To carry or not to carry — Is this the question? Disentangling the carry effect in multi-digit addition

Acta Psychologica 135 (2010) 67–76

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Acta Psychologica j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a c t p s y

To carry or not to carry — Is this the question? Disentangling the carry effect in multi-digit addition Elise Klein a,b,c,⁎,1, Korbinian Moeller c,1, Katharina Dressel a,b, Frank Domahs a,b, Guilherme Wood d, Klaus Willmes a,b, Hans-Christoph Nuerk c a

Section Neuropsychology, Department of Neurology, University Hospital, RWTH Aachen University, Germany Interdisciplinary Center for Clinical Research “BioMAT.”, RWTH Aachen, Germany Department of Psychology, Eberhard Karls University, Tuebingen, Germany d Department of Psychology and Center of Neurocognitive Research, Paris-Lodron University of Salzburg, Austria b c

a r t i c l e

i n f o

Article history: Received 11 April 2010 Received in revised form 31 May 2010 Accepted 3 June 2010 Available online 26 June 2010 PsycINFO classification: 2360 Cognitive Processing Keywords: Mental arithmetic Addition Carry-over Problem size

a b s t r a c t Recent research has suggested addition performance to be determined by both the need for a carry operation and problem size. Nevertheless, it has remained debatable, how these two factors are interrelated. In the current study, this question was pursued by orthogonally manipulating carry and problem size in two-digit addition verification. As the two factors interacted reliably, our results indicate that the carry effect is moderated by number magnitude processing rather than representing a purely procedural, asemantic sequence of processing steps. Moreover, it was found that the carry effect may not be a purely categorical effect but may be driven by continuous characteristics of the sum of the unit digits as well. Since the correct result of a carry problem can only be derived by integrating and updating the magnitudes of tens and units within the place-value structure of the Arabic number system, the present study provides evidence for the idea that decomposed processing of tens and units also transfers to mental arithmetic. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Addition is one of the most basic arithmetical operations. There is evidence that preverbal infants are already capable of comprehending addition relationships within small sets of items (Wynn, 1992). Addition is also the first arithmetical operation taught at school. It is assumed that simple additions may be learned by heart, but more complex additions are supposed to be solved using specific procedures (Dehaene, Piazza, Pinel, & Cohen, 2003). However, the question remains: What properties qualify a given addition problem to be either simple or complex? Some authors define addition problems as being simple when they consist exclusively of single-digit operands and/or sums (e.g., Geary & Widaman, 1987; Stanescu-Cosson et al., 2000), whereas in other approaches, the presence of a carry operation is assumed to distinguish sufficiently between simple and complex additions (e.g., Kong et al., 2005). Moreover, difficulty has been assumed to increase continuously with both the magnitude of the addends involved as well as a decreasing difference between the correct result and a simultaneously presented distractor (e.g., Ashcraft & Stazyk, 1981). However, in recent years more ⁎ Corresponding author. Eberhard Karls University Tuebingen, Institute of Psychology, Friedrichstrasse 21, 72072 Tuebingen, Germany. Tel.: +49 7071 2976112. E-mail address: [email protected] (E. Klein). 1 These authors contributed equally to this manuscript and should therefore be considered shared first authors. 0001-6918/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.actpsy.2010.06.002

and more research has been devoted to the question which particular characteristics of a given addition problem determine its difficulty. As the current study aims at pursuing this question, important sources for problem difficulty in addition will be reviewed briefly. 1.1. Multi-digit addition problem difficulty Previous studies have indicated that addition performance is largely determined by two factors: (a) the magnitude of the addends involved (henceforth referred to as problem size) and (b) the need for a carry operation: (a) Problem size: Response latencies and error rates increase considerably as the size of the addends and the result increases (e.g., 8 + 6 = 14 is less difficult than 38 + 46 = 84). This so-called problem size effect in addition was first reported by Groen and Parkman (1972; see also Ashcraft & Battaglia, 1978; Ashcraft & Stazyk, 1981; Ashcraft, 1995 for a review; LeFevre, Sadesky, & Bisanz, 1996), but has once again received interest in recent years (e.g., Deschuyteneer et al., 2005; Stanescu-Cosson et al., 2000). (b) Carry operation: Addition problems requiring a carry operation (e.g., 28+ 36= 64, henceforth referred to as carry problems) are usually associated with longer response latencies and higher error rates than addition problems not requiring a carry (21 + 43= 64; henceforth referred to as non-carry problems;

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e.g., Deschuyteneer, De Rammelaere, & Fias, 2005; Fürst & Hitch, 2000; Imbo, Vandierendonck, & De Rammelaere, 2007; Kong et al., 2005). Basically, the necessity of a carry operation is determined by the unit digits of the addends in the respective addition problem: Whenever the sum of the unit digits of two addends is equal to or larger than 10 (e.g., 28 + 36; unit sum 8 + 6 = 14), a carry operation is mandatory. Contrarily, no carry is needed whenever the sum of the units is smaller than 10 (e.g., 21 + 43; unit sum 1 + 3 = 4).2 The carry operation is executed by adding 1 (representing the decade digit of the unit sum) to the sum of the decade digits of the addends. For the above example 28 + 36, the sum of the unit digits is 14, so the unit digit of the result (i.e., 64) is 4 and the decade digit of the result is derived by increasing the sum of the decade digits of the addends by 1 (i.e., the decade digit of the unit sum: 2 + 3 + 1 = 6). 1.2. The interrelation of problem size and carry operations So far, research has been mainly focused on evaluating the impact of either the need for a carry operation or problem size on addition performance in isolation. To date there is no well-controlled study investigating the possible interplay between these two factors. For instance, from the study by Kong et al. (2005), no evidence regarding the interrelation of carry operations and problem size can be derived, since problem size was not varied systematically but only matched between the respective item categories. Additionally, the authors based their conclusions on a stimulus set of only 12 items. However, even when problem size was varied, often problem size and the need for a carry were not necessarily manipulated orthogonally, even though the need for a carry had already been identified using regression analyses to determine addition performance (e.g., Widaman, Geary, Cormier, & Little, 1989). In this context, studies either focused on the manipulation of problem size and the split between the correct result and the incorrect solution probe presented (e.g., Ashcraft & Stazyk, 1981), on the manipulation of problem size and the number of single-digit numbers involved (e.g., Widaman et al., 1989), or on the manipulation of problem size and the size of the smallest summand involved (e.g., Groen & Parkman, 1972). In other cases in which problem size was manipulated, it was often confounded with the need for a carry. For instance, in a study by Stanescu-Cosson et al. (2000) all addition problems with large problem size also were carry problems, whereas problems with small problem size never required a carry operation (see also Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999; Kucian et al., 2006; Lemer, Dehaene, Spelke, & Cohen, 2003, for a similar confound). First evidence for a possible interaction between carry and problem size (i.e., a larger carry effect for relatively larger problems), inspiring the present more detailed evaluation of this interrelation, came from a regression analysis in the study by Deschuyteneer et al. (2005). The regression analysis indicated that problem size (measured as the logarithm of the correct result) was a stronger predictor of response latency for addition problems requiring a carry than for non-carry problems, providing evidence for an interactive interrelation. However, in our opinion, this analysis was not fully conclusive. First, their regression analysis was based on a limited and different number of data points for carry and non-carry problems, thereby possibly triggering strategic effects due to an over-representation of certain trial types resulting from the repetition of specific item properties.3 Second, the

