From an early age we learn to live in a world with numbers: on classroom doors, street signs, price tags, our phones, on our work activities. Typically we learn how numbers and quantities relate to each other (e.g., smaller, larger) from an early age; and most of us received some formal training in math, starting from grade school. Grade school training in mathematics coincides with protracted development of the pre-frontal cortex (e.g., [@bib0245]). The pre-frontal cortex is a key brain region, concordant across mathematical-cognition studies in healthy adults ([@bib0035], for meta-analyses). Much progress has been made in understanding brain correlates of mathematical cognition; however, despite the increase in the studies examining children's mathematical problem solving (i.e., quantity discrimination and mathematical operations), a neuropsychological model for children is still not available. We have compiled data from functional magnetic resonance imaging (fMRI) studies and report concordant findings on brain correlates of typically developing children when solving math tasks with and without formal calculations (i.e., operations).

Behavioral protocols with children can be designed using math tasks such as printing or naming numbers, counting and sorting objects ([@bib0005], [@bib0360]). Neuroimaging tasks, however, are largely restricted to the visual domain, because they need to adhere to constraints/limitations of the imaging methodology ([@bib0330], [@bib0025], [@bib0030]). For fMRI studies, task protocols must be as time limited as possible. For instance, stimulus presentation should be brief, a few seconds; longer intervals are harder to control for irrelevant intruding processes (e.g., mind wandering). Manual responses are preferable to verbal responses, being less likely to produce head motion that compromise image quality. Moreover, calculation tasks are typically simple, often 1- or 2- digit operations, so that participants can provide a response within a limited time frame. Most fMRI studies that examine brain correlates of mathematical cognition, either in children or adults, follow these basic task characteristics.

The majority of fMRI studies in the literature investigated mathematical cognition in adults (e.g., [@bib0405], [@bib0230]), and the parietal lobes received the most attention in early studies of mental arithmetic. Indeed, parietal brain regions, such as bilateral intraparietal sulci, left angular gyrus, and bilateral superior parietal cortices, play distinct roles in number processing ([@bib0170]). Although the parietal cortex is fundamental to process mathematical problems, other regions are involved as well ([@bib0015], [@bib0280], [@bib0645], [@bib0650]). Coordinate-based meta-analyses of 53 adult fMRI studies show areas, such as cingulate gyri, insula and the prefrontal cortex that are concordantly active in tasks of numbers and calculation ([@bib0035]).

*Number tasks* are those that contain numbers (e.g., viewing different digits) and quantities (e.g., viewing small or large arrays of dots), but require no formal calculation (e.g., multiplication). They have in common some sort of semantic judgement on numbers or quantities based on stipulated rules. *Calculation tasks* require as well operation rules, such as addition, subtraction and multiplication, applied to numbers. Distinct and common brain areas are active in adults during number and calculation tasks ([@bib0035]). Specifically, they elicit brain responses within regions such as inferior parietal lobule and inferior frontal cortex; however, calculations also recruit prefrontal areas, particularly in middle and superior frontal gyri ([@bib0035]).

The first fMRI study with children was published 16 years ago, by [@bib0220], testing children with or without a velocardiofacial syndrome, on an arithmetic task. fMRI studies of children working on mathematical problems are gradually increasing. As in adult studies, tasks administered to children divide into those investigating numerical processes (e.g., [@bib0015], [@bib0010], [@bib0085], [@bib0320]) and those studying mathematical operations (e.g., [@bib0060], [@bib0140], [@bib0190], [@bib0410]).

Studies that examine numerical processes typically ask children to select the larger number in a set of numbers (e.g., [@bib0015]). The numbers in the set can differ by either small differences/distance (i.e., 1, 2 and 3) or large (i.e., 5, 6, and 7) ones. When the difference is small, children show activity in the superior parietal lobe, medial and inferior frontal gyri, the insula, and subcortical regions − mostly in the right hemisphere ([@bib0015]). A subsequent study by the same researchers shows that when number differences are large, children activate the left hemisphere's dorsolateral prefrontal cortex, inferior frontal gyrus, and intraparietal sulcus ([@bib0010]). Other studies compared brain responses to numbers versus responses to shapes ([@bib0085]); or examined numerical processes versus a control task ([@bib0295]).

Procedural differences in mathematical operations often lead to activity in different cortical regions ([@bib0305], [@bib0545]). Studies examining various operations in the same children are important: [@bib0545] examined brain responses to subtraction and multiplication, and [@bib0305] examined three mathematical operations, addition, subtraction and multiplication. They found several common brain regions associated with them all. In the prefrontal cortex, for instance, addition and subtraction recruit the left middle frontal cortex, whereas multiplication recruits left middle and inferior frontal cortices ([@bib0305]); further, unlike addition and multiplication, subtraction elicited activity in the right intraparietal sulcus. More fMRI studies are needed that examine multiple mathematical operations in the same children.

A meta-analysis by [@bib0275] reports concordance across seven fMRI studies, which tested children with either number or calculation tasks, in right inferior and middle frontal gyri, left superior frontal gyrus and left middle occipital gyrus. This meta-analysis supports the view that prefrontal regions play an important role in mathematical cognition ([@bib0560], [@bib0020], [@bib0035]). [@bib0275] did not detect extensive involvement of parietal cortex, which is critical in mathematical cognition, possibly because of variability in the original studies' methodology and the low number of foci. Another meta-analysis examined 19 fMRI studies that included children ([@bib0300]). Perhaps due to the low number of studies, the authors ([@bib0300]) chose to include studies with fixed effects analyses (e.g. [@bib0290], [@bib0105]), coordinates from contrasts with variable performance (i.e., interaction of brain activity of high and low performers, [@bib0335]), and variable age (i.e., coordinates resulting from a conjunction analysis between children and adults, [@bib0265]); they also included age ranges spanning over participants older than 18 years (i.e., 8.53--19.03 years, [@bib0560]). Although such approach increases the number of studies and coordinates in the meta-analyses, it obscures the reliability of results.

