Early Gray Matter Structural Covariance Predicts Longitudinal Gain in Arithmetic Ability in Children

Abstract Previous neuroimaging studies on arithmetic development have mainly focused on functional activation or functional connectivity between brain regions. It remains largely unknown how brain structures support arithmetic development. The present study investigated whether early gray matter structural covariance contributes to later gain in arithmetic ability in children. We used a public longitudinal sample comprising 63 typically developing children. The participants received structural magnetic resonance imaging scanning when they were 11 years old and were tested with a multiplication task at 11 years old (time 1) and 13 years old (time 2), respectively. Mean gray matter volumes were extracted from eight brain regions of interest to anchor salience network (SN), frontal-parietal network (FPN), motor network (MN), and default mode network (DMN) at time 1. We found that longitudinal gain in arithmetic ability was associated with stronger structural covariance of the SN seed with frontal and parietal regions and stronger structural covariance of the FPN seed with insula, but weaker structural covariance of the FPN seed with motor and temporal regions, weaker structural covariance of the MN seed with frontal and motor regions, and weaker structural covariance of the DMN seed with temporal region. However, we did not detect correlation between longitudinal gain in arithmetic ability and behavioral measure or regional gray matter volume at time 1. Our study provides novel evidence for a specific contribution of gray matter structural covariance to longitudinal gain in arithmetic ability in childhood.


Introduction
With the increasing importance of numerical information in our society, individuals with good mathematical abilities usually have a greater chance of doing well in life [1,2].Especially, achieving basic arithmetic fluency provides a foundation for the development of complex mathematical abilities [3,4].Importantly, successful basic arithmetic ability in childhood is essential to an individual's concurrent academic achievement [5], career success [6], and economic income [7].Therefore, it is of great value to investigate the factors supporting the development of arithmetic ability in children.
Previous behavioral studies have identified several domain-specific cognitive abilities such as working memory [8], inhibitory control [9], task switching [10], and symbolic and non-symbolic number processing [11,12] that contribute to arithmetic ability in children.For instance, working memory capability could predict arithmetic ability throughout primary school [13].Compared with typically developing children, children with arithmetic deficit performed more poorly on working memory, inhibitory control, and symbolic and non-symbolic numerical tasks [14,15].Direct training on working memory task or non-symbolic numerical task could improve performance on untrained arithmetic tasks [16,17].Although these previous studies have intensely investigated the behavioral mechanisms underlying children's arithmetic ability, their brain mechanisms remain largely unknown.
Growing evidence suggests that successful arithmetic ability relies on the activation and deactivation among multiple brain regions [18][19][20].The activated brain regions could be isolated into three dissociable brain networks.Specifically, the anterior insula and anterior cingulate cortex form the salience network (SN) that plays a role in establishing subjective salience of external stimuli and executing complex cognitive processes [21], the dorsolateral prefrontal cortex (dlPFC) and posterior parietal cortex construct the frontal-parietal network (FPN) that plays a role in information retention, manipulation of working memory, and goal-oriented decision-making [22], and the precentral gyrus, premotor cortex, and supplementary motor area form the motor network (MN) that plays a role in mental representation, imagination, and operation [23].On the other hand, the deactivated brain regions, such as the medial prefrontal cortex, posterior cingulate cortex, and angular gyrus, form the default mode network (DMN) that plays a role in internal cognitive processes like arithmetic fact retrieval [24,25].Importantly, the key brain regions of the above-mentioned four networks not only engage in but also interact to support arithmetic processing [19,26].For example, a previous developmental fMRI study found strong causal interactions between the SN and FPN regions during a simple addition performance [19].A resting-state fMRI study reported that children with arithmetic deficit exhibited atypical hyper-coupling among the FPN, MN, and DMN regions [27].
A small number of studies have investigated the brain structural correlates of arithmetic processing in children.Early structural imaging studies found reduced gray matter volume (GMV) mainly in the posterior parietal cortex in children with arithmetic deficit [28].Subsequent studies also reported reduced GMV in the prefrontal cortex [29] and the temporal region [30].A few studies investigated the brain structural correlates of arithmetic ability in typically developing children and found that individual difference in arithmetic ability was positively associated with GMV in multiple brain regions, including the posterior parietal lobe, prefrontal gyrus, precentral gyrus, and temporal cortex [31][32][33].A recent study also reported positive correlation between arithmetic performance and cortical complexity of the left postcentral gyrus, right insular cortex, and left lateral orbital sulcus in children [32].Collectively, existing and limited studies have demonstrated the roles of regionally specific structural measures in the FPN, SN, MN, and DMN in children's arithmetic processing.