2 Note that problem size – conceptualized in terms of the magnitude of the result – is the same in both examples. 3 Please note that the 84 carry and non-carry problems resulted in only 6 and 7 possible sums of which 4 included 5 at the unit position. Thereby, 12 carry resp. 11 non-carry problems summed up to the same result. Since there is only a limited number of possible combinations of units for a given non-carry result (e.g., for 55 only units from 0 to 5) one cannot be sure whether or not participants indeed employed calculation for each and every problem.

proposed interaction of carry and problem size was not directly investigated by orthogonally manipulating these two factors, but was instead inferred rather indirectly from the evaluation of separate and different analyses (i.e., ANOVA vs. regression analysis). Taken together, there are no systematic studies with orthogonally manipulated stimulus material (i.e., matched for potentially confounding factors), which investigates the kind of interrelation, that is, additive or interactive, between carry and problem size. On the one hand, the need for a carry may be a purely procedural factor while problem size may be associated with the processing of the addends' magnitude. On the other hand, there may be an interaction of carry and problem size when, for instance, the need for a carry would also involve the processing of number magnitude information. 1.3. Separate and additive processing of problem size and carry From the triple-code model of Dehaene and Cohen (1995), it can be derived that multi-digit arithmetic problems (including carry and non-carry addition problems) are solved via sequentially processing the single-digit problems at the unit positions, the decade positions, the hundred positions etc. (see also Cohen & Dehaene, 2000). In this context, Dehaene and Cohen (1995) also assumed that this sequential (column-wise) processing of single-digit problems requires cognitive control for planning and monitoring the progression (see also Widaman et al., 1989; Geary & Widaman, 1987). Thus, correctly solving carry problems may require both the processing of single-digit magnitude as well as specifically monitoring the processes involved. Neurocognitive studies have shown that additional (pre)frontal cortical areas are recruited for more difficult problems (e.g., Roland & Friberg, 1985, Gruber et al., 2001) such as the processing of carry operations (e.g., Kong et al., 2005). However, activation within these areas does not necessarily seem specific for number magnitude processing itself (cf. Pesenti et al., 2000). Instead, this activation may rather reflect more general processes involved in complex calculation such as working memory (e.g., Ashcraft & Kirk, 2001; Fürst & Hitch, 2000; Imbo, Vandierendonck and De Rammelaere, 2007; Imbo, Vandierendonck and Vergauwe, 2007; Seitz & Schumann-Hengsteler, 2002) or processes of cognitive control (e.g., Miller, 2000). In contrast, the neural correlate of magnitude processing is more explicitly confined to a specific region: Processing magnitude information has been repeatedly attributed to parietal cortex around the intraparietal sulcus (see Dehaene et al., 2003; Dehaene, 2009 for reviews). Taking into account the different underlying processes and their distinct neural correlates, it seems plausible to assume that problem size and carry may represent two distinct cognitive operations. Moreover, this argument was corroborated by empirical evidence from two-digit addition for which Deschuyteneer et al. (2005) were able to dissociate the effects of carry and problem size behaviourally: the authors observed a standard carry effect but no effect of problem size when explicitly investigating the sequential processing of the single-digit problems constituting their two-digit addition problems. 1.4. An alternative account Generally, it has to be noted that in studies about the effect of carry operations on task performance, the factor carry was frequently confounded with other variables such as problem size (see above, e.g., Stanescu-Cosson et al., 2000; Dehaene et al., 1999; Kucian et al., 2006; Lemer et al., 2003; see Klein, Nuerk, Wood, Knops, & Willmes, 2009 for a more detailed discussion). Therefore, the findings of the studies mentioned above should be interpreted with caution. In turn, the assumed general link of carry operations with increasing frontal activation does not seem mandatory. Corroborating this consideration, there is also empirical evidence suggesting an interactive relation between the need for a carry and problem size. In this view, the sequential solution of single-digit

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problems (in order to determine the result of the overall problem) may not be accounted for entirely by (asemantic) processes of cognitive control. Instead, decomposition of a multi-digit problem into its constituting single-digit problems may reflect a processing strategy provoked by structural attributes of the way numerical magnitude is semantically represented: the base-10 place-value structure of the Arabic number system. According to this place-value structure, the magnitude representation of multi-digit numbers is organized in powers of the basis 10 (i.e., 328 = {3} × 102 + {2} × 101 + {8} × 100). In this conceptualization, adding two numbers basically means to add the multipliers at the corresponding power level of the two addends. Within this framework, a carry operation would become necessary whenever the sum of the multipliers is equal to or larger than 10. In such a case, the decade digit of, for example., the unit sum would have to be carried to the next power level, implying that the need for a carry would result in additional processes of magnitude manipulation and place-value (i.e., unit-decade) integration to compute the correct result (see Fig. 1). Previously, processing number magnitude (for a review see Dehaene et al., 2003) as well as place-value integration have both been associated with IPS activation (Knops, Nuerk, Sparing, Foltys, & Willmes, 2006; Wood, Nuerk & Willmes, 2006, see also Goebel, Johansen-Berg, Behrens, & Rushworth, 2004 for a distinction between the neural correlates of processing single- and two-digit numbers). Thus, the carry from one position (e.g., units) to the next higher power of ten (e.g., tens) within the place-value structure may reflect a special case of number magnitude processing. In line with this argument, Klein et al. (2009) observed that the need for a carry did not result in an increase of frontal activation; instead, the most pronounced change in the fMRI signal was found within the intraparietal sulcus (IPS; see also Menon et al., 2000 for increases in IPS activation with numerical task complexity). Following this rationale, carry and problem size effects should be interactive in nature because both are related to the stage of processing the overall magnitude of a multi-digit number. Again, first empirical evidence for an interaction of carry and problem size comes from the same study on two-digit addition by Deschuyteneer and colleagues (2005, see above). In sum, the fact that the study by Deschuyteneer et al. (2005) has provided evidence for both a procedurally additive as well as an interactive interrelation of carry and problem size, illustrates that to date there is no conclusive empirical evidence distinguishing between these two alternatives. Therefore, the first goal of the present study was to evaluate whether or not a statistically reliable interaction between the need for a carry and problem size can be observed for a balanced and systematically manipulated stimulus set.

