Empirical evidence suggests that working memory (WM) is closely related to arithmetic performance. WM, which is the ability to monitor and update recent information, underlies various cognitive processes and behaviors including planning, selfregulation, and selfcontrol. However, only a few studies have examined whether WM uniquely explains variance in arithmetic performance when other WMrelated domaingeneral factors are taken into account. In this study, we examined whether WM explains unique variance in arithmetic performance when planning, selfregulation, and selfcontrol are considered as well. We used the Tower of London task as a measure of planning, selfrated reports as a measure of selfregulation and selfcontrol, and WM measures, to test which of these domaingeneral functions predicts complex multiplication performance. Results showed that planning predicted multiplication accuracy and selfcontrol predicted response time, while WM and selfregulation did not predict complex multiplication performance. Although WM was not a direct predictor of multiplication performance, it possibly exerted its influence as part of planning ability. We suggest that complex multiplication is not predicted by WM per se, but rather by WMrelated general cognitive and behavioral factors, namely selfcontrol and the planning component of executive functions.
People use numerical skills in their everyday life in various situations such as shopping or paying bills at restaurants. About 20% of adults do not achieve basic levels of mathematic competence required for these life skills (
Various domaingeneral cognitive abilities have been reported to influence complex arithmetic performance. One category of such abilities is executive functions (EF), which include three core cognitive processes: inhibition, shifting, and working memory (WM;
WM contributes to basic arithmetic skills (i.e., addition, subtraction, multiplication, division) because these computations require concurrent storage and processing of digits (e.g.,
Further research suggests that the contribution of different domains of WM to arithmetic skills is influenced not only by the type of problems or domains of WM but also by an individual’s age (
However, most of these studies investigated WM as the only domaingeneral factor predicting arithmetic performance and may, therefore, be misleading because WM is also associated with other EFs such as inhibition and planning (e.g.,
Solving complex multiplication tasks that requires different sequences of computations and cognitive processes likely involves planning skills, for instance, organizing the best strategies step by step to solve the problem. Accordingly, we focused on planning ability as well as WM, because it has been suggested that planning is one of the essential skills for mathematical computations (e.g.,
Therefore, in addition to cognitive components, we assessed selfreported selfregulation and selfcontrol as two behavioral factors closely related to cognitive EF factors (
Previous studies have indicated that both selfregulation and selfcontrol can predict mathematical performance in children (e.g.,
In the present study, we assessed complex multiplication because it requires a variety of cognitive skills (
Furthermore, we investigated the operandrelatedness effect, which has been mostly implicated in simple multiplication. Operandrelatedness within multiplication refers to the solution belonging to another problem, mainly the neighbors of the correct solution in the multiplication table (e.g., 3 × 7 = 24). Studies have shown that this effect leads to slower response times and more errors in simple multiplication (e.g.,
In sum, in the present study, we investigated the relation of the aforementioned domaingeneral factors to complex multiplication problem solving to determine which one is the best predictor of complex multiplication performance in adults, and to test whether any of these domaingeneral factors can predict complex multiplication performance better than WM. Regarding the operandrelatedness effect, we hypothesized that although we usually do not learn complex multiplication from a table, due to stepbystep computations (i.e., first multiplying unit to unit, following to multiplication table, then unit to decade), the operandrelatedness effect may exist in complex multiplication.
To examine our hypotheses, we employed computerized tasks to measure cognitive factors (i.e., WM and planning) and online questionnaires to assess behavioral factors (i.e., selfregulation and selfcontrol). Furthermore, participants performed computerized complex multiplication task in our laboratory.
Forty undergraduate students (33 females, age:
To assess WM, we used visuospatial WM tasks (i.e., NBack and Corsi blocktapping) because they have been shown to have strong relations with multiplication performance (e.g.,
The NBack task had a spatial 2Back design, where in each trial one of the following capital letters: B, F, K, H, M, Q, R, X, (
The computerized version of the Corsi blocktapping task was used (
The computerized version (
In total, 48 complex multiplication problems (
German short versions of the Conners’ adult attentiondeficit/hyperactivity disorder (ADHD) Rating Scale (sample item: “I am easily bored”; CAARS;
This study was part of a larger project which aimed to investigate the effect of selfregulatory training on mathematical performance. First, an online questionnaire consisting of items for assessing demographic information, selfregulation, and selfcontrol, was sent via email to each participant. After responding to the online questionnaire, participants were invited to our laboratory to perform computerized tasks in individual sessions. Subsequently, WM, planning, and multiplication performance were assessed using computerized tasks in the lab. Each task lasted approximately 1015 min. Half of the participants completed the domaingeneral cognitive tasks first, and the other half completed the multiplication task first. Written detailed instructions emphasizing both speed and accuracy were presented before each task.
Response times (RTs) of participants in complex multiplication task were measured by keypress, and defined by the time interval between the presentation of the problems and the response. Only RTs for correct responses were included in the analyses. Furthermore, RTs shorter than 200 ms were not considered. In a second step, RTs outside of the interval of ±3 SD around the individual mean were excluded repetitively until no more outliers remained (for the same procedure see
In order to find the relation between domaingeneral factors and multiplication performance, a bivariate correlation was calculated. Moreover, to uncover which domaingeneral factors predict multiplication performance in adults, two separated stepwise regression analyses were conducted on mean RTs and error rates. It should be noted that prior to regression analysis, the WM variables were aggregated by adding zscores of Corsi blocktapping task and zscores of Nback task accuracy to guarantee adequate statistical power of the model by reducing the number of predictors. The same procedure was conducted with WM nonaggregated measures to provide full insight into the nonaggregated WM measures. Furthermore, there was a statistically significant correlation between WM tasks (see
Variable  1  2  3  4  5  6  7  8  