Targeted meta-analyses were recently performed to identify brain correlates of number processing and notation (i.e., symbolic vs non-symbolic) in adults ([@bib0595]) and children ([@bib0310]). These studies suggest a network of parietal and frontal areas that underlie symbolic and non-symbolic processes. Based mainly on adult data theoretical models of mathematical cognition (e.g., [@bib0160], [@bib0035]) may not be suitable for accounting for developmental data ([@bib0030]). Also, it is challenging to identify developmental theories of cognition that make clear neural predictions on mathematical processes. We used a domain general cognitive theory of development for hypotheses building. The Theory of Constructive Operators ([@bib0505], [@bib0470], [@bib0030]) outlines brain correlates associated with schemes and operation types to predict performance. The theory of constructive operators would predict that brain responses to number and calculations tasks are not material-driven, but process-driven and vary with the trade-off between participants' mental-attentional capacity and the mental demand of the task. Specifically, this trade-off predicts that the right hemisphere is involved in processing of automatized schemes, whereas the left hemisphere is involved in processing problems that involve the child's mental-attentional capacity, and are not automatized yet (details on this account is given in the discussion). Thus, we anticipated that number tasks should favour right frontal and parietal regions, whereas calculation tasks within the child's mental-attentional capacity will recruit additional frontal and parietal regions in the left hemisphere. In the current meta-analyses we explore brain areas involved in mathematical cognition of children younger than 14 years, and provide normative fMRI atlases in standard stereotaxic space for number and calculation tasks.

1. Methods {#sec0005}
==========

1.1. Literature search and article selection criteria {#sec0010}
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The literature was searched, in June 2017, by means of web-of-science (<http://www.isiknowledge.com>), using the terms fMRI and arithmetic and children; fMRI and calculations and children; fMRI and math and children; and fMRI and numerical and children. We have also added five papers using manual search. [Fig. 1](#fig0005){ref-type="fig"} shows the number of articles from this search, and the process we followed to identify eligible articles. Specifically, after eliminating duplicates, the articles were subjected to a series of selection criteria. For inclusion articles needed to: (a) be written in English, (b) have used fMRI and tasks involving numbers and mathematical operations; (c) have healthy children participants as the main or control group; (d) have reported whole-brain, within-group results using random-effects analysis; (e) have reported stereotaxic coordinates in Talairach or Montreal Neurological Institute (MNI) coordinates. Forty-three articles survived these criteria. To maintain data independence, we eliminated articles that reported contrast with analyses involving other age groups (e.g., conjunction between children and adults) and/or other tasks (e.g., conjunction between working memory and arithmetic problem solving). We also eliminated articles that included participants over 18 years in the children group (i.e., age range 8.53--19.03 years. [@bib0560]; age range 7.7--21 years, [@bib0315]; mean age 17 years 11.5 months, [@bib0550]), and one article for including the same experiments (i.e., contrasts) using same participants in different publications ([@bib0395], [@bib0400]). These controls resulted in 32 acceptable articles testing children 14 years or younger.Fig. 1PRISMA flowchart for identification and eligibility of articles (template by [@bib0415]). n = number of papers.Fig. 1