1.5. The carry effect: Categorical or “continuous”? In line with the carry and problem categorical, that is, involving cognitive

rationale favouring the additive interrelation of size, the carry effect is usually assumed to be there is either the need for a carry operation control or not. Due to increased demands on

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cognitive control processes, carry addition problems are assumed to be more difficult to solve than non-carry problems. However, taking into account the decomposition of a multi-digit problem into its constituting single-digit problems, another account may be corroborated, which has not been investigated so far. As introduced above, the characteristic feature of multi-digit numbers is their mandatory link to the place-value structure of the Arabic number system. Considering that, in the case of two-digit addition, a carry operation is required whenever the sum of the units is equal to or larger than 10, we suggest that the carry effect might not be a purely categorical effect. Instead, we suggest that the standard carry effect may also reflect a continuous influence of the unit sum as assumed by the notion of decomposed processing of tens and units in two-digit numbers (Verguts & De Moor, 2005; see Nuerk, Weger, & Willmes, 2001; Nuerk & Willmes, 2005 for a hybrid account assuming decomposed and holistic representations). This would mean that reaction time RT should increase commensurately with the sum of the unit digits. Consequently, unit sum should be a better predictor of RT than a categorical carry predictor. In Fig. 2, hypothetical data patterns indicating either a categorical (Fig. 2A) or a continuous carry effect (Fig. 2B) are illustrated. A purely categorical carry effect should be reflected by a stage-like distribution of data points, whereas in the case of a continuous carry effect, RT should increase incrementally with unit sum. Finally, also a combination of these two options may be present (Fig. 2C). Here, RT should increase as unit sum increases, but a discontinuity between the unit sums 9 and 10 may indicate an additional categorical element of the carry effect. 1.6. Objectives The current study was set up to systematically investigate the interrelation between the need for a carry and problem size using a completely balanced stimulus set. This way, confounds of carry and problem size (as found in some previous studies) were excluded and the interaction of the two factors was examined in a systematic and orthogonal manipulation. Based on our former considerations on the carry effect representing a special case of (column-wise) manipulations of numerical magnitude and on findings suggesting a particular increase of activation in cortex sites assumed to subserve processing of number magnitude information for carry addition problems (e.g., Klein et al., 2009), we expect the carry effect to be more pronounced for addition problems with a large problem size as compared to problems with a small problem size. Moreover, building on the argument of decomposed processing of tens and units of two-digit numbers (see Nuerk & Willmes, 2005 for a review) and transferring this notion to the case of two-digit addition, it is hypothesized that the carry effect may not be purely categorical in nature but may also involve continuous aspects driven by the sum of the unit digits of the addends. 2. Methods 2.1. Participants 20 right-handed volunteers (mean age = 28.8 years; SD = 5.7, 10 female) with normal or corrected to normal vision participated in this study. 2.2. Stimuli

Fig. 1. Execution of a carry operation: the unit digit of the result (here, 54) is 4 and the decade digit of the result is derived by updating the sum of the decade digits of the addends (1 + 3) by the decade digit of the unit sum (here: 1 + 3 + 1 = 5).

In a verification paradigm, 96 different single- and two-digit addition problems as well as 49 filler items were presented twice in Arabic notation: once with a correct solution probe (henceforth correct equations) and once with an incorrect probe (henceforth incorrect equations) with the sequence of the operands being reversed in the latter case (e.g., 13 + 28 = 41 or 28 + 13 = 51,

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Lemaire & Siegler, 1995; Campbell, Parker & Doetzel, 2004). Additionally, this choice implies that half of the probes in incorrect equations differed from the correct result only in the tens position, while in the other half only the unit position was different. Therefore, incorrect equations could not be identified by focusing on either tens or units. Note also that although overall distance to the correct result is large in the second condition (split ±10), it is small if the appropriate base (i.e. decade) position is regarded (split ± 1). Therefore, strategies based on magnitude estimation or guessing should be unlikely. Note that we will refer preferentially to the term “distractor type” instead of “split” due to the specific construction of the distractors. To ensure the validity of the collected data, absolute as well as logarithmic sum, mean magnitude of the unit and decade digit of the correct result and the candidate answer, distance between the correct result and the incorrect solution probe, decade crossing between the correct result and the incorrect solution probe, position (left/right) of the smaller addend within the problem, occurrence of digit 5 at either the units or tens position of the addends as well as the correct/incorrect solution probe, and the parity of the correct result were matched between the four stimulus categories where appropriate (see Appendix B for an overview of stimulus properties and Appendix C for a complete list of all arithmetic problems used). Neither ties nor multiples of ten were included in the critical addition problems as either addends or probes in correct/incorrect equations. Additionally, the unit digits of the addends were different in all critical problems. Finally, no addition problem was part of a multiplication table (e.g., 16 + 24). The set of 49 additional filler items (which was not submitted to any analysis) included both tie numbers and multiples of ten. These were incorporated to approximate their frequency of occurrence to that in an everyday life setting. 2.3. Procedure

Fig. 2. Sketch of models illustrating hypothetical data patterns reflecting a purely categorical carry effect (Panel A), a continuous carry effect (Panel B), and a combination of continuous as well as categorical properties constituting the carry effect (Panel C).

