^{a}

^{b}

^{c}

^{d}

^{e}

Math learning difficulties (MD) correspond to math achievement below the 25th percentile and are cognitively heterogeneous. It is not known precisely how cognitive mechanisms underlie distinct subtypes of MD. A bottom-up, cluster-analytic strategy, based on visuoconstructional, visuospatial and phonological working memory, and non-symbolic and symbolic magnitude processing accuracy, was used to form subgroups of children from 3rd to 5th grades according to their math achievement. All children had nonverbal intelligence above the 20th percentile and presented a broad spectrum of variation in math ability. External validity of subgroups was examined considering intelligence and math achievement. Groups did not differ in age. Two groups with a high incidence of MD were associated, respectively, with low visuospatial/visuoconstructional and low magnitude processing accuracy. One group with average cognitive performance also presented above average intelligence and a small incidence of MD. A fourth group with high cognitive performance presented high math performance and high intelligence. Phonological working memory was associated with high but not with low math achievement. MD may be related to complex patterns of associations and dissociations between intelligence and specific cognitive abilities in distinct subgroups. Consistency and stability of these subgroups must be further characterized. However, a bottom-up classification strategy contributes to reducing the cognitive complexity of MD.

Individuals with math learning difficulties (MD) can have problems in a wide range of numerical skills, such as estimating and comparing magnitudes, reading and writing Arabic numbers, mastering the four basic operations, retrieving the math tables, among others (

One of the main issues about the neuropsychological research on mathematics is the high variability of cognitive factors associated with math achievement, what is reflected in a high heterogeneity of deficits among children with MD (

Different criteria to identify subtypes of MD have been supported by the literature, but this becomes more complex when discussing the difficulty in using an appropriate criterion for the identification of the MD group (see

Classification systems based only on standardized math tests could present some problems since they do not consider, for example, the importance of the stability and persistence of cognitive profiles associated with MD (

This Introduction is divided into four sections. First, we discuss the cognitive heterogeneity underlying math difficulties. Second, we discuss research approaches to subtyping MD, analytically reducing this complexity. Third, we review the literature that used the bottom-up approach of complexity reduction and classification in which subgroups of MD individuals emerged through cluster analysis. Finally, we discuss the approach used in the current study.

Cognitive factors underlying MD have been discussed in the numerical cognitive research and several subtypes of MD have been found (

The search for relevant dimensions to identify subtypes of MD can be based on the analysis of patterns of association and dissociation between the relevant cognitive variables. The most investigated cognitive dimensions in the math domain are discussed below.

Math is considered to be the most difficult subject in school (

Some authors have proposed that EF plays a role also in tasks taping on non-symbolic magnitude representations (

Impairments in WM/EF could explain the comorbidity between MD and attention-deficit/hyperactivity disorder (ADHD). However, impairments in WM/EF are nonspecific, being present in virtually all neurodevelopmental disorders (

The term phonological processing is commonly used to refer to a set of abilities frequently impaired in developmental dyslexia such as rapid automatized naming, phonological working memory and phonemic awareness (

Mental spatial representation, manipulation and visuospatially guided control of action are involved in several math-related abilities such as number representation in the mental number line (

The expression numerical processing is applied to identify the ability to quantify numerosities, such as estimating set sizes, comparing sets and counting; also, to identify the ability to use numerical notations. Non-symbolically represented sets of up to 4 elements are quantified rapidly and accurately through subitizing, which seems to depend on visual attentional, rather than on quantitative, processes (

Workings of the ANS are characterized by the two basic psychophysical laws of Weber and Fechner, which explain several behavioral effects observed in numerical processing (

Accuracy of the ANS has been proposed as a predictor of complex arithmetic abilities, and that ANS represents a core system from which human mathematical thinking emerges. ANS accuracy, assessed using the Weber fraction, has been associated with both typical (

In some studies, the Weber fraction is strongly associated with a specific measure of basic numerical processing (

