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^{b}

^{c}

^{d}

^{e}

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This special issue contains 18 articles that address the question how numerical processes interact with domain-general factors. We start the editorial with a discussion of how to define domain-general versus domain-specific factors and then discuss the contributions to this special issue grouped into two core numerical domains that are subject to domain-general influences (see Figure 1). The first group of contributions addresses the question how numbers interact with spatial factors. The second group of contributions is concerned with factors that determine and predict arithmetic understanding, performance and development. This special issue shows that domain-general (Table 1a) as well as domain-specific (Table 1b) abilities influence numerical and arithmetic performance virtually at all levels and make it clear that for the field of numerical cognition a sole focus on one or several domain-specific factors like the approximate number system or spatial-numerical associations is not sufficient. Vice versa, in most studies that included domain-general and domain-specific variables, domain-specific numerical variables predicted arithmetic performance above and beyond domain-general variables. Therefore, a sole focus on domain-general aspects such as, for example, working memory, to explain, predict and foster arithmetic learning is also not sufficient. Based on the articles in this special issue we conclude that both domain-general and domain-specific factors contribute to numerical cognition. But the how, why and when of their contribution still needs to be better understood. We hope that this special issue may be helpful to readers in constraining future theory and model building about the interplay of domain-specific and domain-general factors.

Numerical cognition encompasses a broad variety of cognitive and neural processes related to the perception, understanding and manipulation of numerical content. Hence, when investigating numerical cognition, we are not looking at an encapsulated cognitive module, supported by a single neural system but rather at a wide-spread network of interrelated cognitive processes with complex neural underpinnings. Much like human behaviour cannot be fully understood when leaving aside the social interactions and influences, we need to understand to what extent the core numerical processes are influenced and/or mediated by domain-general factors and other domains such as spatial skills and language.

Our current understanding of the terms domain-general and domain-specific factors has been shaped by discussions about whether there are domain-specific modules in the mind (

In Fodor’s definition modules need to be innately pre-specified. Along a similar vein, developmental psychologists have proposed that infants possess innately pre-specified domain-specific core knowledge (

‘Modularity’ and ‘domain-specificity’ have often been lumped together, but domain-specificity does not have to imply innateness. Clearly, specified learned systems can also be domain-specific, e.g. cycling, typing, and piano playing do not have to be innate (

More recent discussions suggest that the distinction between domain-general and domain-specific processes might be too crude. In practice, it can be a matter of perspective. While for a researcher interested in numerical cognition, symbolic number processing might be domain-specific and spatial skills might be defined as domain-general (or at least domain-overlapping), for spatial cognition researchers spatial skills might be domain-specific (see

This is also highlighted by the range of topics in this special issue (for an overview see

Schematic of the specific examples (within the dotted ellipse) of domain-general factors (in bold print outside the dotted ellipse) whose relationships to domain-specific numerical competencies (within the grey ellipse) were assessed in this special issue.

At a more fine-grained level, some skills might be more relevant for particular numerical tasks or representations but less so for other numerical processes. For example, working memory capacity influences complex addition, particularly if it includes a carry-procedure, more strongly than it does the retrieval of rote-learned simple multiplication facts (

The present special issue comprises 18 articles that address the question how numerical processes interact with domain-general factors from different angles. To provide the reader with an overview, we subsumed the contributions to this special issue under two core numerical domains with several subdomains (see

The present results make it very clear that mental arithmetic is subject to influences from a broad variety of domain-general factors. These include but are not limited to the following: mathematical language skills, sustained attention, conceptual understanding and creativity. The importance of working memory appears to be particularly controversial since some find that working memory training does not affect arithmetic performance while others report improved numerical understanding after working memory training.

As can be seen from

The names of the first authors of the studies are either depicted in bold or in italic font. Bold font means that this study found an influence, while italic font means, this study did not find an influence. Often, there were different analyses reported within the same paper. In those cases, bold/italic in the table refers to the most complex analysis results (e.g., multiple regression instead of raw correlation). So, if a domain-general factor X had a raw correlation with the target variable (e.g., arithmetic performance), but no influence in the multiple regression, because this variance could be better explained by other variables in the study, then this factor X would be italic, because it does not explain unique variance.