Participants were seated approximately 50 cm from the screen in a dimly lit room. All stimuli were presented in white Arial font (size 26) against a black background using Presentation software (http://nbs. neuro-bs.com/presentation). In each trial, an addition problem was presented in central position above a solution probe. Participants had to decide whether or not the probe presented was the correct result by pressing the corresponding response button. Instructions focused on both speed and accuracy. To familiarize participants with display layout and task requirements, 12 additional practice problems had to be solved before the critical addition problems. All 290 problems were presented in one run. Each problem was presented until one of the response buttons was pressed or the time limit of 10 s was reached. 2.4. Analysis

respectively). For both, correct and incorrect equations the factors carry (carry vs. non-carry; e.g., 27 + 48 vs. 21 + 48) and problem size (sum b 40 vs. N 60; e.g., 13 + 24 vs. 13 + 54) were manipulated in a 2 × 2 within-participant design (please see Table 1). Half of the solution probes in incorrect equations differed from the correct result by ±2, whereas the split for the other half was ±10. These splits were chosen to minimize parity based solution strategies (Krueger, 1986;

Table 1 Systematic variation of the need for a carry and problem size in the 2 × 2 design used. Problem Size

Carry Operation

No Yes

Small

Large

12 + 14 19 + 15

25 + 62 47 + 38

According to Menon et al. (2002) (see also Ashcraft & Stazyk, 1981; Geary & Widaman, 1987; Zbrodoff & Logan, 1990) participants may resort to different solution strategies for verifying correct as compared to incorrect arithmetic equations (e.g., taking into account the split between the incorrect solution probe and the correct result). In the present case, a solution strategy focusing on either tens or units would identify half of the incorrect equations without calculating the whole result. Thus, it cannot be guaranteed that participants actually calculated these items to the end. In this vein, Geary and Widaman (1989) even assumed self-termination of the processing of carry addition problems before the carry is actually executed when the unit of the solution probe is incorrect. Nevertheless, inspection of the data for incorrect equations revealed reliable effects of both problem size and carry-over as well; and yet, the also significant influence of the factor split did indeed indicate additional processing demands for these equations (for a more detailed analysis and discussion of the 96 incorrect equations, the interested reader is referred to Appendix A).

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Based on above considerations the analyses were focused on the 96 critical correct equations. For correct equations, RT was analyzed using a 2 × 2 withinparticipant repeated measures ANOVA with the factors carry (carry vs. non-carry) and problem size (small vs. large). Further, a stepwise multiple regression analysis on mean item RT was conducted for correct equations. The stepwise regression analysis was stopped when the inclusion of another predictor would not increase R2 significantly (at p b .05). The predictors incorporated were decade sum, unit sum, as well as carry-over. The first two predictors, decade sum and unit sum, simply reflect the sum of the digits at the decade or unit position of the two addends, respectively. For instance, the unit sum ranged from 3 (as in 21 + 42) to 17 (as in 19 + 28). Contrarily, the predictor carry-over was coded categorically: +1 in the case the addition problem required a carry and − 1 for problems not requiring a carry. 3. Results Only addition problems that were correctly classified as correct equations were considered for subsequent analyses (for a separate analysis and discussion of the incorrect equations please see Appendix A). In the next step, a trimming procedure first eliminated all RTs b400 ms and N8000 ms; then RTs beyond ±3 standard deviations from the individual mean were excluded. This resulted in a total loss of 6.7% of the data. 3.1. ANOVA: On the interaction between carry and problem size The ANOVA revealed reliable main effects of both carry [RT: F (1,19) = 60.39; p b .001] and problem size [RT: F(1,19) = 140.99; p b .001] on RT. In line with our expectations, non-carry problems were responded to faster (2028 ms) than carry problems (2533 ms). Additionally, problems with large problem size were associated with longer RT than problems with small problem size (2889 ms vs. 1671 ms, respectively). The reliable two-way interaction of carry and problem size [RT: F(1,19) = 7.29; p b .05] indicated that the carry effect was more pronounced for problems with large problem size (difference of 618 ms) as compared to relatively smaller problems (difference of 392 ms, see Fig. 3). Subsequent Bonferroni–Holm corrected t-tests (Holm, 1979) showed that, on the one hand, the carry effect was nevertheless significant for both problem sizes (small problem size: t(19) = 5.59, p b .001; large problem size t(19) = 7.37, p b .001). On the other hand, the effect of problem size was reliable both for problems requiring a carry (t(19) = 10.81, p b .001) and not requiring a carry (t(19) = 11.42, p b .001).

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3.2. Regression analyses: On the continuity of the carry effect The final model comprised the predictors unit sum and decade sum only (R2 = .840, adjusted R2 = .836, F(2, 93) = 243.5, p b .001; see Table 2A), while the categorical predictor carry failed to explain a significant amount of additional variance. Inspection of the beta weights revealed that RT increased as the sum of the decade digits increased. More interestingly, RT also increased continuously as the unit digits of the addends increased; irrespective of the need for a carry-over. This indicated that in the current data the carry effect indeed seemed to be driven by the added magnitudes of the unit digits, rather than being a purely categorical effect. To better illustrate the distinction between a continuous and a categorical carry effect, Fig. 4 depicts the residuals resulting from a regression analysis over all items when the influence of the decade sum on RT was partialled out. A categorical carry effect would then be reflected by two distinct clusters of data points: Residuals smaller than zero for unit sums smaller than 10 and residuals larger than zero for unit sums larger than 10. As this was obviously not the case (see Fig. 4), the carry effect seems to comprise also continuous rather than just categorical properties. However, as the categorical carry predictor is highly correlated with the unit sum, we were interested in what way the results of the regression analysis described above would change when the categorical carry predictor instead of unit sum is included. The model put forward by the regression analysis performed quite similarly to that of incorporating unit sum (R2 = .807, adjusted R2 = .803, F(2, 93) =194.98, p b .001; see Table 2B). Thus, the two models only differed by 4% of explained variance indicating that inclusion of a categorical carry predictor did not result in a substantial loss of prediction accuracy. Taken together, the current data clearly indicated that carry and problem size interacted reliably as the carry effect was more pronounced for addition problems with large problem size. Moreover, the reliable influence of the continuous carry predictor unit sum in two-digit addition suggested that the carry effect in complex addition may be driven by magnitude processing of the unit digits of the addends as well. Yet, as the categorical predictor of carry operation and the continuous predictor of unit sum are highly correlated, we cannot and do not preclude that there are also categorical, procedural aspects determining the carry effect. An additional regression analysis revealed no substantial loss of variance accounted for when the categorical carry predictor was included instead of unit sum. Nevertheless, the current data imply that the carry effect may not be purely categorical as the continuous predictor unit sum was considered in the stepwise multiple regression analysis. More likely, the effect may involve both categorical and continuous characteristics. 4. Discussion The present study was initiated to investigate two main questions: First, we were interested in whether carry and problem size in

Table 2 Outcome of the regression analyses for items with a correct solution probe: Panel A depicts the model specifications of a stepwise multiple regression model in which the three predictors (i.e., decade sum, unit sum, carry) were incorporated. Note that the predictor carry (given in italics) was not considered in the final model. Panel B reflects the model in which only the categorical carry predictor (instead of unit sum) and decade sum were incorporated to explore the possible loss of variance explained. Predictor

A Fig. 3. Carry problems vs. non-carry problems with large and small problem sizes. Error bars indicate standard errors. There are main effects for carry and problem size which also interact; in the most difficult condition (large problem size, carry operation) participants are over-additively slow.

