighbors that can be regarded as intrusion errors divided by all neighbors).

Results The results of this regression analysis are intriguing. In a stepwise regression over number of errors (forward and backward mixed-step procedure), the final model was fairly predictive (R ⳱ .76; adjusted R2 ⳱ .56) with three significant predictors: errors became less likely with a higher number of consistent neighbors (standardized b ⳱ ⳮ.56), with tie (b ⳱ ⳮ.42), and five problems (b ⳱ ⳮ.33). Crucially, problem size was added as the first predictor in this stepwise regression analysis because it correlated most strongly with the number of errors, thus corroborating earlier observations. However, after the number of consistent neighbors was added to the model by the regression procedure, problem size did no longer explain any additional variance and was dropped out of the model (p ⳱ .24 for the partial correlation). Thus, this analysis seems to indicate that problem size explains the likelihood of errors well, but only if neighborhood consistency is not considered. As soon as it is considered, problem size ceases to explain any variance. At first glance, the fact that more consistent neighbors led to fewer errors seems at odds with the analysis just noted, which showed more consistent than inconsistent errors. However, both findings are perfectly in line with the IN-model and can be integrated as follows: high neighborhood consistency leads to fewer errors overall, but for a specific multiplication item and two neighbors with equal distance to this item—one consistent and one inconsistent—the consistent neighbor is more likely than the inconsistent one to be erroneously produced.

Discussion Figure 1. Error rates as a function of decade consistency for answers containing or not containing a potential operand intrusion. 5

6

The present analyses provide first empirical evidence that operand errors in multiplication are much more likely to occur when they are decade-consistent with the correct result than when they are inconsistent with it. Moreover, a

Concerning one-digit operand errors, it seemed unclear whether to treat their “zero decade” as just another (nonexpressed) decade digit. As far as teens are concerned, they mainly differ from other two-digit numbers with regard to the order of their digits in verbal production (e.g., Arabic: 16, with the “6” last; verbal: sixteen, with the “six” first). Furthermore, some of their verbal forms are partly or completely opaque with respect to their decade or unit value or position (e.g., fifteen, twelve). Are consistency effects modulated by opacity or unit– decade inversion? For instance, in solving the problem 5 ⳯ 5, does the 5 in the operand error fifteen produce the same consistency effect as the 5 in the operand error thirty-five? Unfortunately, Verguts and Fias (2005a) do not address these questions in their description of the IN-model. The correct result was used as measure of problem size because this variable was shown to be the best predictor for the problem-size effect (Stazyk, Ashcraft, & Hamann, 1982).



280


Table 2. Overview of regression models for the overall error probabilities per item. See method section for a detailed description of the predictor variables included. Multiple R Model 1

0.57

Corrected

Significance of Change

R2

df

F

p

0.31

62

29.96

0.00

Problem Size Model 2

0.64

0.39

61

8.65

0.71

0.48

60

11.55

0.77

0.56

59

11.73

0.76

0.56

59

Tie Status Five Table Consistent Neighbors

consistency measure of each correct answer’s neighborhood can successfully predict the error rate for that problem without referring to problem size. Given our careful data selection and additional analyses performed, we could exclude relevant factors as possible confounds. These factors, which are known to influence error production in simple multiplication (see the article’s introduction) include a problem’s size or its tie, five, and parity status as well as split, min-, or max- relatedness; “double” activation; or possible operand intrusions into related wrong answers. Repetition or error-priming effects should be minimized because of the randomized stimulus presentation in the original experiment (Campbell, 1997). Our findings of decade-consistency effects in error probability are in line with both the NIT of Campbell (1995) and the IN-model of Verguts and Fias (2005a). Not only should correct results be produced faster and more successfully if they are consistent with operand errors (Verguts & Fias, 2005b), but operand errors should also be more probable if they are decade-consistent with the correct result. Furthermore, the present findings are in line with the notion of different quantity representations for decades and units as proposed by Nuerk and Willmes (Nuerk, Weger, & Willmes, 2001; Nuerk & Willmes, 2005).7 However, other models of fact retrieval in simple multiplication