A performance dissociation was observed by

Early strategies to identify subtypes of MD were top-down or theoretically oriented (

One strategy used to identify subtypes of MD involves cognitive-neuropsychological single-case studies. These studies have helped to identify distinct patterns of performance dissociations, or specific domains of impairment, in MD. Specific patterns of impairment were observed in Arabic number reading (

Analysis of performance dissociations based on group and case studies may be considered a top-down approach to subtyping, as the relevant cognitive dimensions are previously identified and (ideally) pure cases have then sought that fit into the relevant dimensions. The top-down strategy is characterized by several limitations. First and foremost, the subtypes are defined a priori, and this may prevent recognition of patterns when these are not predicted by existing theories. Second, patterns of impairments observed in individual cases are not necessarily consistent with the ideal deficits described in theoretical models. For example, a deficit in ANS would be expected to impair subtraction operations, but this is not always the case (

The limitations of the top-down approach have led some researchers to pursue the data-driven or bottom-up approach, which consists of letting the groups emerge from multivariate techniques of classification, such as cluster analysis (

Results from

Moreover, a few studies have employed a cross-sectional, bottom-up approach focusing on individuals with low math achievement (

Generally, the bottom-up strategy can be an alternative for resolving problems associated with the top-down approach, without using arbitrary cutoff scores to find subtypes of MD. This type of strategy allows examining which cognitive mechanisms are associated with individual differences in math learning. In addition, it offers the possibility of investigating how these mechanisms are associated with working memory, phonological processing, visuospatial/visuoconstructional processing and magnitude processing.

In the current study, a bottom-up, data-driven analytical approach was used to identify profiles of cognitive impairments underlying math difficulties; and, a top-down approach was used for a theoretically-driven analysis of subgroup validity. We wanted to examine the hypothesis that relatively specific impairments in visuospatial/visuoconstructional, phonological and magnitude processing (non-symbolic and symbolic) are associated with standardized math achievement. We also wanted to compare the relative associations of specific cognitive and/or general intelligence factors with math performance within the subgroups. This strategy has not been frequently used in research in numerical cognition. The present study is the first to use a bottom-up approach, to explore different MD subtypes, which also includes the internal Weber fraction and a measure of symbolic magnitude processing efficiency as criterion variables.

The initial sample comprised 290 children, ages from 8 to 11 years, in the 3rd to 5th grades. Participants were assessed in two distinct phases. First, a screening assessment was performed using the Arithmetic subtest from the Brazilian School Achievement Test (TDE; ^{2} < 0.2) or they showed an internal Weber fraction that exceeded the limit of discriminability of the non-symbolic magnitude comparison task (

The study was approved by the local research ethics committee (COEP–UFMG) in compliance with the Helsinki principles. Informed consent was obtained in written form from parents and orally from children.

Different research criteria have been used to identify individuals with MD. According to

The tasks were selected considering the cognitive factors frequently associated with mathematical performance: visuospatial and phonological working memory (

Fluid intelligence was assessed using the age-appropriate Brazilian validated version of Raven’s Coloured Progressive Matrices (

The Brazilian School Achievement Test (

Phonological working memory was evaluated using the backward digit span of the Brazilian WISC-III Digits subtest (

This test is a measure of the visuospatial component of working memory. We used the backward order to assess it, according to the procedure by

The copy of the Rey figure assesses visuospatial and visuoconstructional abilities. It is based on a complex black and white line drawing that the child must copy as accurately as possible. The accuracy score is based on the presence, distortion or malpositioning of each of the 18 elements of the figure. This task assesses visuospatial-representational, executive functions and visuoconstructional abilities (