There is growing evidence that structural covariance could provide new insight into the brain structural correlates of high-order cognitive abilities [34].Generally, structural covariance describes the phenomenon that gray matter properties of a given brain region may covary with those of other broadly distributed brain regions [35,36].Structural covariance is also thought to be associated with individual difference in behavioral performance.For instance, children with higher intelligence were reported to exhibit stronger structural covariance between the frontal and parietal regions [34], and children with higher vocabulary ability were reported to exhibit greater structural covariance between language-related regions such as the parietal, temporal, and frontal regions [37].Given that arithmetic ability is also a high-order cognitive function that requires functional interactions among multiple brain regions of the FPN, SN, MN, and DMN, we speculated that the structural covariance in the brain regions of the four networks would support the development of arithmetic ability in children.
Previous studies mainly used cross-sectional designs to assess relations between brain and arithmetic development [28][29][30]38].However, confounds from individual variability may result in a failure to detect or false conclusion about the neural mechanisms underlying arithmetic processing [39].In addition, cross-sectional designs are limited because they could not answer whether the effects seen in the brain are the cause or the consequence of arithmetic learning.The optimal solution to avoid these issues is to use longitudinal approach.A few previous studies addressed the role of early brain structures in predicting later arithmetic ability.For example, Evans et al. [31] used GMVs in 8-yearold children to predict gains in arithmetic ability 6 years later.They found that the GMVs of the prefrontal and parietal regions at age 8 well predicted longitudinal gain in arithmetic ability.Price et al. [40] investigated longitudinal associations between the GMVs at the first grade and the arithmetic ability at the second grade and found that the left parietal region was the only region showing an association with arithmetic ability at the second grade.
Overall, the present study aimed to examine whether and how the brain structural covariance of the SN, FPN, MN, and DMN predicts the longitudinal gain in arithmetic ability in typical developing children.We analyzed a public longitudinal dataset entitled "Brain Correlates of Math Development" [41].In the dataset, a total of 63 children performed several cognitive tasks inside or outside the MRI scanner at the age of 8 through 14 years old (an average of 11 years old; time 1) and again at an average of 2.2 years later (time 2).We focused on the T1weighted structural MRI data at time 1 and behavioral data of a single-digit multiplication task at time 1 and time 2. Specifically, we speculated that there are significant associations between the structural covariance in brain regions of the SN, FPN, MN, and DMN at time 1 and the arithmetic gain from time 1 to time 2. We took eight seeds in the dlPFC, insular cortex, precentral gyrus, and angular gyrus (all bilateral), which are the core regions of the FPN, SN, MN, and DMN, and investigated whether and how the structural covariance patterns for each given seed could predict the longitudinal gain in arithmetic ability.
First, we predicted that arithmetic gain over time would be positively associated with structural covariance in brain regions underlying cognitive control at time 1, given that cognitive control has been consistently reported to predict future arithmetic outcomes [42,43].Specifically, we expected that greater arithmetic gain over time would be associated with stronger structural covariance between brain regions of the FPN and SN, given that the FPN has been shown to increase its interactions with the SN to support more efficient cognitive control [22].Second, we predicted that arithmetic gain over time would be negatively associated with structural covariance in brain regions associated with magnitude processing and arithmetic facts retrieval at time 1, given that a number of studies have reported that functional connectivity in corresponding regions is negatively associated with individual difference in arithmetic ability [44][45][46].The authors proposed that the lower brain connectivity in better arithmetic performers might be attributed to more efficiency in numerical operations or arithmetic fact retrieval.Specifically, we expected that greater arithmetic gain over time would be associated with weaker structural covariance between brain regions of the FPN, DMN, and MN, as weaker functional interactions among these brain networks have been reported to reflect more efficient problem solving strategies during magnitude processing or arithmetic facts retrieval in individuals with higher ability [27,44,47].