Contrast coordinates were selected based on whether the experimental paradigm was related either to number tasks or calculation tasks. *Number tasks* were defined as those involved in numerical processing without having prescribed operations (e.g., number-distance effects, numerical comparisons). In the number tasks meta-analyses we included data from 18 independent subject-groups that included 30 experiments (i.e., contrasts) using symbolic (i.e., Arabic) stimuli ([@bib0015], [@bib0065], [@bib0085], [@bib0180], [@bib0295], [@bib0345]), non-symbolic (e.g., dots) stimuli ([@bib0010], [@bib0070], [@bib0075], [@bib0260], [@bib0350], [@bib0365], [@bib0400], [@bib0635]), or a combination of both ([@bib0225], [@bib0445]). *Calculation tasks* were defined as tasks that involved prescribed mathematical operations − addition, subtraction or multiplication. Included among calculation tasks were 42 experiments from 18 independent subject groups that used symbolic stimuli, with three exceptions: one paper using non-symbolic stimuli ([@bib0060]) and two others having a combination of symbolic and non-symbolic stimuli ([@bib0340], [@bib0535]). Insufficient experiments (less than 17--20 experiments; [@bib0215]) did not allow us to examine operations separately. Similarly, due to insufficient data of symbolic and non-symbolic experiments in children under 14 years, we could not investigate the hypothesis of whether numbers have abstract or multiple representations (e.g., [@bib0120]). Three experimenters (MA, MPL and MS) performed contrast selection separately, and final decisions were taken in agreement. [Table 1](#tbl0005){ref-type="table"} shows article information, participant demographics, and contrast selection for each category.Table 1Descriptive information of articles and contrasts used in the meta-analyses.Table 1AuthorYearSample (N)FHandAgeNumber tasksFociAnsari200512n/rn/r10.4 (9.2--11.11)Distance effect (small \> large)8Ansari200693R10.4 (9.11--11.11)Distance effect (small \> large)3Berteletti20142011n/r11.5 SD = 1.7 (8.6--13.1)Numerosity Task1Berteletti20153922n/r11:4, SD = 1:6 (8:5--13:7)Numerical Processing Large vs Small4Bugden20121714R105.5 m SD 6.09 m (95--116 m)Distance effect ratio5Cantlon200685n/r4.75 (4.25--4.95)Number \> shape7Demir-Lira20163320n/r10.9 ± 1.5 (8--13.8)Spatial localizer (dot arrays) at Time 14Emerson201224n/rn/r8.24 SD = 2.26 (4.32--11.86)Number network5Gullick2013166R10y;8 m (9;11--11;9)Fifth graders: Positive \> Negative numbers11Fifth graders: Negative \> Positive numbers1Fifth graders: Positive distance effect9Fifth graders: Negative distance effect19Fifth graders: Positive \> Negative distance effect5Fifth graders: Negative \> Positive distance effect14155R12y;8 m (11;9--13.5)Seventh graders: Negative \> Positive numbers1Seventh graders: Positive distance effect12Seventh graders: Negative distance effect9Seventh graders: Positive \> Negative distance effect9Seventh graders: Negative \> Positive distance effect5Kaufmann2008124R8.6 SD = 1.2Nonsymbolic numerical processing2Kersey20173522n/r5.45 (3.6--6.99)Conjunction:Adaptation and numerical deviant effect5Kucian2011157n/r10.6 SD 1.5Non-Symbolic Numerical distance effect14Kucian201116n/rn/r9.5 SD 1.1Order vs. control task11Libertus2009157n/r8.8 (8y 11 days--9y 1 mon)Digits \> (Letters and Faces)3Meintjes201018n/rR(8--12)PJ \> control task17Park20142112R5.55 (4.82--6.59)All Number Tasks16Numerical \> Nonnumerical7Symbol \> Dots9Close \> Far2Vogel20151913n/r10.2 SD 2.55 (6--14)Parametric modulation for number adaptation5AuthorYearSample (N)FHandAgeCalculation TasksfociAshkenazi20121711n/r8.12 SD = 6.59monsComplex \> Simple Addition23Bertelletti20153922n/r11:4, SD = 1:6 (8:5--13:7)Subtraction large vs small6[@bib0095]20162514R8.8 SD 0.7(7.7--10.7)Subtraction − Control14Davis20092714n/r8.1 (7.1--9.4)Single-digit exact calculation5Double-digit exact calculation13Single Digit Approximation18Double Digit Approximation8Davis20092412n/r8.2 (8y1m--9y1m)Exact calculation task2Approximate calculation task.7[@bib0150]20111812R11.77 (10.08--12.92)Small \> large11Large \> small8Addition \> subtraction1Subtraction \> addition15Du201319n/rn/r10.62 SD 0.31 (10.12--11.26)AP addition v EX addition (non trained group)17[@bib0285]2015158n/r(7.5--9.6)Arithmetic problem solving before tutoring3Kawashima200484R11.6 SD = 1.6 (9--14)Addition6Subtraction6Multiplication6Krinzinger2011209n/r108 SD = 21 m (69--105 m)Symbolic exact addition12Non-symbolic exact addition12Meintjes2010b18n/rR(8--12)Exact addition \> control task25Metcalf20137440R7.8 (7--9)Complex \> Control6[@bib0420]20112413R9.6 SD = 0.9 (7.8--11.4)16n/rR3-Operand (Simple) Addition or Subtraction216n/rR4-Operand (Complex) Addition or Subtraction38n/rR3-Operand (Simple) Addition or Subtraction58n/rR4-Operand (Complex) Addition or Subtraction6Peters201622105L10.73 SD = 0.87 (9--12)Subtraction: Digits − Fixation6Subtraction: Words − Fixation10Subtraction: Dots − Fixation8Subtraction: Words − Digits2Subtraction: Dots − Digits10Subtraction: Dots − Words10Subtraction: Digits − Dots6Subtraction: Words − Dots5Prado20143421n/r11.54 (8.47--13.56)Smaller subtraction2Smaller subtraction \> smaller multiplication3Larger \> smaller problems2Larger \> smaller multiplication,4Smaller \> larger (multiplication and subtraction combined).4[@bib0555]20142813n/r(7--11)Addition \> Control collapsing across Time-1 and Time-217[@bib0565]20119039R2nd graders 7.67 SD = 0.4; 3rd graders 8.67 SD = 0.4Addition: Complex \> Simple5Addition: Simple \> Complex5[^1]

1.2. Meta-analyses {#sec0015}
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Activation Likelihood Estimate (ALE) is a coordinate-based meta-analytic method ([@bib0615], [@bib0205], [@bib0210]). Foci from experiments (i.e., contrast coordinates) from selected articles are used to generate 3D maps indicating the likelihood of activation within a given voxel of a template brain ([@bib0205]). Data are compared against a random spatial distribution (i.e., noise) to identify the likelihood of significant clusters. A whole-brain statistical map of ALE scores is generated that estimate the probability of the significance of a brain region being active during a given cognitive function.

ALE meta-analyses were computed using GingerALE 2.3.6 (<http://brainmap.org/ale/>). First, MNI coordinates were transformed into Talairach space using the best-fit MNI-to-Talairach transformation ([@bib0355]). Each meta-analysis (i.e., number and calculation tasks) contains foci from all possible relevant experiments as the analyses algorithm minimizes within-group effects and provides increase power ([@bib0620], [@bib0215]). We report brain areas that survived an uncorrected voxel-level threshold of p \< 0.001 with a cluster-level correction at p \< 0.05 for multiple comparisons ([@bib0210], [@bib0215]). Contrast analyses are performed on images corrected for multiple comparisons with cluster level p = 0.05 and uncorrected p = 0.001, thus the threshold for contrast number tasks vs calculation tasks is set to p = 0.01 uncorrected, with 5000 permutations and minimum volume 50 mm^3^ (e.g., [@bib0595]). For displaying results from ALE maps we employed AFNI ([@bib0125]).

For comparison we performed meta-analyses of fMRI data that examine number and calculation tasks with adults (methods and result presented in supplementary material).