In the computerized non-symbolic magnitude comparison task, participants were instructed individually to compare two simultaneously presented sets of dots, indicating which was more numerous. Black dots were presented in a white circle over a black background. In each trial, one of the two white circles contained 32 dots (reference numerosity), and the other contained 20, 23, 26, 29, 35, 38, 41 or 44 dots. Each magnitude of dot sets was presented eight times. The task comprised of 8 learning trials and 64 experimental trials. The maximum stimulus presentation time was 4,000 ms, and the intertrial interval was 700 ms. Between trials, a fixation point appeared on the screen for 500 ms; the fixation point was a cross printed in white with height and width of 3 cm. As a measure of ANS accuracy, the Weber fraction (

In the computerized symbolic magnitude comparison task, participants were instructed individually to judge if an Arabic digit presented on the computer screen was larger or smaller than 5. The digits presented on the screen were 1, 2, 3, 4, 6, 7, 8 or 9 (with numerical distances from the reference varying from 1 to 4), printed in white over a black background. If the presented digit was smaller than 5, children should press a predefined key on the left side of the keyboard. Otherwise, if the presented digit was greater than 5, children should press a key on the right side of the keyboard. The task comprised a total of 80 trials, 10 trials for each numerosity. The presented number was shown on the screen for 4000 ms, and the time interval between trials was 700 ms. Before each test trial, there was a fixation trial (a cross) with duration of 500ms. As a measure of symbolic magnitude processing efficiency, we used an RT index penalized for inaccuracy:

This task comprised addition (27 items), subtraction (27 items), and multiplication (28 items) operations for an individual application, which were printed on separate sheets of paper. Children were instructed to answer as quickly and as accurately as possible, with the time limit per block being 1 min. Arithmetic operations were organized into two levels of complexity and were presented to children in separate blocks: one consisted of simple arithmetic table facts and the other of more complex ones. Simple additions were defined as those operations having results below 10 (i.e., 3 + 5), while complex additions were those having results between 11 and 17 (i.e., 9 + 5). The problems (i.e., 4 + 4) were not used for addition. Simple subtractions comprised problems in which the operands were below 10 (i.e., 9 − 6), while in complex subtractions the first operand ranged from 11 to 17 (i.e., 16 − 9). No negative results were included in the subtraction problems. Simple multiplications consisted of operations with results below 25 or belonging to the 5-table (i.e., 2 × 7, 6 x 5), while in complex multiplication, the results ranged from 24 to 72 (6 × 8). Previous evidence regarding the validity of this task was obtained by

Children were assessed in their schools, in two sessions of approximately 30 minutes each, by specially trained undergraduate psychology students. Intelligence and school achievement assessments were applied to groups of approximately 6 children during the first session, and the other tasks were individually assessed in the second session. The order of the neuropsychological tests was pseudo-randomized in two different sequences.

We performed hierarchical cluster analysis (Ward method with squared Euclidean distance) using measures of phonological and visuospatial working memory, visuospatial and visuoconstructional processing, and symbolic and nonsymbolic magnitude accuracy as the criterion variables for cluster formation. The Ward method considers all possible combinations of clusters and combines clusters which minimize the increase in the error sum of squares in each iteration (

To characterize the neuropsychological profile of the clusters, we performed a series of variance analysis (ANOVA) with each of the neuropsychological measures as the dependent variables. We also compared the distribution of age and intelligence among clusters. To examine cluster validity, we investigated the frequency of MD (based on TDE scores) in each cluster, and we performed a series of ANCOVAs, with intelligence as a covariate, using the z-scores of the TDE Arithmetic subtest, as well as the single digit operations as dependent variables. We reported appropriate effect sizes indexes (Cohen’s

To identify possible subgroups of cognitive performance that could be eventually associated with math achievement, we used a bottom-up strategy. This type of strategy consists of letting candidate subgroups emerge through cluster analysis and afterwards interpreting and examining their validity.

As criteria for cluster membership, we used the performance in the backward forms of digit span and Corsi blocks, Rey figure copy, internal Weber fraction (

Four clusters were identified. Each cluster was interpreted and characterized according to performance on the respective criterion variable.