Participants
We analyzed structural imaging data from a public neuroimaging dataset entitled "Brain Correlates of Math Development."This dataset provides functional and structural MRI data acquired using a 3T scanner in a sample of 132 typically developing children when they were about 11 years old (time 1, 62 boys, mean age: 11.26 ± 1.46 years, range: 8.36-15.00years).Sixty-three of the children underwent repeated assessments with an interval of approximately 2 years (time 2, 21 boys, mean age: 13.69 ± 1.53 years, range: 10.91-16.47years).The data have been deposited in the OpenNeuro (https://openneuro.org),and a detailed description is provided in a previous study [41].Parent self-report indicated that all subjects were native English speakers and healthy without neurological or psychiatric disorders.The present study only analyzed the sixty-three subjects who received arithmetic tests at both time points.Fifteen subjects were excluded, of whom seven had poor structural imaging data quality at time 1, six had accuracy of the arithmetic task being below 50% either at time 1 or at time 2, and two had performance gain of the arithmetic task beyond two standard deviations, yielding 48 children in the final sample (time 1, 21 boys, mean age: 11.35 ± 1.44 years, range: 8.47-14.09years).

Behavioral Tasks Arithmetic Ability
The task involved seventy-two single-digit multiplication problems: forty-eight with correct solutions and twenty-four with false solutions.False solutions were the result of multiplying the first operand plus or minus 1 and the second operand (e.g., 6 × 8 = 40 or 56).Among these problems, half problems were simple (i.e., both operands being smaller or equal to 5), and the other half problems were difficult (i.e., both operands being larger than 5).Participants performed twenty-four practice trials before entering the formal testing.Problems involving 0 or 1 and ties (e.g., 4 × 4) were not included in the final testing but were used in the practice trials.For each trial, a single-digit multiplication problem was presented for 800 ms, followed by a blank screen for 200 ms.Subsequently, a proposed solution was presented for 800 ms.Then a red fixation square was displayed for 2,200, 2,600, or 3,000 ms (shown in Fig. 1).Participants were asked to judge whether the proposed solution was true or false.They were required to respond with the index finger if the proposed solution was true and with the middle finger if proposed solution was false.In order to control for basic processing speed, twenty-four control trials were included, for which a blue square was presented for 800 ms, followed by a red fixation square lasting 2,200, 2,600, or 3,000 ms (shown in Fig. 1).

Structural Covariance Predicts Arithmetic Development
Participants were required to respond with their index finger when the blue square turned red.Reaction time and accuracy were recorded for each trial.
Previous research has suggested that the use of arithmetic operation is more frequent for solving hard problems, whereas easy ones can be easily solved by simple memory retrieval [47,48].Hence, behavioral performance of the hard multiplication problems was chosen as the measure for arithmetic ability in the present study.In addition, several behavioral and neuroimaging studies have suggested reaction time as a more reliable measure than accuracy for single-digit numerical tasks due to limited variation in accuracy [47,49].Hence, to measure performance gain in arithmetic ability, we first calculated difference score in reaction time from time 1 to time 2 for correctly solving hard multiplication problems.In order to control for the possible effect of outliers in task performance, trials with reaction time above or below 3 SD from the individuals' mean were excluded when calculating mean reaction time for each participant at each time point.In order to account for initial difference in arithmetic ability, the reaction time at time 1 was regressed out from the difference score over time.The remaining residual of each child was standardized and used to represent individual longitudinal gain in arithmetic ability.Additionally, the reaction time of the control trials was used as a measure of basic processing speed.
Working Memory There were two working memory tasks.One task was the listening recall subtest of the Automated Working Memory Assessment [50].In this task, a series of sentences were presented, and participants were asked to judge whether each sentence was true or false and to remember the final word of each sentence.This task involved simultaneous storage and processing of verbal information and was used to measure verbal working memory.The other task was the spatial recall subtest.In this task, two pictures were presented simultaneously, with a red dot near the one on the right.The participants would need to identify whether the shape on the right is the same as the shape on the left after rotation.At the end of the trial, the participants were asked to remember the location of the red dot.This task involved simultaneous storage and processing of visuospatial information and was used to measure visuospatial working memory.

Intelligence
Intelligence was measured by the Wechsler [51] Abbreviated Scale of Intelligence.There are two scales: verbal intelligence scale and performance intelligence scale.The verbal intelligence scale includes vocabulary and similarity subtests, while the performance intelligence scale includes block design and matrix reasoning subtests.

Imaging Data
High-resolution structural images provided by the public dataset were used for regional morphometry.Details of imaging data acquisition have been described by a previous study [52].For the sake of completeness, the main descriptions were repeated here.For each subject, a high-resolution T1-weighted 3D structural Ren/Li/Wang/Li image was acquired using a Siemens 3T TIM Trio MRI scanner (Siemens Healthcare, Erlangen, Germany) at CAMRI, Northwestern University's Center for Advanced MRI.The following parameters were used: repetition time = 2,300 ms, echo time = 3.36 ms, matrix size = 256 × 256, field of view = 240 mm, slice thickness = 1 mm, number of slices = 160.
In the present study, structural imaging data at time 1 were preprocessed using the Computational Anatomy Toolbox (http:// dbm.neuro.uni-jena.de/cat12/)for SPM12 (Wellcome Trust Centre for Neuroimaging; http://www.fil.ion.ucl.ac.uk/spm).First, the Template-O-Matic toolbox [53] was used to generate a customized age-and sex-matched tissue probability map, which was used for the following segmentation.Second, the T1-weighted images were segmented into gray matter, white matter, and cerebrospinal fluid.Third, the Diffeomorphic Anatomical Registration through Exponentiated Lie Algebra tool (DARTEL) was applied to normalize each participant's high-resolution structural image to the standard Montreal Neurological Institute space with 1.5 mm isotropic voxels.Compared to the conventional algorithm, the DARTEL approach provides more precise spatial normalization to the template than standard registration methods [54].Fourth, images were visually inspected for sample homogeneity of the unsmoothed data.Mahalanobis distance, combining the mean correlation and weighted overall image quality, was inspected, and outliers (>2 standard deviations) were excluded from further analyses.Finally, spatial smoothing was conducted using 8-mm full-width half-maximum Gaussian kernel.The smoothed gray matter images were subjected to the following analyses.