What can be seen immediately from the overview in

At this point, it is very important to note that this does not mean that studies, which report opposite results (i.e., bold and italic names in the same cell), are necessarily contradicting each other. On the contrary, the studies differ in multiple aspects. They often use different operationalisations of the underlying constructs, consider different co-variates in the analyses, use different paradigms, stimuli and effects for the to-be-predicted target variables, investigate different age groups, employ different designs (e.g., experimental, correlational, interventions) and different uni- and multivariate analyses; in sum, they reflect the variety present in the field studying domain-general and domain-specific influences on numerical processing. We will discuss this in more detail later. However, what can be said after a short inspection of

Although

Domain-general influence | Numerical and arithmetic effect/capability |
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Extension: |
Extension: |
Direction: |
Direction: |
Arithmetic | Numerical Identification | Counting | Arabic Magnitude Comparison | |

Working Memory | ^{g} |
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Executive Functions | ^{g} |
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Intelligence | ||||||||

Attention | ^{d} |
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Visuo-Spatial Processing | ^{a} |
^{b} |
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Visuo-Motor Integration | ||||||||

Language | ||||||||

Mathematical Language^{c} |
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Ordinality (non-numerical) | ^{f} |
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Processing/Perceptual Speed | ||||||||

Face Recognition | ||||||||

Self Control | ^{g} |
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Self Regulation | ||||||||

Social Power | ^{e} |
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Creativity | ||||||||

Socio-Economic Status |

^{a}Note that Crollen et al. is a review of existing studies, not presenting new empirical data. Therefore, parentheses were used.

^{b}Georges et al. examined correlations, regressions, moderations. One significant raw correlation disappeared in regression and moderation analysis. In one moderation analysis an interaction between SNARC and arithmetic prevailed, but arithmetic itself did not predict SNARC.

^{c}Purpura et al.: "Mathematical language has been classified in this study as a domain-general variable, but it should be noted that it also highly overlaps with domain-specific skills as it is comprised of content-specific language."

^{d}Katz et al. found attention correlated with OM effects in the non-symbolic, but not in the symbolic condition.

^{e}Huber et al. found an influence of social power in a number line length estimation task in the "increase" condition, but not in the "decrease" condition.

^{f}Schröder et al. found non-significant correlations of non-numerical (weekdays) and numerical SNARC in three of four analyses. In the remaining analysis, they observed

^{g}In Nemati et al.'s paper Planning (Tower of London) predicted accuracy, but not RT, while Self-control predicted RT, but not accuracy. Working memory was not a significant predictor in the regression, but was a predictor in the mediation analysis.

Domain-specific influence | Numerical and arithmetic effect/capability |
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Extension: |
Extension: |
Direction: |
Direction: |
Arithmetic | Numerical Identification | Counting | Arabic Magnitude Comparison | |

Extension Approximate | ^{b} |
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Extension Exact | ||||||||

Direction Implicit Cardinal | ^{a} |
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Arithmetic Performance | ^{a} |
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Counting Abilities | ||||||||

Knowledge of Arabic Numbers | ||||||||

Procedural Skills | ||||||||

Conceptual Knowledge | ||||||||

Ordering | ^{c} |
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Symbolic Magnitude Comparison |

^{a}Georges et al. examined correlations, regressions, moderations. One significant raw correlation disappeared in regression and moderation analysis. In one moderation analysis an interaction between SNARC and arithmetic prevailed, but arithmetic itself did not predict SNARC.

^{b}Purpura et al. presents different response cart tree analyses for different age groups and for high and low performance prediction. Only mathematical language was (almost) consistently predictive in all analyses.

^{c}The OM modulation was absent for size ordering, but present for ordering symbolic and non-symbolic sequences.