2. Results {#sec0020}
==========

Mean age and/or age range of children in the meta-analyses for all experiments are shown on [Table 1](#tbl0005){ref-type="table"}. The average age of children included in the meta-analyses was calculated to be 9.34 ± 2.18 years for number tasks and 9.58 ± 1.4 years for calculation tasks. Of the articles reporting gender, 44% and 52% of participants were female for number and calculation tasks, respectively. About 75% of the all articles reported handedness and tested right-handed participants with the exception of five left-handed participants in the sample for calculation tasks.

2.1. ALE maps {#sec0025}
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### 2.1.1. Number tasks {#sec0030}

In children, number tasks are associated with significant ALE values in the inferior parietal lobule Brodmann area (BA 40) extending to parts of the inferior parietal sulcus, claustrum and insula (BA 13) in the right hemisphere ([Table 2](#tbl0010){ref-type="table"}, [Fig. 2](#fig0010){ref-type="fig"}).Table 2Concordant areas for processing number and calculation tasks in children.Table 2Volume mm^3^ALE ValuexyzBrain regionChildren: Number tasks117360.01938−4854Right Inferior Parietal Lobule BA 400.01838−4642Right Inferior Parietal Lobule BA 400.01234−3448Right Postcentral Gyrus BA 3210720.01830188Right Claustrum0.01536160Right Insula BA 13  Children: Calculation tasks142960.02901050Left Superior Frontal Gyrus BA 60.02422044Right Medial Frontal Gyrus BA 80.02041042Right Cingulate Gyrus BA 32231280.04432186Right Insula BA 13328480.030−30−6238Left Precuneus BA 190.021−42−4842Left Inferior Parietal Lobule BA 40[a](#tblfn0005){ref-type="table-fn"}0.018−28−7242Left Precuneus BA 19413680.034−30166Left Claustrum512400.026−46436Left Precentral Gyrus BA 6[b](#tblfn0010){ref-type="table-fn"}69200.0222−7048Right Precuneus BA 7  Children: Number tasks \> Calculation tasks  No suprathreshold clusters  Children: Calculation tasks \> Number tasks19843.5402840Right Cingulate Gyrus BA 323.156−31344.5Left Medial Frontal Gyrus BA 63.03621246Right Medial Frontal Gyrus BA 625282.9483−7348Right Precuneus BA 731602.489322410Right Insula BA 1341042.512−30126Left Claustrum2.447−32166Left Insula BA 13  Children: Calculation tasks AND Number tasks19680.01830188Right Claustrum0.01536160Right Insula BA 13[^2][^3][^4]Fig. 23D rendered ALE activation maps superimposed on an anatomical brain. All regions survived cluster level correction p = 0.05 for multiple comparison control at an uncorrected p = 0.001. All coordinates are listed in [Table 2](#tbl0010){ref-type="table"}.Fig. 2

### 2.1.2. Calculation tasks {#sec0035}

When solving mathematical operations children activate a diverse set of areas ([Table 2](#tbl0010){ref-type="table"}; [Fig. 2](#fig0010){ref-type="fig"}). The largest cluster is centered on the left superior frontal gyrus (BA 6) and extends to the right medial frontal gyrus (BA 8) and cingulate gyrus (BA 32). The second largest cluster with the highest likelihood for being detected is found in the right insula (BA 13). Concordance in parietal areas is observed bilaterally; left precuneus (BA 19) and inferior parietal lobule (BA 40) including parts of the inferior parietal sulcus and angular gyrus BA 39, and right precuneus BA 7. The left claustrum has the second highest likelihood of being detected. A cluster centered on the left precentral gyrus (BA 6) encompasses also grey matter in the inferior and middle frontal gyri (BA 9).

### 2.1.3. Contrast: number tasks vs calculation tasks {#sec0040}

In children, compared to calculation tasks, number tasks show no suprathresholed clusters. Compared to number tasks, calculation tasks show increase concordance in the right cingulate gyrus (BA 32), bilateral medial frontal gyrus (BA 6), right precuneus (BA 7), bilateral insula and left claustrum, ([Table 2](#tbl0010){ref-type="table"}). Areas that show concordance in conjunction between number and calculation tasks are the claustrum and insula (BA 13) in the right hemisphere ([Table 2](#tbl0010){ref-type="table"}).

3. Discussion {#sec0045}
=============

Brain activity related to mathematical cognition in typically developing children was examined using coordinate-based ALE meta-analyses. Children activate a varied set of areas in established parietal and frontal regions when solving problems with numbers and operations. We highlight the role of insula and claustrum in mathematical operations because these regions have not been previously emphasized in this sort of cognition. To facilitate discussion, we organized findings by cortical regions, and for each we indicate characteristics of activity (e.g., its extent or magnitude) and the categories (i.e., number and calculation tasks), which elicit the activity. We build on previous models ([@bib0160], [@bib0035]) to offer a topographical model and a theoretical interpretation of mathematical cognition in children.

Number and calculation tasks elicit responses in the parietal lobes in children. This is a critical region linked to mathematical problem solving ([@bib0170]). Specifically, the first and most concordant cluster for number tasks is in the right inferior parietal lobule (BA 40), where complex (concrete or abstract) effort-demanding (non-automatized) objects are found (e.g., [@bib0515], [@bib0470]). The left inferior parietal lobule (BA 40) is activated during calculation tasks − sites for "active" complex objects that can receive transformations/operations (e.g., [@bib0470]). The critical role of the inferior parietal lobule in mathematical cognition has been discussed extensively in adults ([@bib0165], [@bib0170], for reviews) and children ([@bib0085], [@bib0145]). The current findings are consistent with previous reports; and highlight a hemispheric dominance for number tasks in the right hemisphere, in contrast to calculation tasks driven by the left hemisphere.