Statistical Analysis
We first investigated whether behavioral measures at time 1 could predict longitudinal gain in arithmetic ability.The relationships between behavioral measures at time 1 and longitudinal gain in arithmetic ability were examined by both conventional correlation analysis and multivariate stepwise regression analysis.Then we investigated whether regional GMV at time 1 could predict longitudinal gain in arithmetic ability.Smoothed gray matter images at time 1 were submitted to a second-level multipleregression analysis.Longitudinal gain in arithmetic ability was modeled as the covariate of interest while controlling for gender, age, total intracranial volume (TIV), and total intelligence.The results were thresholded at a voxel-wise p < 0.001 (uncorrected) and then FWE-corrected for multiple comparisons at a cluster level p < 0.05 based on the Gaussian random field (GRF) theory.
Then we applied a seed-based approach to investigate brain structural covariance.According to a previous study investigating gray matter structural covariance in the developing brain [55], the right frontal-insular cortex (38, 26, −10), the right dlPFC (44,36,20), and the right precentral gyrus (28, −16, 66) were selected as three seed regions of the SN, FPN, and MN, respectively.The left angular gyrus (−52, −64, 28) was selected as a seed of the DMN according to a previous fMRI study [25], where brain deactivation of this region was found to be correlated with individual difference in multiplication ability.Analyses using contralateral regions of interest (obtained by changing the sign of the x-coordinate for each seed) were also performed.Additionally, the left and right calcarine sulcus (±9, −81, 7) of the primary visual network were selected as two control seeds [55].The mean GMV of each seed was extracted from a 5-mm-radius sphere around those coordinates from the modified gray matter images and was used as a regressor of interest in the following general linear models.
To map structural covariance networks of each seed region, smoothed gray matter images at time 1 were submitted to a secondlevel regression analysis.Ten separate regression models were conducted by entering the extracted GMV from each seed as a regressor of interest.In each regression model, age, gender, and TIV were added as covariates.Each regression model was conducted as below: Y GMV f or each voxel of the brain ( ) Specific contrasts were set in order to identify voxels that expressed a positive correlation with each seed across all participants.Resulting correlation maps were set at a voxel-wise p < 0.001 (uncorrected) and then FWE corrected for multiple comparisons at a cluster level p < 0.05, based on the GRF theory.
To assess the relationship between gray matter structural covariance and longitudinal gain in arithmetic ability, smoothed gray matter images at time 1 were submitted to a second-level regression analysis again.Ten separate regression models were conducted by entering mean GMV of each seed region, longitudinal gain in arithmetic ability, and their parametric interaction term as three regressors of interest.In each model, age, gender, TIV, and general intelligence were added as covariates.Each regression model was conducted as below: Y GMV f or each voxel of the brain ( ) Specific T contrasts were established to identify voxels that expressed a significant interaction between mean GMV of each seed and longitudinal gain in arithmetic ability.The threshold for the resulting statistical parametric maps was established at a voxelwise p < 0.001 (uncorrected) and then FWE-corrected for multiple comparisons at a cluster level p < 0.05 based on the GRF theory.While a positive interaction indicates stronger structural covariance at time 1 in children with greater arithmetic gain over time, a negative interaction indicates weaker structural covariance at time 1 in children with greater arithmetic gain over time.Additional statistical analyses were performed using SPSS and G * power 3.1.9.7 to calculate effect size for significant interactions between GMV of seed of interest and arithmetic gain over time in the regression models.The effect size was calculated as Cohen's f 2 , and the f 2 values were categorized as small (0.02-0.14), medium (0.15-0.34), and large (≥0.35) sizes.

Longitudinal Gain of Arithmetic Ability
We first examined individual difference in the arithmetic gain of children from 11 years old (time 1) to 13 years old (time 2).As shown in Figure 1, the participants exhibited considerable variation in the arithmetic gain over time, with gains ranging from −1.90 to 1.62.We then divided the participants into two groups: a low-progress group (n = 24) and a high-progress group (n = 24), based on the median split of the longitudinal gain.The two groups were matched in terms of age (low-progress: 11.34 ± 1.31 years; highprogress: 11.35 ± 1.60 years; t 46 = −0.02,p = 0.98, Cohen's d = 0.01) and gender distribution (low-progress: 11 boys, 13 girls; high-progress: 10 boys, 14 girls; χ 2 = 0.09, p = 0.77, Cramer's = 0.04).A confirmatory analysis was carried out to test whether the longitudinal arithmetic gain measure displayed the expected pattern of behavioral changes over time.We calculated a repeated-measure ANOVA for the reaction time of the hard multiplication trials, using time (time 1 vs. time 2) as the within-subjects factor and group (low-vs.high-progress) as the between-subjects factor.The similar ANOVA was performed for the reaction time of the control trials.
For the reaction time in the hard multiplication trials, a significant interaction between time and group was observed (shown in Fig. 1; F 1, 46 = 46.55,p < 0.001, partial η 2 = 0.63).As shown in Figure 1, the high-progress group had a significant decrease in the reaction time in the multiplication trials from time For the reaction time in the control trials, there was a significant main effect of time (shown in Fig. 1; F 1, 46 = 28.02,p < 0.001, partial η 2 = 0.38) but no interaction between time and group (F 1, 46 = 0.06, p = 0.81, partial η 2 = 0.001).The main effect of time indicated that all participants (either the low-progress or high-progress groups) became faster at solving the control trials.Hence, the longitudinal change in the reaction time in the hard multiplication trials reflected more change in arithmetic ability than in basic processing speed.