B

Decade sum Unit sum Carry-over Decade sum Carry-over

Regression weight Raw

Standardized

201.92 101.27

.88 .46 .11 .84 .42

192.77 312.71

t

Changes in R2

p

20.79 10.90 1.33 18.31 9.13

.64 .21

b.001 b.001 .19 b.001 b.001

.64 .17

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Fig. 4. Residual RT when decade sum is partialled out by a regression analysis on item RT based on decade sum only (i.e., overall RT–RT as predicted by the magnitude of the decade sum), thereby, indicating in which way residuals are influenced by the unit sum.

complex mental addition are interrelated, and if so, in what way. Second, also the nature of the carry effect was studied, that is, whether the carry effect has categorical and/or continuous properties. In the following, these two issues will be discussed in turn. 4.1. On the interrelation of carry operations and problem size representations The current data clearly indicates that carry and problem size interacted reliably: The carry effect was more pronounced for addition problems with large problem size (i.e., sums larger than 60) as compared to problems with small problem size (i.e., sums smaller than 40). This observation is of particular importance, as the present study was the first to systematically investigate the interaction between carry and problem size using a stimulus set properly controlled for possible confounds. Unlike in previous studies, investigating single- and two-digit mental addition (e.g., StanescuCosson et al., 2000), carry and problem size were manipulated orthogonally. The carry effect in mental addition seems to increase with increasing problem size of the addends. The current data indicate that the carry effect is influenced by the processing of number magnitude information rather than representing a purely procedural asemantic sequence of processing steps. Such an interpretation is in line with the notion of decomposed processing of multi-digit numbers, complying with the base-10 place-value structure of the Arabic number system, for which adding two-digit numbers reflects separate additions of tens and units. Yet, in case a carry is required, the decade digit of the unit sum would have to be carried to the tens position where updating the sum of the decade digits is required. This results in additional processes of magnitude manipulation and placevalue (i.e., unit-decade) integration. As already outlined in the introduction, such an interpretation is supported by recent neuroimaging data, showing that both the processing of number magnitude (for a review see Dehaene et al., 2003) as well as place-value integration may be associated with IPS activation (Knops et al., 2006; Goebel et al., 2004; Wood et al., 2006). In this vein, the carry from one position (e.g., units) in the place-value structure to the next position (e.g., tens) may be indicative also of a particular instance of number magnitude processing. This relates directly to the second question pursued in the current study. 4.2. The nature of the carry effect In the introduction, we suggested that the carry effect might also be considered a continuous effect of the unit sum. Following this

rationale, RT should increase as unit sum increases for addition problems. This prediction was corroborated by the regression analysis. Indeed, unit sum was incorporated as a significant predictor indicating the carry effect to be determined by the continuous factor unit sum. Inspection of Fig. 4 illustrates this interpretation. If the carry effect were of a purely categorical nature, one would expect the regression residuals for unit sums smaller than 10 (reflecting noncarry problems) to be consistently smaller than 0; whereas the regression residuals for unit sums no less than 10 should be larger than 0 (comparable to a data pattern as depicted in Fig. 2A). However, this was obviously not the case. Instead, the complementary notion of the carry effect also reflecting a special case of ‘semantic’ magnitude processing rather than just a purely procedural (asemantic) effect was supported4. Thus, our data suggest that the carry effect might not only be of a purely categorical nature; rather it seems to involve continuous properties as well which are reflected by the influence of the unit sum. Since the categorical carry predictor and the linear unit sum are highly correlated, we were interested in whether the influence of unit sum was present within both categories of carry as well as non-carry problems. If inclusion of the predictor unit sum were only due to its high correlation with the categorical carry predictor, unit sum should not be a reliable predictor for RT in separate regression analyses for carry and non-carry problems incorporating decade sum and unit sum as the only predictors. However, unit sum was a reliable and comparable predictor in both cases (carry problems: standard regression coefficient b = .19, t = 2.97, p b .01, R = .90, adjusted R2 = .81; non-carry problems: b = .20, t = 3.22, p b .01, R = .92, adjusted R2 = .84). This indicates that the predictor unit sum may be correlated but not collinear with the categorical carry predictor and thus accounted for a specific part of the variance. Therefore, our data indicates that the carry effect indeed seems to be driven by the unit sum suggesting a concomitant continuous origin of the carry effect. Taken together, one should be careful not to regard the effect of a carry operation as exclusively categorical (as it is usually done in the literature). More likely, the effect may involve both, categorical and continuous characteristics.

4 Please note that the interpretation of the carry effect as a continuous effect of the unit sum is consistent with the results by Imbo, Vandierendonck and De Rammelaere (2007) who observed the value of the carry to influence performance. Since the value of the carry is a direct result of the unit sum of the addends (e.g., in 14 + 28 + 9 unit sum equals 21), these two factors are directly related.