When children solve mathematical operations, areas with the highest likelihood of being detected appear in the right insular cortex for calculation tasks. The left claustrum also shows one of the highest values of significant concordance for calculation tasks. The insula and claustrum are also concordant in a smaller cluster for number tasks. The insula is a structure that connects the temporal and frontal lobes deep within the lateral fissure; the claustrum borders the insula medially. Initially, the insula became known for its involvement in affective processes (for meta-analyses, see [@bib0255], [@bib0195]). Its role in the interaction of cognition, emotion and interoception has also been discussed ([@bib0630]). Such interpretations of insular role are consistent with the hypothesis that, together with the anterior cingulate, insula may be involved in intrinsically motivated behaviours ([@bib0600], [@bib0625], [@bib0495], [@bib0030]). The insula, together with subcordical structures such as the basal ganglia, have been implicated in studies that involve learning and training ([@bib0100]). The role of the claustrum is often bundled with the functions of the insula; however, this is a distinct region anatomically ([@bib0385]), and in terms of structural connectivity ([@bib0440]). Recent reviews suggest the claustrum to function as a cross-modal integrator to create conscious percepts ([@bib0130], [@bib0250], for reviews). Thus, we propose that for children solving mathematical operations, high activation in the insula may express their *intrinsic motivation* in the task, with reference to learning and training. For the claustrum, we propose that it helps integrate motivated top-down and bottom-up processes. Importantly, neither the insula nor the claustrum have been implicated in models of mathematical cognition, although the current findings suggest that these areas are critical.

Another critical region for calculation tasks is the cingulate gyrus (BA 32), which extends medially from superior frontal (BA 6) and medial frontal (BA 8) gyri. The cingulate gyrus is also a key region for mental-attention and working memory, since it may be where affective intentions convert into cognitive goals ([@bib0045], [@bib0495]). Significant clusters of concordant activity in the dorsal cingulate gyrus (BA 32) are observed for calculation tasks in children. The cingulate gyri have been related to various higher order cognitive activities. Progressively more evidence highlights its role as a coordinator of activity in multiple attentional systems ([@bib0540]), multimodal functions ([@bib0580]), and as coordinating activity based on task complexity ([@bib0610]). We suggest that in children cingulate gyri integrate affective motives and available information to generate solutions involving specific cognitive goals.

Calculation tasks also elicit concordance in a cluster in the left precentral gyrus (BA 6) that extends to inferior and middle frontal gyri (BA 9). The middle frontal gyri, often referred to as dorsolateral prefrontal cortex, are associated with executive-driven mental attention and working memory ([@bib0045], [@bib0110], [@bib0115]). Compared to inferior frontal cortices, which are associated with simpler cognitive actions, the dorsolateral prefrontal cortex implication in coordinated cognitive control of complex processes is important, and it was identified as such early on ([@bib0575], [@bib0110], [@bib0115]). Studies of mathematical cognition do associate activity in middle frontal gyri with procedural complexity and working memory processes ([@bib0175], [@bib0230], [@bib0325], [@bib0590], [@bib0650]). This hypothesis is supported by adult data, which show that middle frontal gyri are implicated in calculation tasks but not in number tasks ([@bib0035]). We have replicated the latter finding with adult meta-analyses of number and calculation tasks (Table S2).

3.1. Topographical model of mental-arithmetic in children {#sec0050}
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There is a close agreement on brain locations implicated in children and adults during mathematical problem solving (Table S2--S3; [@bib0035]), and predictions by the triple-code model ([Fig. 3](#fig0015){ref-type="fig"}). The only unaccounted brain region by either the triple-code model ([@bib0160]) and the adult meta-analyses is the claustrum, which was concordant for children. A critical difference we observe is that children implicate the right insula (BA 13) more extensively than adults in calculation tasks, whereas adults implicate more prefrontal areas (BA 44, 46, 10; Table S3). We provide a theoretical framework that may explain these findings in the next section.Fig. 3Mapping results on children meta-analyses (in red), on triple-code model (green), and adult meta-analyses (orange). We illustrate in green the schematized cortical locations of the triple-code model proposed by [@bib0155], [@bib0160]: (1) Inferior parietal cortex: quantity representation, (2) Temporal cortex: visual-computational number symbols, (3) Articulatory loop, (4) Verbal system, (5) Basal ganglia: arithmetic facts, (6) Thalamus: arithmetic facts, and (7) Prefrontal cortex: strategy choice and planning. In orange are additional schematic locations of areas concordant among adult studies, as demonstrated by meta-analyses ([@bib0035]): (a) Superior frontal BA 10: formulates complex goals, sub-goal creation, (b) Middle frontal BA 46: in more or less misleading situations it monitors more than a few items, (c) Inferior frontal BA 9: monitor simple rules or a few items, (d) Precentral gyrus: eye movements, (e) Insula: interoceptive motivation of goal-directed and default-mode processes, (f) Cingulate gyrus: converts affective goals into cognitive goals to be implemented, (g) Right angular gyrus: visual-spatial fact retrieval (i.e., spatial-temporal schemes with non-verbalizable configural relations), and (h) Cerebellum: goal directed, visual motor sequencing. Sub-cortical regions specific to meta-analyses of number or calculation tasks were not depicted. Here we added the (i) right basal ganglia: coordination of top-down and bottom-up operative/motor processes. In red are schematic locations of areas concordant among children studies, as demonstrated by the current meta-analyses. (j) Claustrum: integration of motivated top-down and bottom-up processes.Fig. 3