Predicting Arithmetic Gain by Behavioral and Gray Matter Measures
No significant correlations were found between arithmetic gain over time and age (r = 0.06, p = 0.71), basic processing speed (r = −0.15,p = 0.33), verbal intelligence (r = 0.05, p = 0.07), performance intelligence (r = 0.73, p = 0.65), verbal working memory (r = 0.16, p = 0.28), and visuospatial working memory (r = −0.02,p = 0.92) at time 1.No significant gender difference was detected in arithmetic gain over time (t 46 = 0.28, p = 0.79, Cohen's d = 0.08).To test whether multiple behavioral measures together could predict arithmetic gain over time, we performed multivariate stepwise regression with arithmetic gain over time (dependent variable) and all behavioral measures (independent variables) at time 1.Similarly, this multivariate analysis did not reveal any significant behavioral correlates of arithmetic gain over time (p > 0.05).
Next, we investigated whether regional GMV at time 1 could predict arithmetic gain over time.We performed a whole-brain regression analysis using the arithmetic gain over time as dependent variable and the smoothed GMV as independent variable and revealed no significant correlation between the whole-brain GMV and arithmetic gain over time.

Mapping of Gray Matter Structural Covariance
We applied a seed-based structural covariance approach to examine the SN, FPN, MN, and DMN, respectively (Table 1).For the SN, the structural covariance of the right frontal-insular cortex encompassed the middle/inferior frontal gyrus, insula, and putamen.The structural covariance of the left frontal-insular cortex was similar to the pattern seen for the right frontal-insular cortex but additionally encompassed the superior parietal lobule, superior/inferior temporal gyrus, middle/inferior occipital gyrus, lingual gyrus, fusiform gyrus, precuneus, postcentral gyrus, precentral gyrus, and cingulate gyrus.
For the FPN, the structural covariance of the right dlPFC includes a large portion of the frontal cortex (i.e., medial/superior/middle/inferior frontal gyrus), a small part of the superior temporal cortex, inferior parietal lobule, insula, and postcentral gyrus.The structural covariance of the left dlPFC resembled that of the right dlPFC but additionally encompassed the inferior frontal gyrus and precuneus.For the MN, the structural covariance of the right precentral gyrus includes the frontal cortex (i.e., superior/middle frontal gyrus), precentral gyrus, parietal cortex (i.e., superior parietal lobule and angular gyrus), middle temporal gyrus, and lingual gyrus, whereas the structural covariance of the left precentral gyrus includes only the precentral gyrus.

Prediction of Arithmetic Gain by Gray Matter Structural Covariance
Finally, we investigated whether gray matter structural covariance of the seeds of interest at time 1 could predict longitudinal gain in arithmetic ability (Table 2).When using the right frontal-insular cortex of the SN as the seed region, a significant and positive interaction between GMV of the frontal-insular cortex and arithmetic gain was detected in predicting GMV of the left medial frontal gyrus (voxel-wise p < 0.001, cluster-level GRF corrected, R 2 = 0.20, f 2 = 0.25, Fig. 2) and GMV of the right precuneus (voxel-wise p < 0.001, cluster-level GRF corrected, R 2 = 0.12, f 2 = 0.17, Fig. 2), indicating stronger structural covariance of the right SN seed with the frontal and parietal regions in the highprogress group relative to the less-progress group.The similar analysis with the left frontal-insular cortex as seed region showed no association in the structural covariance depending on longitudinal arithmetic gain.When using the right dlPFC of the FPN as the seed region, a significant and positive interaction between GMV of the right dlPFC and arithmetic gain was detected in predicting GMV of the right insula (voxel-wise p < 0.001, cluster-level GRF corrected, R 2 = 0.24, f 2 = 0.32, Fig. 3a), indicating stronger structural covariance of the right FPN seed with the right insula in the high-progress group relative to the less-progress group.In contrast, a significant and negative interaction between GMV of the right dlPFC and arithmetic gain was detected in predicting GMV of the left supplementary motor area (voxelwise p < 0.001, cluster-level GRF corrected, R 2 = 0.18, f 2 = 0.22, Fig. 3b), indicating weaker structural association of the right FPN seed with the motor region in the highprogress group relative to the less-progress group.When using the left dlPFC as the seed region, a significant and negative interaction between GMV of the left dlPFC and arithmetic gain was detected in predicting GMV of the left superior temporal gyrus (voxel-wise p < 0.001, cluster-level GRF corrected, R 2 = 0.18, f 2 = 0.22, Fig. 3b), indicating weaker structural association of the left FPN seed with the temporal region in the highprogress group relative to the less-progress group.
When using the right precentral gyrus of the MN as the seed region, a significant and negative interaction between GMV of the right precentral gyrus and arithmetic gain was detected in predicting GMV of the right middle frontal gyrus (voxel-wise p < 0.001, cluster-level GRF corrected, R 2 = 0.14, f 2 = 0.16, Fig. 4a), indicating weaker structural association of the right MN seed with the frontal region in the high-progress group relative to the less-progress group.When using the left precentral gyrus as the MN seed, a significant and negative interaction between GMV of the left precentral gyrus and arithmetic gain was detected in predicting GMV of the right precentral gyrus (voxel-wise p < 0.001, cluster-level GRF corrected, R 2 = 0.17, f 2 = 0.20, Fig. 4b) and GMV of the left precentral gyrus (voxel-wise p < 0.001, clusterlevel GRF corrected, R 2 = 0.16, f 2 = 0.19, Fig. 4b), indicating weaker structural association of the left MN seed with bilateral motor regions in the high-progress group than the less-progress group.
When using the left angular gyrus of the DMN as the seed region, a significant and negative interaction between GMV of the left angular gyrus and arithmetic gain was detected in predicting GMV of the left middle temporal gyrus (voxel-wise p < 0.001, cluster-level GRF corrected, R 2 = 0.12, f 2 = 0.14, Fig. 5), indicating weaker structural association of the left default parietal seed with the left temporal region in the high-progress group when compared to the less-progress group.Considering the covariance patterns of the right angular gyrus, no effects were detected in the structural covariance depending on longitudinal arithmetic gain.Bilateral calcarine sulcus of