The metaphor of the mental number line (MNL), a spatially ordered representation of numerical magnitude, is often used to describe the mental representation of cardinal values and the interaction between representations of number and space has been an active research area for decades now (

According to this taxonomy, the approximate number system can be categorised as an extensive SNA since the activation of an approximate numerosity is conceptualised as an activation of a magnitude range on the MNL with the peak activation representing the most probable output to other cognitive systems. According to the number sense hypothesis, numerical information is internally represented by an analogue magnitude code in an approximate manner that allows for a numerical estimation of a set of items (e.g., a set of dots). The analogue magnitude code is invariant to input modality, format, and to non-numerical stimulus aspects such as density or the overall surface covered by the items. According to this approach, numerosity is a principal feature of our environment that can be directly sensed, comparable to color, contrast, or brightness. A concurrent model proposes that numerosity is derived indirectly from non-numerical stimulus dimension such as density, for example (

When the association between numbers and space entails the relative position of one object with respect to another,

While there is no doubt that representations of number and space interact, one question that remains controversial is how the spatial layout of the hypothesized MNL can be assessed best. That is, what paradigm can be used to obtain the best possible measure of the MNL metrics? Since the SNARC effect can be explained by at least five alternative, not necessarily spatial accounts (for an overview, see

Alternatively, one may conceive of reach trajectories as being modulated by the amount of response competition. That is, reach trajectories to either of several and simultaneously competing targets may reflect the confidence in the particular choice made, which in turn can be understood as the difference in accumulated evidence for the present options.

It becomes clear that we are still at the very beginning of our understanding of the contribution of domain-general and domain-specific influences to directional space-number associations. While some papers in our special issue suggest that numerical representations are part of one common and much more general mental magnitude representation, others suggest that even within the numerical domain, cardinal and ordinal associations or explicit and implicit magnitude associations are not part of one common construct. Clearly, there is still much work ahead to build a framework which incorporates all these findings.

The combination of numerical information is thought to represent a directional SNA with implicit coding of

To sum up, the results show that numerical and spatial processes interact with each other. Yet, these spatial-numerical associations are not a unitary construct. We need to differentiate between different regimes of numerosity perception (subitizing, estimation, texture perception) that are governed by the overall number of items in a scene and their spatial layout. Further, the proposed taxonomy provides a useful framework to organize the different SNAs. Implicit directional associations between number and space need to be dissociated from explicit ones, as shown for the SNARC effect in parity and magnitude judgment tasks, respectively, and different numerical representations such as cardinality or ordinality need to be dissociated. However, not only numerical attributes are associated with space, but also numerical functions. The operational momentum effect provides an exciting test bed for investigating the interaction between numerical functions (e.g. approximate arithmetic) and spatial capacities.

To sum up this section, first, our special issue shows the need for distinctions in the associations between domain-specific number capabilities (cardinality, ordinality, functions) in their relation with the more domain-general processing of space. Second, however, this special issue also seems to suggest that the number magnitude system may be part of a more general mental magnitude system (see for example,

(Early) prediction of arithmetic capabilities by more basic domain-specific factors (e.g., approximate number system) or domain-general factors (e.g., working memory) has been a long-time dream of cognitive and educational researchers and practitioners in arithmetic research. Identifying such building blocks and cornerstones of arithmetic development and functioning would have important consequences for education and intervention. Education – even much before formal schooling – could focus on mastering elementary building blocks of arithmetic to improve arithmetic performance and learning at large. Moreover, diagnostics could identify children who have trouble mastering the basic building blocks of arithmetic, before formal schooling, and targeted interventions may then help to improve these building blocks of later arithmetic development and the long-term outcome of those children. To identify such building blocks, prediction and intervention studies are essential – most researchers seem to agree that both domain-specific and domain-general factors predict (later) arithmetic performance. However, there is still no clear consensus on which of those factors are fundamental to arithmetic performance and arithmetical development. As we will see, about half of our special issue is devoted to the question of predicting and improving arithmetic performance and development.