3.2. Developmental theory: the theory of constructive operators (TCO) {#sec0055}
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The Theory of Constructive Operators (TCO) is a general, constructivist theory of development and learning ([@bib0515], [@bib0520], [@bib0530], [@bib0450], [@bib0470], [@bib0475], [@bib0480], [@bib0490]). It provides a mathematical model for predicting developmental stages ([@bib0505], [@bib0485], [@bib0495], [@bib0040]). The TCO is framed in terms of organismic *operators* (content-free brain utilities, i.e., regulations or general controls), *schemes* (contextual information carriers − cell assemblies or networks), and, finally, organizing *principles* ([@bib0500], [@bib0505], [@bib0515], [@bib0455], [@bib0470], [@bib0475]; [Fig. 3](#fig0015){ref-type="fig"}). Operators are general-purpose and content free resources, which can apply on schemes in any domain. [Table 3](#tbl0015){ref-type="table"} provides a short description and basic brain location of operators in the TCO. [Fig. 4](#fig0020){ref-type="fig"} suggests how operators interact with schemes to elicit performance. Psychologically, schemes are self-propelling information-bearing units (cell assemblies or networks), which can be classified into three groups: executive, operative (action blue-prints), and figurative (object or feature representations). Thus, schemes carry information that can appear under three distinct categories: figurative, operative, and executive (Pascual-Pascual-Leone and Baillargeon, 1994; [Fig. 3](#fig0015){ref-type="fig"}): (a) *figurative schemes* are perceptions and representations of concepts or objects (e.g., in the case of numbers: symbolic 3 or ★★★); (b) *operative schemes* correspond to actions and specific procedures, applicable to objects or concepts (e.g., in the case of math, the act of multiplying 3 × 3); and (c) *executive schemes,* a subdivision of operative schemes, that carry general contextualized procedures (e.g., in the case of math, to plan the order of steps for solving 3 + 3/(3 × 3)). The currently activated and dominant set of compatible executive schemes can function as an operator (which we call *E* for executive). *E* can regulate or control the current functioning of other operators (e.g., *M-* or *I-*operator; [Table 3](#tbl0015){ref-type="table"}) as they apply to modify schemes in their here-and-now dynamism (e.g., [@bib0455], [@bib0470], [@bib0475]).Table 3Description of Operators and their Corresponding Brain Regions in their Likely Evolutionary Order (after [@bib0470]; [@bib0470]).Table 3OperatorDescriptionBrain Region*A*Set of **affective** processes that intervene in motivation and attentive arousal.Limbic Lobes*C*Both the process of **content** learning and the schemes derived from associative content.Primary & secondary association areas*F*The **field** operator together with the Schemes' Overdetermination Principle (*SOP*, see below) act as the brain\'s *binding mechanism* bringing closure to mental representations in a neo-Gestaltist manner.All Areas*LC*The process of automatized **logical-structural** learning derived from **content** learning through over-practice.Right Hemisphere*T***Temporarily** and effortlessly collates *sequences* of schemes, thus facilitating coordination of temporally-structured invariants that constitute distal objects.Occipito-temporal*S*Facilitates emergence of **spatial** schemes by coordinating *relations of coexistence* among activated schemes effortlessly within the situation.Occipito-parietalBSocial '**Being**' Schemes − related to personal, self/other referential schemes.Default-mode areas*I*The attentional **interrupt**: It produces the central *active inhibition* of unwanted schemes, which were activated by the situation or the mind.Prefrontal*M*Effortful **mental-**attentional *activation* of simple or complex (functional-structure) schemes.Prefrontal*LM***Logical-structural** learning caused by effortful use of **mental**-attentional capacity.Left hemisphere tertiary areas*E***Executive,** i.e., dominant *set of activated and compatible executive schemes* in the person's repertoire that are useful for the task at hand.PrefrontalFig. 4A simplified illustration of the Theory of Constructive Operators (TCO): Operators (in green; operator definitions are listed in [Table 3](#tbl0015){ref-type="table"}), schemes (in blue), and the principle of schematic over-determination of performance − or SOP (in red).Fig. 4

A principle of Schematic Overdetermination of Performance (*SOP −* [@bib0455], [@bib0460], [@bib0470], [@bib0475], expresses both the spreading of activation in the brain, and its *final common path* of neuronal resolution into a performance − which is a generalized version of Sherrington's original idea ([@bib0585], [@bib0390]). This *SOP* determines which set of compatible and dominant schemes eventually will apply to generate an outcome. Mutually incompatible schemes compete, and those more strongly activated eventually apply.

3.3. Integrating neuroimaging data and the TCO {#sec0060}
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Our findings have important theoretical and practical implications. Practically they provide coordinates in stereotaxic space that future studies may benefit from. Theoretically, they offer support for developmental theories that follow process-specific rather than content-specific approaches ([@bib0505], [@bib0515], [@bib0475], [@bib0480], [@bib0430]). To clarify this issue we distinguish processes across four main dimensions of variation: (a) operative versus figurative processing, (b) complexity levels of processing, (c) left- versus right-hemisphere processing, and (d) motivation and self-control.

### 3.3.1. Operative versus figurative {#sec0065}

We suggest that operative processes and executive schemes are expressed in frontal and prefrontal areas, respectively, whereas representations and figurative/object schemes are expressed in parietal, temporal and occipital areas ([@bib0515], [@bib0470]). This is in agreement with early neurologists who viewed the brain as separating distinct, different modes of processing (e.g., [@bib0370]). It is important to explain why more operative processes and executive schemes are needed when solving calculations, expressed by additional brain activity in frontal and prefrontal areas. A clear distinction between the operative processes implicated in calculation tasks versus number tasks is found in adults (Table S2--S3), consistent with past meta-analyses ([@bib0035]). Operative processes are also important in children's math tasks, as expressed by the involvement of prefrontal and sub-lobar regions, including insular cortex, during these tasks.