Structural Covariance Predicts Arithmetic Development
the visual network was used as a control seed region.We did not find any structural association depending on longitudinal arithmetic gain (data not shown).

Discussion
In the present study, we used a seed-based structural covariance approach to examine the neural correlates of arithmetic gain over time in typical developing children.The bilateral frontal-insular cortex, bilateral dlPFC, bilateral precentral gyrus, and bilateral angular gyrus were selected as seeds to represent the SN, FPN, MN, and DMN, which have been reported to be engaged in arithmetic processing [18,19,25,56].We found that the longitudinal gain in arithmetic ability was associated with stronger structural covariance of the SN seed with the frontal and parietal regions and stronger structural covariance of the FPN seed with the insula.In contrast, the longitudinal gain in arithmetic ability was associated with weaker structural covariance of the FPN seed with the motor and temporal regions, weaker structural covariance of the MN seed with the frontal and motor regions, and weaker structural covariance of the DMN seed with the temporal region.However, no behavioral measures or regional GMV were found to be associated with the longitudinal gain in arithmetic ability.These findings provide evidence for the importance of structural covariance in supporting arithmetic development in children.

Stronger Structural Covariance of the SN and FPN Predicts Arithmetic Gain
The present study found that children with stronger structural covariance of the insula seed with frontal and parietal regions and stronger structural covariance of the frontal seed with the insula region became faster over time on the arithmetic task.The insula region, as well as the frontal and parietal regions, are often co-activated during performance of a wide range of high-order cognitive tasks [57][58][59].Previous task-based imaging studies, including two meta-analyses, have consistently demonstrated the crucial engagement of the insula region in arithmetic processing [19,56,60].In a developmental fMRI study, Supekar and Menon used multivariate approaches and demonstrated a causal interaction between the insula and the frontal-parietal regions during addition-problem solving in both children and adults [19].In line with the task-based imaging studies, intensified intrinsic functional connectivity of the insula region with the frontal and parietal regions has also been reported to be positively associated with better arithmetic performance in both children and adults [61,62].The insula region has been suggested to be responsible for detecting stimulus saliency in the outside world and can interact with the frontalparietal system for top-down cognitive control [22].Thus, greater functional connectivity of the insula region with the frontal and parietal regions may facilitate arithmetic processing by implementing cognitive control more efficiently.Importantly, previous research has confirmed the role of a b Fig. 3. Relationships between arithmetic gain and structural covariance of the FPN seed region at 11 years old (time 1).The left panel indicates the right and left dlPFC as the FPN seed.The middle panel shows each cluster that is significant for the seed-by-gain interaction.The right panel illustrates structural covariance between GMV in the FPN seed and GMV in each cluster.The x-axis represents GMV in the seed region, and the y-axis represents GMV in each cluster.Slope differences show the different patterns of structural covariance across participants depending on the arithmetic gain.a In comparison with the low-progress group, the high-progress group showed a stronger structural covariance between the right dorsolateral prefrontal cortex (dlPFC) and the right insula but a weaker structural covariance between the right dlPFC and the left supplementary motor area (SMA).b In comparison with the low-progress group, the high-progress group showed a weaker structural covariance between the left dlPFC and the left superior temporal gyrus (STG).