To begin with, the capacity to judge the ordinal relation between objects has recently been suggested to be an important stepping stone for arithmetic performance. The paper by

Moving on from factors specific to the numerical domain, such as Arabic digit order, to domain-relevant skills,

One first step towards the development of such a taxonomy is taken by

Several studies in this special issue took the laudable approach to investigate the relative contribution of domain-specific and domain-general factors towards arithmetic and mathematical performance within the same study.

Two further papers in this special issue investigated the role of working memory for mathematical performance in the context of other domain-general factors.

In contrast, in

Finally,

As we laid out in the introduction, prediction studies are essential to identify targets for early education, instruction, and intervention. However, we also believe, there are some serious shortcomings currently in the literature as whole. First, in our special issue alone over twenty different predictors were tested and many more potential predictors are out there. In general, each study uses its own set of predictors. Consequently, different studies reveal different sets of predictors, which are relevant for good (later) arithmetic performance or arithmetical development. However, it is important to note that the results of a study do not only depend on the predictors included, but also on the predictors not included. For instance,

This leads us to the second point: the power of prediction studies. Large-scale longitudinal studies are hard to conduct and require a lot of effort. This is even more so the case for the age range for which finding predictors is arguably most important and of most practical relevance: from kindergarten to school. Therefore, either the sample size is often quite small for the number of predictors or the set of predictors is very limited. Both solutions can lead to misleading results and this might be one of the reasons why different prediction studies have often very different outcomes. What is needed, is a large-scale multi-center prediction study, which incorporates a large number of children and all relevant domain-specific and domain-general predictors so far found in the literature.

Our final point is that the outcome measure is often either one of many available standardized mathematics test or a curriculum-based test of mathematics. Those tests are often ‘umbrella tests’, i.e. measuring a large range of numerical, arithmetical and mathematical abilities without an option to distinguish between them. Consequently, those tests are frequently used without a model of its underlying representations and processes. In numerical cognition, it is virtually undisputed nowadays that different (neuro-cognitive) representations and networks are supporting different numerical processes and operations (e.g.,

Compared to prediction studies, intervention studies, in addition to their obvious practical implications, have an important theoretical advantage. Prediction studies are by its very nature correlational (when they are longitudinal over different time points) and all variables assessed are dependent variables. In contrast, in intervention studies the variable of interest, e.g. type of training, is manipulated as independent variable and therefore findings from intervention studies allow (cautious) causal rather than only correlational conclusions. Like in every other study, this does not preclude the influence of confounded or mediating variables. However, if a particular intervention leads to a training effect, this allows the conclusion that either the variable of interest (the training) or a variable confounded with it were instrumental for the intervention outcome, e.g., for improvement in arithmetic performance.

In this special issue, two papers evaluated the effect of working memory training on numerical and arithmetic performance.

The papers in this special issue show that domain-general as well as domain-specific abilities influence numerical and arithmetic performance virtually at all levels. This special issue thus makes it very clear that for the field of numerical cognition a sole focus on one or several domain-specific factors, like the approximate number system or spatial-numerical associations, is not sufficient. Vice versa, in most studies that included domain-general and domain-specific variables, domain-specific numerical variables predicted arithmetic performance above and beyond domain-general variables. Therefore, a sole focus on domain-general aspects, such as, for example, working memory, to explain, predict and foster arithmetic learning is also not sufficient. In the two intervention studies of this special issue, effects of domain-general interventions were weak or even not existent. Therefore, by acknowledging the importance of domain-general factors for arithmetic we do most certainly not advocate a restriction on domain-general factors. Rather, we are convinced that to understand numerical and arithmetic performance, development and learning, the contribution of both domain-general and domain-specific factors must be considered. However, these contributions may not simply be linearly additive or even independent; rather their interplay and their interactions must be studied more thoroughly both concurrently and longitudinally in future research. We believe that only then the full picture of arithmetic performance, development and learning can be understood.

What is still missing in our view are thoroughly developed models that specify how and to which extent domain-general and domain-specific factors contribute to numerical and arithmetical performance, development and learning and how those factors interact. As shown in

HCN was supported by the Deutsche Forschungsgemeinschaft (DFG: NU 265/3-1).

The authors have declared that no competing interests exist.

The authors have no support to report.