The descriptive analysis for each cluster with means and standard deviations of each criterion variable is shown in

Measure | Cluster 1 ( |
Cluster 2 ( |
Cluster 3 ( |
Cluster 4 ( |
ANOVA |
Post-hoc (Bonferroni test) | η^{2}_{p} |
|||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Criterion variables of cluster formation | ||||||||||||

Digit span (backwards) | -0.78 | 0.66 | -0.22 | 0.71 | -0.45 | 0.63 | 1.14 | 0.71 | 75.24 | < .001 | 4 > 2 > 3 = 1 & 1 = 2 | 0.546 |

Corsi Blocks (backwards) | -1.00 | 1.08 | -0.48 | 0.70 | 0.01 | 0.73 | 0.72 | 0.91 | 28.95 | < .001 | 4 > 3 > 2 = 1 | 0.316 |

Rey Figure (copy) | -1.94 | 0.52 | 0.03 | 0.76 | 0.25 | 0.76 | 0.42 | 0.69 | 73.85 | < .001 | 2 = 3 = 4 > 1 | 0.541 |

Weber fraction^{a} |
0.22 | 0.89 | 1.01 | 1.28 | -0.18 | 0.69 | -0.49 | 0.65 | 24.76 | < .001 | 2 > 1 > 4 = 3 & 1 = 3 | 0.283 |

Symbolic magnitude efficiency^{a} |
0.32 | 0.82 | 1.15 | 1.42 | -0.31 | 0.57 | -0.44 | 0.42 | 35.25 | < .001 | 2 > 1 > 3 = 4 | 0.360 |

Age and intelligence | ||||||||||||

Age (month) | 115.91 | 10.56 | 116.78 | 10.03 | 119.11 | 10.37 | 117.70 | 8.73 | 0.86 | 0.450 | -- | 0.014 |

Raven’s CPM ( |
-0.12 | 0.45 | 0.50 | 0.57 | 0.59 | 0.68 | 1.11 | 0.52 | 24.20 | < .001 | 4 > 2 & 3 > 1 | 0.278 |

^{a}Weber fraction and Symbolic magnitude efficiency have an inverse interpretation, the higher the values, the worse the accuracy.

Cluster 1 was characterized by low visuospatial performance, both in the Rey figure and Corsi blocks, compared to other clusters. Compared to Cluster 2, this Cluster showed lower performance on the digit span (

Cluster 2 showed the lowest performance on the tasks which assessed non-symbolic (

Cluster 3 presented average or near average performance for all variables and was labeled "Average performance". Compared to Cluster 1, Cluster 3 did not differ significantly on the digit span and

Cluster 4 showed higher performance than all other clusters on the digit span, the Corsi blocks and

To examine the validity of the emerging clusters, we analyzed age and performance differences in measures of intelligence, standardized math achievement and single digit operations in each cluster.

We calculated ANOVAs to analyze age and intelligence differences among the clusters.

All participants had general intelligence scores above the 20th percentile. The Raven’s CPM was correlated statistically with the digit span (

We calculated ANOVAs to investigate differences in intelligence among the clusters. As shown in

Clusters 1 and 2 were characterized by a higher frequency of children with math learning difficulties, defined as performance below the 25th percentile in the Arithmetic subtest of the TDE. Only one child with math difficulties was observed in Cluster 4. The frequency of MD children was 56.5% in Cluster 1, 38.9% in Cluster 2 and 17.7% in Cluster 3. Cluster 4 comprises Control children (see ^{2} = 14.80,

As the TDE Arithmetic subtest was used to categorize individuals according to typical or atypical achievement, performance on a different set of single-digit operation tasks was used as an external criterion. ANCOVAs comparing the clusters in single-digit operations were calculated using intelligence as a covariate. Clusters 1 and 2 presented lower performance than Cluster 4 in all operations. Cluster 4 exhibited scores above the mean in all single-digit operations. Even though Clusters 1 and 2 had different cognitive impairments, they presented a similar profile in single-digit operations. Cluster 3 performed slightly below the mean and significantly below Cluster 4 in the more complex single-digit subtraction and multiplication tasks (see