Structural Covariance Predicts Arithmetic Development
the insula in cognitive control in the bilateral and rightward hemisphere [19,59].Similarly, the arithmetic-related insula engagement has been extensively localized in the right hemisphere [56,61], suggesting that the right insula may be more essential than the left insula in facilitating arithmetic processing.Interestingly, our results also showed an association between arithmetic gain and structural covariance of the right but not the left insula.We therefore argue that the right insula may be more active and important in interacting with frontal and parietal regions to support cognitive control involved in arithmetic learning.

Weaker Structural Covariance of the FPN, MN, and DMN Predicts Arithmetic Gain
We found that longitudinal gain in arithmetic ability was associated with weaker structural covariance of the left dlPFC seed with the superior temporal region.It has Ren/Li/Wang/Li been reported that children with low achievement in arithmetic tasks found it difficult to directly retrieve the arithmetic facts from long-term memory and had to adopt numerical operations such as a backup strategy [63].For example, in a multiplication problem such as "6 × 7 = ?," the children might have to retrieve the answer to an easy multiplication problem such as "6 × 6 = ?"and then perform an additional arithmetic operation such as "36 + 6 = ?" to get the correct answer.Such strategy has been frequently reported for children to use to solve multiplication problems with operands larger than 5 [48].Given that the backup strategy is an inefficient strategy that requires greater cognitive control to coordinate the multiple steps needed to get the correct answer, it has been reported to be associated with greater functional connectivity between the frontal and temporal regions [47].Thus, the present result that the stronger structural covariance between the frontal and temporal regions was associated with less arithmetic gain provides supporting evidence for the utilization of this inefficient strategy.Moreover, the frontal-temporal structural covariance finding was specific to the left hemisphere.The left superior temporal region is thought to be responsible for the storage of phonological representations of arithmetic facts [47], and the left dlPFC is considered to be involved in the cognitive control involved in the retrieval of arithmetic facts [64].Hence, the left hemispheric dominance in frontal-temporal structural covariance may also reflect more laborious retrieval processes of arithmetic facts from long-term memory in the low-progress children.Overall, although the two groups showed comparable arithmetic performance at time 1, the utilization of an inefficient backup strategy or effortful retrieval of arithmetic facts might hinder longitudinal arithmetic gain from time 1 to time 2.
We also found that longitudinal gain in arithmetic ability was associated with weaker structural covariance between the frontal and motor regions and weaker structural covariance within the motor regions.The motor regions have been reported to be involved in arithmetic problem solving [65].A possible explanation is that participants often use a finger counting strategy for arithmetic problem solving [66].Interrupting finger movement has been reported to disrupt arithmetic processing in both children and adults [66,67].During the initial stage of arithmetic learning, the finger counting and backup strategies have been reported to be frequently used by children, especially for those with arithmetic deficits [63,68,69].We speculate that the children with lower progress might also utilize the finger counting strategy to achieve upregulated arithmetic performance at time 1.Accordingly, they might rely on the strong coupling between the frontal and motor regions and within the motor regions as well.The inefficient finger counting strategy might also be detrimental for the longitudinal arithmetic gain from time 1 to time 2.
Moreover, the present study found that longitudinal gain in arithmetic ability was associated with weaker structural covariance between the default parietal and temporal regions.Previous task-based imaging studies

Structural Covariance Predicts Arithmetic Development
have highlighted the critical role of the default parietal region in arithmetic processing [25,70].For example, using positron emission tomography, Zago et al. [70] observed stronger activation in the default parietal region for single-than two-digit multiplication problems.Given that the answers for single-digit problems are mostly retrieved by memorizing table facts, the stronger activation in the default parietal region could be interpreted as a stronger reliance on automatic arithmetic fact retrieval.When examining the intrinsic functional connectivity in children, researchers found that the functional connectivity between the default parietal and temporal regions in the first grade was negatively correlated with arithmetic ability in the second grade, indicating that hyper-connectivity between the two regions could be considered a neural signature of low arithmetic ability in the future [44].The authors interpreted that the hyper-connectivity may reflect more efforts in the processing and retrieval of highly learned arithmetic facts.In the present study, the greater structural covariance between the default parietal and temporal regions with less arithmetic gain may provide additional evidence for the role of the hyper-connectivity in arithmetic disability.Additionally, our results showed significant associations for structural covariance of the left but not the right parietal regions with arithmetic gain, which is consistent with previous findings suggesting left hemispheric dominance of the parietal region in retrieval of arithmetic facts [40].