1. Multiplication Error (%)  0.18  0.11    
2. Multiplication RT (ms)  3.10  0.56  .27*    
3. SelfRegulation  13.62  5.57  .17  .00    
4. SelfControl  41.42  8.74  .04  .34*  .17    
5. NBack Accuracy  0.87  0.09  .30*  .16  .28*  .04    
6. CorsiBlock Forward  5.50  0.65  .27*  .05  .22  .18  .35*    
7. CorsiBlock Backward  6.05  0.81  .18  .15  .22  .20  .31*  .47**    
8. TOL Accuracy  0.73  0.13  .43**  .13  .23  .45**  .53**  .36*  .25   
9. WM^{a}  0.00  5.27  .33*  .03  .31  .15  .50** 
^{a}WM = aggregated WM tasks including Corsi BlockTapping Forward, Corsi BlockTapping Backward, and NBack Accuracy. Ranged between 5.65 and 5.11.
*
Descriptive statistics for all study variables are provided in
Stepwise multiple regression analysis was conducted to test whether selfregulation, selfcontrol, WM, and TOL accuracy significantly predict participants' performance in the multiplication task. Two series of stepwise regression analyses were separately conducted for multiplication error rate and RT as dependent variables. The stepwise model of total error rate,
Dependent variable  Predictor  Excluded variable  Standardized 


Total errors (%)  TOL Accuracy  0.35  2.93  .01*  0.43  
SelfRegulation  0.08  0.52  .60  
SelfControl  0.19  1.15  .26  
WM^{a}  0.15  0.90  .37  
RT (ms)  SelfControl  0.02  2.26  .03*  0.34  
SelfRegulation  0.06  0.41  .69  
WM  0.08  0.52  .61  
TOL Accuracy  0.03  0.20  .84 
^{a}WM = aggregated WM tasks including Corsi blocktapping forward, Corsi blocktapping backward and NBack accuracy.
*
Dependent variable  Predictor  Excluded variable  Standardized 