Measure | Cluster 1 ( |
Cluster 2 ( |
Cluster 3 ( |
Cluster 4 ( |
ANCOVA |
Post-hoc (Bonferroni test) | η^{2}_{p} |
|||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Simple addition | -0.98 | 1.57 | -0.33 | 1.33 | 0.20 | 0.57 | 0.33 | 0.45 | 10.03 | < .001 | 3 = 4 > 1 = 2 | 0.139 |

Complex Addition | -0.83 | 1.02 | -0.35 | 1.01 | 0.03 | 0.84 | 0.54 | 0.84 | 8.59 | < .001 | 4 > 1 = 2; 4 = 3; 2 = 3; 3 >1 | 0.121 |

Simple subtraction | -0.53 | 0.99 | -0.33 | 0.99 | -0.02 | 0.97 | 0.47 | 0.82 | 4.46 | < .005 | 4 > 2 = 1; 4 = 3; 1 = 2 = 3 | 0.067 |

Complex subtraction | -0.77 | 0.49 | -0.41 | 0.83 | -0.10 | 0.81 | 0.75 | 1.02 | 15.07 | < .001 | 4 > 1 = 2; 4 > 3; 3 > 1; 3 = 2 | 0.195 |

Simple multiplication | -0.84 | 0.97 | -0.28 | 1.04 | -0.02 | 0.85 | 0.57 | 0.81 | 7.51 | < .001 | 4 > 1 = 2; 4 > 3; 3 > 1; 3 = 2 | 0.108 |

Complex multiplication | -0.47 | 0.75 | -0.19 | 1.09 | -0.19 | 0.83 | 0.62 | 0.97 | 6.96 | < .001 | 4 > 1 = 2 = 3 | 0.101 |

In

Single-digit operations across clusters. Bars indicate the standard errors of the means.

The present study investigated the heterogeneity of mathematics difficulties and its relation to domain-specific and domain-general cognitive skills using a bottom-up or data-driven approach. Using cluster analysis, and based on neuropsychological performance, four groups with specific cognitive profiles were formed. Next, we investigated how these groups performed on math school achievement and single-digit operation tasks, and the effect of general intelligence on performance.

To our knowledge, this is the first study to use a bottom-up approach to explore different math difficulties (MD) subtypes that also includes measures of both non-symbolic and symbolic comparison accuracy as criterion variables. Our final solution was composed of four clusters: Cluster 1 presented low visuospatial performance; Cluster 2, low magnitude processing accuracy, and Clusters 3 and 4 had, respectively, average and high performance in all tasks. Age was comparable across groups and all of the children had normal intelligence. Nevertheless, the clusters presented differences in math performance.

Children with MD were concentrated in Cluster 1 (56.5%) and Cluster 2 (38.9%). Interestingly, Cluster 1 (visuospatial deficits) and Cluster 2 (magnitude processing deficit), performed similarly in the arithmetic school achievement test. The other two clusters were predominantly composed of Control children. In Cluster 3, only 17.7% of children were classified as MD, and in Cluster 4 this percentage decreased to 1.8% (just one child). Arithmetic performance of Clusters 3 and 4 was comparable. It should also be noted that Cluster 4 was composed mostly of children with higher intelligence, which might be associated with the generally higher performance presented in all of the cognitive and mathematical tasks.

A similar scenario of cluster differences was observed for the single-digit operation tasks. Clusters 1 and 2 presented difficulties in the single-digit tasks, while Clusters 3 and 4 presented, respectively, average and high performance. None of the clusters presented selective deficits in any specific kind of single-digit operation. It is important to mention that between-cluster differences regarding standardized math and single-digit operations remained significant after statistically controlling for the effects of intelligence.