### 3.3.2. Complexity levels in processing {#sec0070}

Association areas may in some sense express level of *processing* required by a task. Consider that both *operative* (motor, efferent associative) and *figurative* (sensorial, afferent associative) schemes can be organized in context-sensitive *heterarchical* ways (i.e., functionally-flexible hierarchies). These heterarchies were first described by classic neuropsychologists (e.g., [@bib0375], [@bib0200]) as functionally nested sequences of *primary* areas (afferent or efferent − e.g., BA 4, and BA 1, 2, 3 and BA 17), *secondary* areas (coordination of local unimodal processes − e.g., BA 6, and BA 5, and BA 18), *tertiary* areas (coordination of *regional multimodal* areas − e.g., BA 8 and BA 7, BA 19), and finally *general multimodal* areas, also called *high tertiary or quaternary* (which are functionally generic, applicable across domains, and integrating the totality of regional multimodal processing − e.g., BA 9, BA 47, BA 46, BA 10, BA 39, BA 40). Distinguishing levels of processing complexity of a task is key for predicting and interpreting mathematical performance. In our data we find, for instance, that number tasks in children implicate secondary and tertiary association areas, whereas calculation tasks elicit activity in more tertiary areas.

### 3.3.3. Left hemisphere versus right hemisphere processing {#sec0075}

Traditionally semantic-pragmatic differences between left and right hemispheres, have been verbal-analytical versus visuospatial-global processing, respectively ([@bib0240]). However, this is no longer tenable as a main distinction between left versus right hemispheres − because we find that both hemispheres activate with both content types (i.e., verbal and visual-spatial). Within a constructivist-developmental viewpoint, we claim that propensities fostered by left hemisphere are best characterized as *analytical mental-attentional processing* (or novel, effortful working-memory) − often in *demanding* problem-solving situations; whereas right hemisphere fosters propensities of *overlearned or automatized processing* that engage in either very easy or very difficult tasks ([@bib0050], [@bib0515], [@bib0475]). [Table 4](#tbl0020){ref-type="table"} shows circumstances that give rise to right or left hemisphere dominance by considering the novelty of task, mental-demand (*Md*) of the task and mental-attentional capacity (*Mc*) of the individual ([@bib0450], [@bib0475]). A trade-off between *Md/Mc* can be used to predict hemispheric dominance such that when *Md \<\< Mc* processing would favour the right hemisphere, when *Md *≤ *Mc* processing would favour the left hemisphere and when *Md \>\> Mc* processing would favour the right hemisphere. This distinction is novel with respect to the function of right hemisphere, which would be implicated in two different sorts of instances: (a) In more or less automatized (very easy) processes, or (b) when the needed mental-attentional resources are above and beyond what the individual has available (i.e., above his/her mental-attentional capacity; [@bib0510]). In the latter case, when a task's mental attentional demand is too high for the left hemisphere to cope with alone, the right hemisphere is mobilized in search of potentially useful overlearned or automatized schemes. For instance, in the current data ([Table 2](#tbl0010){ref-type="table"}) we see that the largest cluster elicited in children when solving number tasks is in the right parietal cortex (since figurative schemes for number tasks are easier). In contrast, calculation tasks elicit activity mainly in the left parietal cortex (calculation involves harder − relational − figurative schemes to be coordinated using the child's mental attentional capacity). Similarly, activity for the simple number tasks is observed in the right hemisphere, whereas calculation tasks implicate the left frontal cortex more extensively. It would be interesting for future studies to examine brain correlates of incorrect trials, to identify whether right hemisphere is a main contributor to attempts to solve trials that are too difficult for the children to solve.Table 4Model of Right-Left-Right hemispheric dominance.Table 4Mc/Md trade offFamiliarity/NoveltyHemisphere − strategyFactors of transformation − HemisphereMd \<\< McHigh Familiarity**Right** − associative heuristic to cope fast with easy taskwith experience remains familiar −processing at RightMd ≤ McNovel**Left** − mental attentional heuristics for problem solvingwith experience becomes familiar, overlearned − transfer to RightMd \>\> McHigh novelty**Right** − associative heuristic to cope with a too-difficulty taskwith maturation Mc increases, and with experience task becomes less novel-- transfer to Left[^5]

### 3.3.4. Motivation and self-control {#sec0080}

Motivation is the process whereby *affective* tendencies are expressed in conscious or unconscious affective goals (motives); which are then converted into conscious or unconscious cognitive goals ([@bib0465], [@bib0030], [@bib0055]). *Affective goals* are dispositions to pursue desirable vital/life outcomes or consequences, whereas *cognitive goals* are explicit intentions to do what is believed congruent with one's affective goals. Motivation crosses borders between affective and cognitive goals, and is critical for successfully doing difficult mental tasks. We propose that these affectively-driven *endogenous* cognitive goals rely critically on the insular cortex.

The insula is a structure that connects temporal and frontal lobes deep within the lateral (Sylvian) sulcus; it is active in many diverse task situations, including cognition, interoception, perception and emotion ([@bib0090], [@bib0630], for meta analyses). It is also implicated in cognitive processes such as inhibitory control ([@bib0080]), and speech and language processing ([@bib0435]). In a study examining math anxiety, data showed that anticipating an upcoming math-task elicited increased activity in the posterior insular cortex ([@bib0380]). Together with sub-cortical structures like the basal ganglia, the insula is involved in learning and training tasks ([@bib0100], [@bib0605], [@bib0235]). It has been shown to participate in the affective − aversive − feeling of difficult effort ([@bib0135], [@bib0425], [@bib0640]). The insula integrates interoceptive affect feedback and is related to the orbitofrontal cortex − involved in exteroceptive feedback ([@bib0270]). Regarding interoceptive feedback, Damasio writes: "... relative to the visceral and internal milieu, the insula is the equivalent of the primary visual or auditory cortices" ([@bib0135]). The insula is proposed to have a generic role in problem solving ([@bib0035]), and mathematics may serve as a domain for targeted testing of the understanding of interceptive feeling of effort when comparing children and adults.