7

1.38

Raw Correlation

0.57

5.47

0.00

0.57

0.60 ⳮ0.29

6.08 ⳮ2.94

0.00 0.00

0.57 ⳮ0.23

0.58 ⳮ0.32 ⳮ0.31

6.34 ⳮ3.48 ⳮ3.40

0.00 0.00 0.00

0.57 ⳮ0.23 ⳮ0.32

0.17 ⳮ0.40 ⳮ0.33 ⳮ0.51

1.18 ⳮ4.59 ⳮ3.85 ⳮ3.42

0.24 0.00 0.00 0.00

0.57 ⳮ0.23 ⳮ0.32 ⳮ0.57

ⳮ0.42 ⳮ0.33 ⳮ0.66

ⳮ4.86 ⳮ3.95 ⳮ7.58

0.00 0.00 0.00

ⳮ0.23 ⳮ0.32 ⳮ0.57

0.00

Problem Size Tie Status Five Table Consistent Neighbors Model 5

p

0.00

Problem Size Tie Status Five Table Model 4

t value

0.00

Problem Size Tie Status Model 3

Standardized Beta

0.24

in their present state do not explicitly predict the kind of systematic patterns found in our present analyses (Ashcraft, 1987; McCloskey & Lindemann, 1992; Rickard, 2005; Sharer & Siegler, 1998; Siegler, 1988). In principle, though, it seems possible to adapt most of these models to incorporate some device leading to the described decadeconsistency effects. Both the IN-model and the NIT correctly predict decadeconsistency effects in multiplication. However, the two models clearly differ in their predictions regarding the relationship between problem size and neighborhood-consistency measures. In fact, the problem-size effect could be accounted for by neighborhood consistency, as predicted by the IN-model. However, the NIT, which accounts for the problem-size effect by a compressed magnitude representation (used for almost any numerical task), cannot explain why the problem-size effect disappears when neighborhood consistency is considered. This is also true for other current models of fact retrieval despite their different accounts for the problem-size effect. Importantly, the problem-size effect in multiplication seems to be different from the magnitude effects in other numerical tasks (e.g., number comparison). Thus, although the problemsize effect in multiplication and the magnitude effects in different tasks (e.g., number comparison) bear a similar

Decomposed processing of tens and units in magnitude comparison (as reviewed by Nuerk & Willmes, 2005) and decade–unit consistency effects in multiplication may—to some extent—be related. Both findings suggest that two-digit numbers are not represented holistically, but rather decades and units (and their neighborhood) are represented separately. In this sense, the present study sheds new light not only on the problem-size effect in multiplication, but also on the nature of the representation of two-digit numbers in general.




name, their representational grounding may be very different from one another.8 At which level of processing can the consistency effect be located? Verguts and Fias (2005b) themselves state that it is hard to discriminate between a semantic and a phonological locus, and the IN-model was compatible with either of these possibilities. Although the present data are far from being decisive, the fact that all critical error pairs in our analyses were operand-related points to a semantic involvement. A purely phonological account would predict a substantial number of unrelated errors, as long as they share digits with the correct result. According to the IN-model, large tie-problems may lead to faster and more accurate answers than expected based on their problem size—because they have less inconsistent neighbors than comparable non-tie problems. Similarly, five problems should have a processing advantage because of the higher unit consistency of their neighbors. Although this explanation of Verguts and Fias (2005a, 2005b) may be qualitatively correct, it is not consistent with the quantitative regression analyses of the present data. Why did the tie and five effects not disappear once neighborhood consistency entered into the regression? With respect to the five effect, the answer is quite obvious: whereas Verguts and Fias (2005a) claim that this effect is due to higher unit consistency of five-problems, we have focused on decade consistency in our analyses (see footnote 3). However, the answer is less clear concerning the tie effect. Maybe this effect is not only semantic (affecting the actual retrieval of arithmetic facts) but also encoding-based, as indeed argued by several authors (Blankenberger, 2001). To conclude, the typical error in multiplication is not only operand-related and close to the correct result. Rather, it tends to be decade-consistent with the correct answer. Moreover, the classical problem-size effect in arithmetic fact retrieval was—at least in the data at hand—confounded with neighborhood consistency, which completely explained its variance. Thus, as predicted by Verguts and Fias (2005a), the hallmark effect of simple arithmetic fact retrieval, the problem-size effect, may just be an artifact. Acknowledgments This project was supported by funding from the Interdisciplinary Centre for Clinical Research “BIOMAT” (project VVZ 51), by the START program of the RWTH Aachen University, by the DFG grant KFO 112/TP2 supporting Hans-Christoph Nuerk, and the EU Marie-Curie RTN “NUMBRA” proposal Nr. 504927.

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Frank Domahs Universita¨tsklinikum der RWTH Aachen Abteilung Neuropsychologie Pauwelsstr. 30 D-52074 Aachen Germany Tel. Ⳮ49 24 180 89909 E-mail [email protected]