These results raise several points of discussion. In the following, we are going to examine the cognitive specificity of clusters, as well as the role of visuospatial/visuoconstructional abilities, magnitude processing and intelligence on math performance.

Cluster 1 presented the lowest performance in the standardized math achievement test. Intelligence was also lower in this group, as compared to the other clusters, but still in the normal range. This group was interpreted as having a significant deficit in visuospatial and visuoconstructional abilities, with the lowest mean scores in the Rey figure copy and backward Corsi blocks. It is possible that this visuospatial impairment is attributable to deficits in the executive components of these tasks, as performance on the backward digit span was also below the sample mean. One interesting feature of Cluster 1 is that the Weber fraction and symbolic magnitude efficiency were average. Also, noteworthy, Cluster 1 presented the highest frequency of MD individuals. Intelligence in Cluster 1 was normal and evenly distributed around the mean. This suggests that intelligence and school achievement may dissociate in a group of individuals with low visuospatial/visuoconstructional performance.

Cluster 2 showed the most specific pattern of cognitive deficits, with low performance only in magnitude processing accuracy (ANS and symbolic magnitude efficiency measures). Performance on the other three cognitive markers was average, and mean intelligence was above the population mean.

Cluster 3, the largest one, was composed of individuals with average to high average performance in most of the cognitive tasks, including intelligence. Interestingly, this cluster presented lower phonological working memory, but still in the normal range. This could be associated with the small MD group allocated in this cluster. Cluster 3 presented MD children in a smaller proportion than Clusters 1 and 2.

Cluster 4 was composed mostly of individuals with high performance in neuropsychological and math tests. The highest cognitive performance in this cluster was related to phonological working memory and intelligence. This cognitive profile is associated with the high math performance in this cluster.

We believe that the clusters that emerged represent relatively specific, consistent and theoretically interpretable patterns of cognitive performance and associations with intelligence and math achievement. As cluster analysis is an exploratory technique, it is not possible to attest to the replicability of these patterns in other samples, their temporal stability or their epidemiological relevance. Cluster analysis is interpreted in this context as a device to identify patterns of association and dissociation between psychological processes, in the same vein as the role played by quasi-experimental studies in cognitive neuropsychology (

Important results have emerged in the analysis so far and will be discussed in further detail. Some specific patterns of cognitive performance related to visuospatial and magnitude processing accuracy are related to math performance. At the same time that math achievement is related to intelligence (Clusters 3 and 4) it seems also to dissociate from general cognitive ability in some cases (Clusters 1 and 2).

In our study, Cluster 1 with low performance in the Rey figure copy and backward Corsi blocks was the one with the lowest achievement in math and the one that aggregated the largest portion of kids with MD. As intelligence in this group was in the normal range and also partially dissociated from math achievement, one may suppose that specific visuospatial and visuoconstructional deficits are detrimental to math achievement.

As both Rey copy and backward Corsi tasks impose important demands in terms of executive functioning, it is not possible to distinguish between visuospatial representational and access deficits. Low performance could thus be related to task requirements on executive functions. It is important, however, to underline the visuospatial nature of the most severe difficulties, as this group performed below but not far from average on digit span tasks, which also tap into executive functions (

A cluster of low visuospatial working memory performance associated with math difficulties was also observed by

Supporting evidence for an association between visuospatial abilities and ANS-related performance was also obtained by

In our results, no such association between visuospatial and the non-symbolic numerical processing signature was observed in a second cluster. The discrepancy between the results found by

The influence of visuospatial abilities on math achievement and the existence of subgroups of children with MD who are visuospatially impaired has been postulated since the inception of neuropsychological interest in the area (

A role for visuospatial processing in single-digit operations has been found in preschool children (

Our results suggest that relatively specific impairments in visuospatial and visuoconstructional processing, partially dissociated from general intelligence and working memory performance, could be associated with difficulties in learning arithmetic. This hypothesis could be corroborated by single-case observations in which visuospatial and visuoconstructional processes would be the only forms of impairment.