Self-control is related to motivation, in the sense that a motivated action requires self-control to be carried out, particularly in complex situations. When task problem-solving appears complicated and misleading, self-consciousness may become particularly necessary (e.g., [@bib0135], [@bib0525]). The operative aspect of this self (i.e., the functional organization that William James would call I-self) is possibly in medial prefrontal cortex and anterior cingulate cortex; this is the operative aspect of self-control found in the *medial frontal cortex* ([@bib0050], [@bib0570], [@bib0495]). The figurative/representational self-processes (the me-self of William James) would be located (see [@bib0135]), in the *posteromedial cortex*, i.e., the combination of posterior cingulate cortex, retrosplenial cortex, and precuneus.

The active coordination between motivation and self-control would elicit activation in the anterior cingulate and insular cortices. Both anterior cingulate and insular cortices are prevalent clusters in children's mathematical problem-solving. In adults, the cingulate and insula cortices activate bilaterally in number and calculation tasks (Table S2). Importantly, these activations in adults are listed further down in the ALE-obtained coordinate list (Table S2 cluster 3 and 4 for number tasks and cluster 5 and 7 for calculation tasks), possibly because adults' prior practice has already transformed into cognitive goals and cognitive circuits/networks the original affective goals (motives) *and* interoceptive feelings of effort, within the math domain. In contrast, the cingulate gyrus and the insula occupy in children respectively the first and second larger clusters of activity for calculation tasks; moreover the insula has the highest likelihood of being detected during calculations in children. In our view this finding is important, suggesting that intrinsic (interoceptive) motivation and self-control in children are critical to perform difficult calculations − but much less so during number tasks, which are more automatized.

3.4. Limitations and considerations for future studies {#sec0085}
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The current meta-analyses focus on functional correlates of brain regions in children relating to number and calculation tasks. We report data representing concordance across these types of task. There are limitations associated with the meta-analysis method, and with choices we had to make, due to the methodology in the original studies. Shortcomings of the ALE method have been previously discussed (e.g., [@bib0185], [@bib0050]). However, ALE meta-analysis remains superior to traditional review approaches, because it permits investigating over-arching patterns of activity across studies. An advantage is the automated computational steps that permit quantification of locations of common activation across studies. In terms of article selection, we chose a conservative approach; we eliminated many articles that appeared to use same participants (see method section), and articles that used children in large age ranges. Eliminating these studies may have lowered strength of concordance detected, but the conclusions drawn from such conservative approach are likely to be more representative of brain responses in school age children, which was our main concern. Further, because of a lack of studies, we have not specifically examined brain responses to mathematical operations separately (i.e., only addition, only multiplication). Methodological choices in some studies lacked developmental theoretical grounding; for instance, since we know from behavioural research that performance on mathematical abilities improves with age, studies should specify a theoretical reason for averaging data over large age ranges. Similar behavioural performance on a task does not imply that the same set of brain regions support the particular performance across age groups ([@bib0030]). For future studies, in addition to demographics such as gender and age range of participants (missing in some articles), it is critical to report within-group contrast coordinates for healthy children − even when this is not the main focus of the paper. Such practices would be beneficial for future meta-analyses on the topic.

4. Conclusion {#sec0090}
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These meta-analyses investigate brain activity in children that underlies processing of number and calculation tasks. These are the first meta-analyses in children younger than 14 years distinguishing between number and calculation tasks. Based on these results we sketched a neuropsychological developmental model of mathematical cognition in stereotaxic-space. We find that mathematical performance in children emerges from known core-regions associated with number processing, such as parietal and frontal areas; but it also emerges from regions not previously recognized in a mental-arithmetic network, such as the insula, the claustrum, and the cingulate gyrus. The insula, in particular, may play a critical role in children's mathematical calculation, because children need strong intrinsic motivation and affective goals to cause their effort in attention and complex processing. Future behavioural and neuroimaging work on children's mathematical cognition should benefit from a refined topographical atlas of mathematical processes in healthy children.

Conflict of interest {#sec0095}
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None.

Appendix A. Supplementary data {#sec0105}
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The following is Supplementary data to this article:

Support is gratefully acknowledged from the Natural Sciences and Engineering Research Council of Canada to M.A. and J.P-L, and from the Russian Science Foundation (\# 17-18-01047) to M.A.

Supplementary data associated with this article can be found, in the online version, at [http://dx.doi.org/10.1016/j.dcn.2017.08.002](10.1016/j.dcn.2017.08.002){#intr0015}.

[^1]: N = number of participants; F = females; Hand = handedness of participants; n/r = not reported; R = right handed; L = left handed Age = mean age, SD = standard deviation, and or age range in parenthesis.

[^2]: Note: Single-study clusters survived a voxel-level threshold of uncorrected p \< 0.001 with a cluster-level threshold for multiple comparisons at p \< 0.05 ([@bib0215]). Contrast threshold was set to p = 0.01, 5000 permutations, \>50 mm^3^. Coordinates (x, y, z) are reported in Talairach convention; L, Left; R, Right; BA, Brodmann area; ALE, Activation likelihood estimate.

[^3]: Encompasses gray matter within angular gyrus BA 39.

[^4]: Encompasses gray matter in the inferior and middle frontal gyri (BA 9).

[^5]: *Note:* Mc = mental-attentional capacity of the individual; Md = mental-attentional demand of the task.