E. Klein et al. / Acta Psychologica 135 (2010) 67–76

4.3. Decomposed processing of the digits constituting a number The finding of the carry effect being driven by the unit sum is relevant regarding the notion of decomposed processing of tens and units (Nuerk et al., 2001; Nuerk & Willmes, 2005; Verguts & de Moor, 2005). In this view, the magnitudes of tens and units constituting two-digit numbers are represented separately. To activate the overall magnitude of a two-digit number, the single-digit magnitudes of tens and units need to be assigned their respective value and subsequently to be integrated into the base-10 place-value structure of the Arabic number system. In the case of two-digit addition, increased unit-decade integration demands occur whenever a carry operation is required. Because in this case the sum of the unit digits is equal to or larger than 10, the decade digit of the unit sum has to be carried to the tens position to yield the correct result. In this context, the current data indicate that the carry effect in the verification of two-digit addition problems can be accounted for by the integration of these decomposed representations into the placevalue structure of the Arabic number system. So far, the concept of a decomposed representation of two-digit numbers has only been invoked for the case of number comparison (Verguts & de Moor, 2005) and not for the case of addition. Therefore, the present study provides first empirical evidence that decomposed processing of tens and units can be successfully applied to mental arithmetic as it accounts for a standard effect (i.e., the carry effect). Considering that – within the framework of a decomposed representation of two-digit number magnitude – the carry effect is driven by the magnitude of the unit digits, it may also be informative regarding the first aim of the current paper: Processes of unit-decade integration may index a particular case of processing a number's magnitude (Wood et al., 2006), corroborating that both the problem size effect as well as the carry effect can be associated with the processing of number magnitude information as indicated by a significant interaction of these two factors. 4.4. Limitations and perspectives Generally, it should be noted that we are well aware that the current study is only a first step towards a more comprehensive understanding of the underlying processes in multi-digit addition and that there are still many more steps to take. Therefore, in the remainder of this discussion some points requiring further investigation will be addressed. Consider first the interrelation of the need for a carry and problem size. While both factors interacted reliably for correct equations, no such interaction was observed for incorrect equations. However, in our view the interrelation of carry and problem size can be verified most accurately by the results of the correct equations. A first point is the known fact that the evaluation of incorrect equations requires additional and/or different cognitive processes (Ashcraft & Battaglia, 1978; Ashcraft & Stazyk 1981; Zbrodoff & Logan, 1990; see Menon et al., 2002 for neuro-functional evidence), including the possibility that the calculation process in incorrect equations may even be self-terminated before any carry operation is executed whenever participants notice an incorrect unit digit of the solution probe (Geary & Widaman, 1987; Widaman et al., 1989). The present data corroborate this assumption. As the predictor distractor type was included in the regression model for incorrect equations, it is evident that additional and/or different cognitive processes were involved in the processing of these equations (e.g., evaluating the difference between the solution probe and the correct result). Moreover, a solution strategy focusing on either tens or units would identify half of the incorrect equations without calculating the whole result. Thus, for incorrect equations it cannot be guaranteed that participants actually calculated all items to the end, suggesting another possible explanation for the lack of interaction between

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carry and problem size in incorrect equations. Taken together, for incorrect equations, the calculation process may be self-terminated even before any carry operation is executed (e.g., Geary & Widaman, 1987), and such equations are assumed to require additional and/or different cognitive processes which are not principally related to either problem size or the need for a carry. Therefore, we focused our interpretation of the interrelation of carry and problem size on the results observed for the correct equations as these should be more informative for the purpose of the current study. However, please see Appendix A for a detailed description and discussion of the results for incorrect equations. Consider next that in paradigms comprising single- and two-digit addition (as has been the case in part for the small problem size condition of the current study), the number of digits of the addends is positively correlated with problem size (e.g., Widaman et al., 1989). To rule out the possibility that the inclusion of single-digit problems might have biased the finding of the carry×size interaction, an additional regression analysis for problems with a small problem size was carried out on the residuals between the observed RT and the RT predicted by a criterion variable accounting for the influence of the presence or absence of a single-digit summand. The final regression analysis was highly predictive (R=.82, adj. R2 =.66) incorporating the predictors of problem size (standardized β=−.21, t=2.43, pb .05) and, most importantly, also the interaction term of problem size and carry (standardized β = .80, t=9.41, pb .001). Thus, the interaction of problem size and carry was still observed when controlling for the presence of single-digit summands in the addition problems used. Based on these analyses, we are confident that the current results are not driven by the particular properties of the stimulus set. However, this finding does not resolve other important questions regarding the processing of single-digit summands such as the use of fact retrieval to solve addition problems. Future studies employing exclusively two two-digit addition problems could possibly shed further light on the issue of single- vs. two-digit addition. Another issue which should be subject to further research is the use of strategies actually applied by the participants. For instance, strategies such as rounding up or down might have an impact on the results regarding the carry effect, as their goal is to avoid the execution of the carry itself (i.e., 49+ 17 decomposed into 50 + 17− 1). However, as we did not ask participants about their strategies used and obtained only behavioural data (i.e., response latencies and error rates), conclusions on specific strategies would remain highly speculative. Future studies employing more fine-grained techniques (e.g., eye fixation behaviour, neuroimaging), as well as systematically using participants' self reports on the strategies they employed, may shed light on this issue. Consider next the question of the paradigm type (e.g., verification vs. production). Our motivation to use a verification paradigm (and not a production paradigm) was well-considered. On the one hand, response time measurement in production paradigms (via voice key or keyboard) also has some potential methodical pitfalls. Moreover, using a production paradigm may produce additional variance due to the process of generating a verbal output of the result; of course, we wanted to reduce such additional variance to a minimum. On the other hand, we have taken great care to select the stimuli such that potential shortcomings in verification paradigms – in particular the use of different solution strategies – can be considered to be small in the present design: Guessing and estimation are unlikely as problems were complex (pushing for calculation rather than for direct memory retrieval) and incorrect probes were not very deviant from the correct result in terms of their base position. More importantly, the split of the incorrect probes was always ± 2 or ± 10, so focussing on the unit digit (or the decade digit) did not provide any advantage or disadvantage because all digits were equally likely determining the correctness of the solution. Note that in this way also the use of a short-cut strategy based on parity judgements was prevented since the correct and the incorrect solution probe for each problem did not differ in parity. However,

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though we are therefore confident that by employing this careful stimulus selection, verification is an appropriate paradigm for the investigation of the questions addressed in this study, an additional future examination of the present results by a production paradigm may nevertheless be useful, especially with regard to possible interactions between number representation and number word production. Finally, regarding the nature of the carry effect we do not attempt to claim that the carry effect is purely continuous; we cannot and would not preclude that under certain conditions the nature of the carry effect may also comprise categorical elements such as fact retrieval strategies (e.g., Ashcraft & Stazyk, 1981; Geary & Widaman, 1987). Differences in presentational format (e.g., simultaneous vs. sequential presentation of addends and the result), stimulus set (e.g., including multi-digit numbers such as four-digit numbers), display layout (e.g., presenting the result below the addends), instructions (e.g., exact calculation vs. approximation), or active production/ choice reaction time paradigms may result in heterogeneous findings regarding the nature of the carry effect, that is, the specification of categorical or continuous characteristics.5 In summary, even though there are still a number of questions to be answered as regards the processes underlying mental addition, we nevertheless think that the present results provide first evidence on the interrelation of carry and problem size as well as the nature of the carry effect. 5. Conclusions The current study addressed two research issues: First, we were interested in the interrelation of carry and problem size. Unlike previous studies (e.g., Deschuyteneer et al., 2005; Kong et al., 2005), we orthogonally manipulated the factors' carry and problem size. It was observed that the magnitude representation underlying the problem size effects is not independent from carry procedures as indexed by an interaction between these two factors: The carry effect is stronger for larger problem sizes. Consequently, using only small problem sizes (e.g. single-digit or teen numbers) may lead to an underestimation of the carry effect. Second, we aimed at disentangling properties of the carry effect: Here we suggested that the categorical carry effect may in part be viewed as a continuous effect of the unit sum which is based on decomposed processing of the magnitude of the constituent unit and decade digits in multi-digit additions. The current data corroborated this assumption. In a regression analysis unit sum turned out to be a significant predictor of item RT, whereas the categorical carry predictor was not considered for the final model. Assuming that the magnitudes of tens and units are represented separately, this implies that integrating the decomposed representations into the place-value structure of the Arabic number system may be crucially involved in evaluating carry addition problems, extending the notion of decomposed processing of tens and units to mental arithmetic. In sum, representations and processes underlying the carry effect and their interplay seem to be more complex than assumed previously. Future studies may disentangle under which perceptual, strategic, and language conditions which processes are of particular importance. In either case, exploring the representations underlying the carry effect is a promising way to learn more about the nature of multi-digit arithmetic.