The most specific and consistent group emerging in our study was Cluster 2, which was characterized by normal intelligence in the face of a large percentage of children presenting MD. The cognitive deficits presented in Cluster 2 were a higher Weber fraction and a higher

It is remarkable that intelligence was only weakly correlated with the Weber fraction (

A role for possibly innate, non-symbolic and approximate numerical representations in the acquisition of arithmetic skills, has been proposed in the triple code model (

Empirically, the role of the ANS in math learning has been also contentious. Some researchers have obtained data suggesting a moderate effect of ANS-related performance on math achievement both in typical (

Recent studies have also proposed that ANS is important for mathematical achievement even when considering other basic symbolic numerical processes, such as counting and cardinality comprehension (

As Cluster 2 was the only one exhibiting impairments in both the ANS accuracy and symbolic magnitude efficiency, together with a large proportion of individuals with MD, our results support the hypothesis that low resolution of non-symbolic and symbolic magnitude processing should be considered an important risk factor for MD.

Intelligence and other highly complex cognitive abilities, such as working memory, are clearly implicated in math learning at every age and ability level (

Our results clearly point to the importance of nonverbal fluid intelligence, at least as measured by the Raven’s CPM, as an important correlate of math achievement, both in typical and atypical individuals. The interactions between intelligence, specific cognitive deficits, and math achievement could be quite complex. Results suggest both associations and dissociations. The most salient association was found in the high performing Cluster 4. Only one individual in Cluster 4 exhibited MD and the performance in all math, intelligence and cognitive tasks was well above average. This supports a positive, mutually reinforcing loop between general and specific cognitive abilities and math achievement. This positive association contrasts with the negative association between intelligence and math achievement in one of the clusters described by

Dissociations between intelligence and specific cognitive abilities were observed in Clusters 1 and 2. As a group, both Clusters 1 and 2 were characterized by low average intelligence and low math achievement. However, this association does not hold for all individuals. A significant proportion of individuals in both groups presented MD, although their intelligence was well above average. This suggests math difficulties in highly intelligent individuals in Clusters 1 and 2 could be explained, respectively, by specific deficits in executive visuospatial abilities and poor resolution of numerical magnitude processing.

A role for intelligence in math difficulties was emphasized by the results of the children with specific cognitive deficits in Clusters 1 and 2.

Both patterns of associations and dissociations were observed, suggesting the existence of specific mechanisms and complex interactions among them.

In conclusion, this study supports the hypothesis that MDs are a cognitive heterogeneous phenomenon. At the same time, our data suggest that single mechanisms may play specific roles. Other authors, using different clustering criteria, were able to identify a host of distinct subgroups, varying from one study to another (

Another salient feature in this study was the role of intelligence. Data suggested both associations and dissociations. Higher intelligence was associated with typical cognitive profiles and higher math achievement. Higher math performing groups presented both higher intelligence and lower incidence of cognitive deficits compared to other groups. A cognitive explanation of math difficulties thus requires the concomitant consideration of both general and specific factors, paying attention to complex interactions underlying interindividual variability.

Arithmetic is a very complex subject from the cognitive point of view. Single-case studies in adults and children show that, to a certain degree, arithmetic is composed of relatively segregated modules (

Specific cognitive requirements, such as number sense, visuospatial or phonological processing, may change according to the level of development or difficulty. Moreover, as the child masters performance on a certain level, based on certain specific abilities, new requirements are imposed in order to reach a higher level. This would result in very complex and recurrent interactions between general and specific cognitive mechanisms throughout development.

We would like to thank the children, parents, and principals of the schools for collaborating with this research. We especially thank Mr. Peter Laspina, from ViaMundi Idiomas e Traduções for reviewing the manuscript.

The authors have declared that no competing interests exist.