Total errors (%)  TOL Accuracy  0.35  2.93  .01*  0.43  
SelfRegulation  0.08  0.52  .60  
SelfControl  0.19  1.15  .26  
Corsi Block Forward  0.13  0.83  .41  
Corsi Block Backward  0.08  0.52  .60  
NBack Accuracy  0.10  0.60  .55  
RT (ms)  SelfControl  0.02  2.26  .03*  0.34  
SelfRegulation  0.06  0.41  .69  
Corsi Block Forward  0.11  0.73  .47  
Corsi Block Backward  0.08  0.51  .61  
NBack Accuracy  0.15  0.96  .34  
TOL Accuracy  0.03  0.20  .84 
*
Mediation analyses were conducted following the guidelines by
*
*
The effects of nonaggregated WM measures on multiplication errors without controlling for TOL accuracy were not significant (Path
Furthermore, additional mediation analyses were conducted to test the extent to which the relationship between TOL accuracy and multiplication error is mediated by WM measures and to explore the common and specific multiplication accuracy variance predicted by TOL and WM measures (
The operandrelatedness effect in multiplication was replicated in this study by using paired
The present study investigated the role of domaingeneral factors including WM, planning, selfregulation, and selfcontrol abilities in complex multiplication performance in adults. Consistent with previous studies in children (e.g.,
Although most previous studies emphasized the major role of WM in multiplication performance (e.g.,
The other reason for the differences between the study by
Finally, there might be a methodological reason for the differences between the study by
In summary, on the basis of the current data, we can be confident that planning plays a major role in predicting complex multiplication performance, and that WM exerts influence on complex multiplication performance indirectly via planning. However, what requires further investigation is whether WM explains the unique variance of complex multiplication performance in addition to the indirect influences exerted via planning. In our data, this is not the case, but it remains possible for other WM tasks.
Regarding multiplication RT, selfcontrol was the only significant predictor: more selfcontrol was associated with faster responses. One reason could be that procedural arithmetic computations (i.e., stepbystep computation during arithmetic problem solving) require concentrating and ignoring irrelevant information, which in turn rely on selfcontrol (e.g.,
However, unexpectedly, selfregulation did not show any relation to complex multiplication performance in the present study. One possible reason could be the insufficiency of our measurement instrument (i.e., ADHD symptoms selfreport) that was used for assessing selfregulation in healthy adults in this study. Although a strong association between ADHD symptoms and selfregulation deficits exists in individuals with ADHD (
Interestingly, in the present study, the operandrelatedness effect was found in complex multiplication, which suggests an extension of this effect beyond the multiplication table. Although several studies in adults (e.g.
An important limitation of the current study is that participants were all university students and not representative of the general population. An additional limitation was the lack of latent variable approach regarding EFs as we did not assess different indicators for EF core components (i.e., inhibition, shifting, updating WM;
Furthermore, in the current study, we used a visuospatial NBack task which included letters as stimuli in the twosplit grid frames. Participants were asked to remember the letters as well as their location to answer correctly. Hence, performance in our NBack task might also recruit verbal WM capacity to some extent. It could not predict complex multiplication performance when other domaingeneral factors such as planning and selfcontrol were considered. However, verbal and visuospatial WM were not disentangled in our study, which is a limitation, because some previous studies suggested a stronger involvement for verbal WM than visuospatial WM in multiplication performance (e.g.,
In addition, consistent with previous research indicating that people with selfregulation deficits such as people with ADHD may have difficulties in visuospatial WM (e.g.,
Another point is that since both complex multiplication and planning were the most complex tasks, a stronger correlation between the two is not surprising. This assumption needs to be tested in future studies that investigate the correlation between simple arithmetic and planning or consider additional complex tasks to examine the uniqueness of the relationship between planning and complex multiplication. Finally, as outlined above, there was a ceiling effect in our NBack task (
The findings of the present study show that planning is a better predictor for multiplication accuracy than other domaingeneral factors (i.e., WM, selfcontrol, and selfregulation). This might be traced back to procedural processes that are required for both planning and complex multiplication problem solving, and may also be due to the multicomponential nature of planning which involves other domaingeneral factors such as WM. For both dependent variables, the hallmark construct of domaingeneral factors, WM, no longer explained any unique variance when other cognitive and behavioral domaingeneral factors were considered. However, due to the strong association between WM and planning and the results of mediation analyses, we cannot conclude that WM has no influence on multiplication performance, but rather that it influences complex multiplication performance through planning performance. The mediation analyses seem to suggest that WM is part of the planning component of EF, which may be most relevant for more complex arithmetic computations. We conclude that more domaingeneral factors engaged in arithmetic processing need to be taken into account when the total influence of one factor like WM is examined. However, as we argued in our limitation section, our findings are restricted to welleducated adults, namely university students, and to complex multidigit multiplications as well as particular assessment tasks and the complexity of the version used. Whether these results generalize to other age groups, less skilled individuals, and less complex problems and other ways to assess WM and planning as well as other control measures remains an important question for followup studies. Nevertheless, we argue that our results strongly suggest that not only WM, but other domaingeneral factors need to be considered to better understand the foundations of arithmetic performance.
Variables^{a}  Entire Sample ( 