Acknowledgements The research and the preparation of this article were supported by a START-programme grant (AZ 160/05) of the Faculty of Medicine at the RWTH Aachen University to Hans-Christoph Nuerk and Frank Domahs supporting Katharina Dressel. Appendix A. Analysis of incorrect equations For incorrect equations, a 2 × 2 × 2 ANOVA comprising carry (carry vs. non-carry), problem size (small vs. large), and distractor type (i.e., the distance between incorrect solution probe and correct result, 2 vs. 10) was run to identify whether solution probes were rejected depending on their distance to the correct solution. Further, a stepwise multiple regression analysis on mean item RT was conducted for incorrect equations. In addition to the predictors decade sum, unit sum, and carry-over from the analysis of correct equations, the predictor distractor type was included (coded as − 1 for distance 2 between incorrect solution probe and correct result and +1 for distance 10). Results ANOVA Similar to the ANOVA on correct equations, also the 2 × 2 × 2 ANOVA on incorrect equations revealed main effects of both carryover [RT: F(1,19) = 33.37; p b.001] and problem size [RT: F(1,19) = 130.00; p b.001], while there was no significant main effect of distractor type [RT: F(1,19) = .43; p = .52]. Again, non-carry problems were responded to faster (2128 ms) than carry problems (2388 ms) and problems with a large problem size faster than problems with a small problem size (2745 ms vs. 1772 ms, respectively). However, for incorrect equations no carry × problem size interaction was observed [F(1, 19) = 0.69, p = .42]. A two-way interaction of carry and distractor type [RT: F(1,19) = 10.06; p b.01] indicated that the carry effect was more pronounced for problems with distractor type +/−2 (288 ms) as compared to problems with distractor type +/−10 (230 ms). Subsequent t-tests (Bonferroni–Holm corrected) showed that on the one hand, the carry effect was significant for both distractor types (2: t(19) = 4.57, p b.001; 10: t(19) = 5.43, p b.001). On the other hand, t-tests also revealed that distractor type was reliable for both carry (t(19) = 11.33, p b.001) and non-carry problems (t(19) = 12.39, p b.001). Regression The final model of the stepwise multiple regression incorporated the predictors decade sum, unit sum, and distractor type (R2 = .69, adjusted R2 = .68, F(3, 92) = 67.2, p b.001; see Table 3), while the categorical predictor carry failed to explain a significant amount of additional variance. Please note that the positive beta weight of the predictor distractor type indicated that probes with an incorrect decade digit (+/−10) were harder to reject than probes with an incorrect unit digit (+/−2).

Table 3 Outcome of the regression analyses for incorrect equations: final model of the stepwise multiple regression. Predictor

5

Please note that the presentation format does not seem to determine the result pattern, as identical findings regarding the interrelation of problem size and carry were observed when presenting the addition problem and the solution probe horizontally in the same line (Moeller & Nuerk, 2009).

Decade sum Unit sum Distractor type Carry-over

Regression weight Raw

Standardized

161.36 66.46 200.00 −.04

.75 .32 .29

t

Changes in R2

p

12.68 5.44 4.88 −.37

.48 .12 .08

b.001 b.001 b.001 .71

E. Klein et al. / Acta Psychologica 135 (2010) 67–76

Discussion of results for incorrect equations The results of the present study indicate that processing correct and incorrect equations seems to involve both similar (regarding the influence of the continuous predictor unit sum), but, even more importantly, also different cognitive processes (reflected by the importance of the predictor distractor type for incorrect equations). In particular, the inclusion of the predictor distractor type in the regression model for incorrect equations suggests that the rejection of incorrect equations is driven by additional processes such as evaluating the split between the solution probe and the correct result. Since these additional cognitive processes are more likely to be related to the solution of incorrect equations (Ashcraft & Battaglia, 1978; Ashcraft & Stazyk 1981; Menon et al., 2002; Zbrodoff & Logan, 1990) rather than to the factors problem size or the need for a carry, inferences on the lack of an interaction between the latter two factors in incorrect equations may be problematic. It has even been suggested that in incorrect verification equations, the calculation process may be self-terminated before any carry operation is executed whenever an incorrect unit digit of the solution probe is noticed (Geary & Widaman, 1987; Widaman et al., 1989). Taking into account above considerations, we focused our interpretation of the interrelation of carry and problem size on correct equations. Nevertheless, the present regression analysis of incorrect equations again corroborated the notion that the carry effect in complex addition may not be purely categorical, but may involve also continuous aspects driven by magnitude processing of the unit digits of the addends.

Appendix C. List of arithmetic problems used. For means of stimulus properties for the different stimulus groups please see Appendix B. Incorrect equations (given in italics) were used as a kind of filler item (for results and discussion of these items, the reader is referred to Appendix A). Further filler items (49) containing also 0 at units position are not depicted.