Behavioral and Regional Gray Matter Measures Did Not Predict Arithmetic Gain
The present study did not find any behavioral measures correlated with the longitudinal gain in arithmetic ability.A number of previous behavioral studies have reported that behavioral measures, such as intelligence, processing speed, working memory, could predict longitudinal gain in children's arithmetic ability [42,43,71].However, these studies often employed a relatively large sample size (n > 100).When restricted to a small sample size, as in the present study (n < 50), two previous studies reported that arithmetic gain over time was significantly predicted by brain measures but not by behavioral measures [31,33], and one study reported that arithmetic gain over time was better predicted by brain measures than behavioral measures [72].Additionally, a previous longitudinal study of reading development reported that brain measures such as right prefrontal activation during a reading task significantly predicted future reading gain in children with dyslexia (n = 25) [73].However, none of 17 widely used standardized measures of reading and language significantly predicted reading gain in those children.Taken these and our present findings together, it suggests that the brain measures may outperform behavioral measures in predicting future academic ability when a small sample size (n < 50) is administered.Future research should consider if there are other behavioral measures not included in this study that may predict future arithmetic gain over and above initial brain measures.
Previous imaging studies have reported mixed evidence regarding the longitudinal association between GMV and arithmetic ability in children [31,40].For example, Evans et al. [31] found that GMV in the frontal, parietal, and temporal regions predicted future gains in arithmetic performance.Price et al. [40] reported that GMV in the parietal region was the only region showing an association with later arithmetic performance.Supekar et al. [33] found that GMV in the hippocampus but not the parietal region was related to arithmetic gain over time.In the present study, we failed to find any regional GMV measures in predicting arithmetic gain over time.It has been suggested that the discrepancies in the literature on the association between brain measures and arithmetic skill could be partly attributed to the different demands of the tests used to measure arithmetic ability [74].Notably, the present study only focused on response time in a single-digit multiplication task, whereas most previous studies used standardized tests that included a wide range of arithmetic skills [31,40].Only one study has examined the specific role of GMV in predicting multiplication ability in children who were about the same age as our participants [74].This study only found a significant concurrent association between GMV in the left temporal region and multiplication ability at age 11 but not at age 13 and failed to find any regional GMV measures at age 11 in predicting multiplication gain over time.Given the fact that the left temporal region is critical to arithmetic fact retrieval, GMV in this region only explaining gains in early stages of multiplication processes is likely due to the younger children relying on repeatedly reciting multiplication tables to solve the calculation problems, while older children may have automatized these procedures.Hence, regional GMV may play a timelimit role in explaining multiplication achievement.Future studies should consider longitudinal studies with multiple time points including younger children to address this issue more comprehensively.

Fig. 1 .
Fig. 1.Multiplication task and longitudinal gain in arithmetic ability from 11 (time 1) to 13 years old (time 2). a Procedure for a hard multiplication trial.b Procedure for a control trial.c Individual difference in the longitudinal gain of arithmetic ability.d Change in reaction time of the control trials for the low-progress and high-progress groups.e Change in reaction time of the hard multiplication trials for the low-progress and highprogress groups.

Fig. 2 .
Fig. 2. Relationships between arithmetic gain and structural covariance patterns of the SN seed region at 11 years old (time 1).The left panel indicates the right frontal-insular cortex as the SN seed.The middle panel shows each cluster that is significant for the Seed-by-Gain interaction.The right panel illustrates structural covariance between GMV in the SN seed and GMV in each cluster.The x-axis represents GMV in the SN seed, and the y-axis represents GMV in each cluster.Slope differences show the different patterns of structural covariance across participants depending on the arithmetic gain.In comparison with the low-progress group, the high-progress group showed a stronger structural covariance of the right frontalinsular cortex with both the left medial frontal cortex (mPFC) and the right precuneus.

Fig. 4 .
Fig. 4. Relationships between arithmetic gain and structural covariance of the MN seed region at 11 years old (time 1).The left panel indicates the right and left precentral gyrus as the MN seed.The middle panel shows each cluster that is significant for the Seed-by-Gain interaction.The right panel illustrates structural covariance between GMV in the MN seed and GMV in each cluster.The x-axis represents GMV in the seed region, and the y-axis represents GMV in each cluster.Slope differences show the different patterns of structural covariance across participants depending on the arithmetic gain.a In comparison with the low-progress group, the high-progress group showed a weaker structural covariance between the right precentral gyrus and the right middle frontal gyrus (MFG).b In comparison with the lowprogress group, the high-progress group showed a weaker structural covariance between the left precentral gyrus and the bilateral precentral gyrus.

Fig. 5 .
Fig. 5. Relationships between arithmetic gain and structural covariance of the DMN seed region at 11 years old (time 1).The left panel indicates the left angular gyrus as the DMN seed.The middle panel shows the cluster that is significant for the seedby-gain interaction.The right panel illustrates structural covariance between GMV in the DMN seed and GMV in the significant cluster.The x-axis represents GMV in the seed region, and the y-axis represents GMV in the significant cluster.Slope differences show the different patterns of structural covariance across participants depending on the arithmetic gain.In comparison with the low-progress group, the highprogress group showed a stronger structural covariance between the left angular gyrus and the left middle temporal gyrus (MTG).

Table 1 .
Mapping of covariance of gray matter structure in each seed point