20.95 (1.98)  
Gender  
Female  33 
Handedness  
Right  39 
Field of study (%)  
Psychology  40 
Cognitive Science  37 
Medicine  8 
Media  2 
Educational Science  8 
Environment Science  5 
Participation payment  
Course credit  16 
Money  24 
Math score in University entrance exam  
below 8  7 
8 to 11  12 
12 to 15  21 
^{a}Information obtained from background online questionnaire.
First operand  Second operand  Correct Solutions  Incorrect Solutions 


Operandrelated^{a} Errors  Operandunrelated Errors  
4  19  76  72  90 
13  7  91  84  86 
14  6  84  98  80 
17  5  85  90  82 
18  3  54  57  59 
3  19  57  76  64 
5  13  65  70  68 
16  3  48  51  34 
7  14  98  84  79 
15  6  90  75  94 
18  4  72  68  91 
5  19  95  90  88 
^{a}operandrelated errors are matched by Mean, Mean distance difference from correct solutions and parity to operandunrelated errors.
Variable  

Multiplication Error (%)  0.18 (0.11)  .03  1.02  1.18 
Multiplication RT (ms)  3100 (560)  .20  0.27  0.19 
SelfRegulation  13.62 (5.57)  .13  0.82  1.15 
SelfControl  41.42 (8.74)  .20  0.18  0.36 
NBack Accuracy  0.87 (0.09)  .01  1.79  3.98 
Corsi Forward  5.50 (0.65)  .13  0.22  0.68 
Corsi Backward  6.05 (0.81)  .00  0.95  1.39 
Planning Accuracy  0.73 (0.13)  .00  0.80  0.02 
^{a}KolmogorovSmirnov pvalues.
Dependent Variable: Multiplication Errors (%) 


Variable  Path Identifier  Standardized 

TOL^{a} Accuracy  
b  .29  .35  .13  .03  
WM^{b}  
a  .03  .50  .01  .00  
c  .01  .33  .01  .04  
c´  .01  .15  .01  .36 
^{a}TOL = Tower of London. ^{b}Aggregated WM measures.
Dependent Variable: Multiplication Errors (%) 


Variable  Path Identifier  Standardized 

TOL^{a} Accuracy  
b  .28  .35  .14  .05  
NBack Accuracy  
c  .27  .24  .17  .12  
c´  .66  .48  .19  .00  
c´´  .09  .08  .20  .66  
CorsiBlock Forward  
d  .03  .18  .03  .24  
d´  .04  .20  .03  .14  
d´´  .02  .11  .03  .54  
CorsiBlock Backward  
e  .00  .03  .02  .87  
e´  .00  .01  .02  .93  
e´´  .00  .02  .02  .90 
^{a}TOL = Tower of London
*
Variable  Dependent Variable: Multiplication Errors (%) 


Path Identifier  Standardized 

TOL^{a} Accuracy  
a  8.51  .50  2.38  .00  
c  .35  .43  .12  .00  
c´  .29  .35  .13  .03  
c´´  .28  .35  .14  .05  
WM^{b}  
b  .01  .15  .01  .36  
NBack Accuracy  
d  .37  .53  .09  .00  
d´  .09  .07  .20  .67  
CorsiBlock Forward  
e  1.76  .36  .72  .01  
e´  .02  .11  .03  .53  
CorsiBlock Backward  
f  1.50  .25  .94  .11  
f´  .00  .02  .02  .90 
^{a}TOL = Tower of London. ^{b}Aggregated WM measures.
This research was funded by a grant from German Academic Exchange Service (DAAD): Research Scholarship for Doctorate students and young researchers for more than 6 months, 2014/15 (57048249) to Parvin Nemati. Mojtaba Soltanlou is supported by the Science Campus Tübingen, Project 8.4 given to HansChristoph Nuerk.
The authors have declared that no competing interests exist.
We would like to thank Barbara Peysakhovich and Julianne Skinner for language proofreading of this manuscript and all participating students in our study.