Small Problem Size

Non-carry

Appendix B. Stimulus properties (means) for different stimulus groups in the factorial design of the experiment Carry

Small problem size

Large problem size

Noncarry

Noncarry

Summand 1 9.92 Summand 2 10.42 Problem size categorical 1 Carry yes/no 0 Correct sum (identical with problem 20.33 size) Log correct sum 1.28 Decade sum 1.25 Unit sum 7.83 Summand smaller left 0.50 Parity Summand 1 1.67 Parity Summand 2 1.33 Parity correct sum 1.25 5 in unit position of Summand 1 0.17 5 in unit position of Summand 2 0.17 5 in unit position of correct sum 0.08 5 in decade position of Summand 1 0.00 5 in decade position of Summand 2 0.00 5 in decade position of correct sum 0.00 Decade of Summand 1 0.58 Decade of Summand 2 0.67 Unit of Summand 1 4.08 Unit of Summand 2 3.75 Decade of correct sum 1.25 Unit of correct sum 7.83 Distractor* 22.00 log distractor* 1.31 Distance between sum/distractor* −1.67 Absolute value distance sum/distractor* 6.00 Decade crossing sum/distractor* 0.50 Decade distractor* 1.42 Unit distractor* 7.83 Parity distractor* 1.25 5 in unit position of distractor* 0.04 5 in decade position of distractor* 0.00

Carry 10.38 10.29 1 1 20.67

Carry

41.29 41.58 2 0 82.88

38.46 43.21 2 1 81.67

1.29 1.92 0.75 7.58 13.17 7.04 0.50 0.50 1.29 1.46 1.46 1.42 1.33 1.38 0.17 0.17 0.17 0.17 0.08 0.08 0.00 0.04 0.00 0.04 0.00 0.00 0.38 3.75 0.38 3.83 6.63 3.79 6.54 3.25 1.75 7.58 3.17 7.04 22.33 84.54 1.33 1.92 −1.67 − 1.67 6.00 6.00 0.50 0.50 1.92 7.75 3.17 7.04 1.33 1.38 0.04 0.04 0.00 0.00

1.91 6.83 13.33 0.50 1.46 1.46 1.33 0.17 0.17 0.08 0.04 0.04 0.00 3.21 3.63 6.38 6.96 7.83 3.33 83.33 1.92 − 1.67 6.00 0.50 8.00 3.33 1.33 0.04 0.00

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Large Problem Size

Correct

Incorrect

Correct

Incorrect

4+3 =7 6 + 13 = 19 7+2 =9 12 + 7 = 19 5 + 14 = 19 12 + 13 = 25 16 + 12 = 28 4 + 13 = 17 14 + 15 = 29 13 + 16 = 29 5+3 =8 12 + 15 = 27 14 + 12 = 26 15 + 13 = 28 13 + 14 = 27 4 + 15 = 19 2 + 13 = 15 2 + 17 = 19 14 + 4 = 18 12 + 5 = 17 16 + 3 = 19 5 + 13 = 18 14 + 3 = 17 17 + 12 = 29 5 + 7 = 12 13 + 8 = 21 8 + 6 = 14 9 + 4 = 13 8 + 7 = 15 17 + 6 = 23 7 + 17 = 24 19 + 5 = 24 6 + 15 = 21 4 + 17 = 21 7 + 6 = 13 13 + 18 = 31 9 + 5 = 14 5 + 16 = 21 5 + 8 = 13 7 + 9 = 16 18 + 7 = 25 17 + 19 = 36 14 + 17 = 31 19 + 15 = 34 15 + 8 = 23 8 + 9 = 17 9 + 14 = 23 7 + 4 = 11

3+4 =9 13 + 6 = 17 2 + 7 = 19 7 + 12 = 29 14 + 5 = 29 13 + 12 = 35 12 + 16 = 38 13 + 4 = 19 15 + 14 = 27 16 + 13 = 19 3+5 =6 15 + 12 = 29 12 + 14 = 28 13 + 15 = 18 14 + 13 = 17 15 + 4 = 29 13 + 2 = 17 17 + 2 = 29 4 + 14 = 16 5 + 12 = 19 3 + 16 = 17 13 + 5 = 16 3 + 14 = 27 12 + 17 = 19 7 + 5 = 14 8 + 13 = 23 6 + 8 = 12 4 + 9 = 23 7 + 8 = 25 6 + 17 = 21 17 + 7 = 34 5 + 19 = 34 15 + 6 = 23 17 + 4 = 31 6 + 7 = 23 18 + 13 = 21 5 + 9 = 12 16 + 5 = 23 8 + 5 = 23 9 + 7 = 14 7 + 18 = 27 19 + 17 = 26 17 + 14 = 21 15 + 19 = 24 8 + 15 = 21 9 + 8 = 27 14 + 9 = 21 4 + 7 = 13

25 + 62 = 87 37 + 41 = 78 24 + 62 = 86 42 + 34 = 76 21 + 74 = 95 24 + 45 = 69 24 + 71 = 95 41 + 35 = 76 65 + 32 = 97 43 + 25 = 68 65 + 21 = 86 42 + 32 = 74 41 + 38 = 79 74 + 23 = 97 31 + 45 = 76 65 + 31 = 96 24 + 43 = 67 61 + 32 = 93 32 + 47 = 79 53 + 36 = 89 28 + 51 = 79 36 + 43 = 79 67 + 32 = 99 26 + 43 = 69 47 + 38 = 85 26 + 65 = 91 29 + 48 = 77 39 + 26 = 65 37 + 46 = 83 46 + 35 = 81 34 + 57 = 91 36 + 47 = 83 52 + 29 = 81 34 + 49 = 83 27 + 45 = 72 38 + 25 = 63 25 + 67 = 92 65 + 28 = 93 35 + 27 = 62 45 + 39 = 84 46 + 36 = 82 29 + 67 = 96 39 + 28 = 67 49 + 32 = 81 64 + 28 = 92 24 + 68 = 92 29 + 38 = 67 28 + 69 = 97

62 + 25 = 89 41 + 37 = 76 62 + 24 = 96 34 + 42 = 86 74 + 21 = 93 45 + 24 = 79 71 + 24 = 97 35 + 41 = 74 32 + 65 = 95 25 + 43 = 78 21 + 65 = 96 32 + 42 = 76 38 + 41 = 89 23 + 74 = 87 45 + 31 = 78 31 + 65 = 98 43 + 24 = 69 32 + 61 = 91 47 + 32 = 89 36 + 53 = 79 51 + 28 = 69 43 + 36 = 89 32 + 67 = 89 43 + 26 = 67 38 + 47 = 83 65 + 26 = 93 48 + 29 = 87 26 + 39 = 63 46 + 37 = 73 35 + 46 = 91 57 + 34 = 93 47 + 36 = 93 29 + 52 = 71 49 + 34 = 81 45 + 27 = 82 25 + 38 = 73 67 + 25 = 94 28 + 65 = 91 27 + 35 = 72 39 + 45 = 94 36 + 46 = 92 67 + 29 = 94 28 + 39 = 65 32 + 49 = 83 28 + 64 = 82 68 + 24 = 94 38 + 29 = 69 69 + 28 = 87

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