The study complied with the Helsinki guidelines for research with human participants and was previously approved by the ethics board of the Universidade Federal de Minas Gerais (COEP-UFMG). Informed consent was obtained in written form from parents/guardians and orally from children.

Data were obtained from a demographically based sample of children from 6 to 14 years attending public schools in Belo Horizonte, Brazil. Initially, 2,195 children participated in a screening phase, responding to group-administered tests of intelligence and school achievement ( Table 1). Children with intelligence above PR15 were invited to participate in a second phase of individual neuropsychological assessment, using the following exclusion criteria: gemelarity, chronic diseases (e.g., diabetes, sickle cell anemia, and epilepsy), autism spectrum disorder, monogenic or chromosomal genetic syndromes, and maternal alcohol/drug use during pregnancy. A total of 378 pupils participated in the second phase of the study and were genotyped for the FMR1 CGG repeat.

Table 1
Sociodemographic characteristics of participants.

MD: Math difficulties.

Performance in the Arithmetic subscale of the Brazilian School Achievement Test (TDE) ²² allowed to classify participants into a group of broadly defined MD with performance below PR25 and a control group with performance at above PR25. In the second phase, stricter diagnostic criteria (PR<6–7) informed by the individual neuropsychological assessment were applied to classify math achievement.

A total of 378 children completed both phases of the neuropsychological evaluation, 121 children (61 girls) were identified as having MD. To the control group were assigned 257 children (146 girls).

Neuropsychological testing was conducted at the schools by specially trained psychology research assistants. Tests were applied in groups in the first phase and individually in the second phase. The neuropsychological protocol is shown in Table 2. A few additional tasks were applied in individuals with FMR1 expanded alleles.

Table 2
Instruments used in the neuropsychological assessment.

TDE: Teste de Desempenho Escolar (School Achievement Test); WISC-III: Wechsler Intelligence Scale for Children III.

Frequencies of gray zone and premutation alleles in the MD and control groups were compared using the Fisher’s exact test. Neuropsychological results of each individual with expanded FMR1 alleles were compared to those of subgroups of controls, referred to as comparison groups, controlling for sex, age, school grade, and socioeconomic status. The z metric was chosen for comparisons, with a cutoff of normality at z=-1.5 (PR6–PR7) ³².

The results of the FMR1 genotyping for the 378 participants on the individual neuropsychological assessment are shown in Table 3.

Table 3
Alleles identified in Math difficulties and control groups (n=378).

MD: math difficulties.

One premutation allele was detected in a girl from the MD group (1:61 considering only girls and 1:121 considering boys and girls). Three children presented alleles in the gray zone: one boy and one girl in the control group (1:111 and 1:146, respectively) and one boy in the MD group (1:60 considering boys and 1:121 considering boys and girls). In addition, four children in the control group (4:146) presented alleles with 41–44 CGG, which belong to the expanded gray zone range ( Table 3). These four children have normal results in the neuropsychological assessment and will not be further discussed. The most frequent allele had 30 CGG repeats, except for the boys in the MD group for whom the 31 CGG repeat allele was the most frequent (Supplementary Figures 1–4). As expected, considering the inclusion criteria of intelligence above PR15, no FMR1 full mutations were observed.

Quantitative results of the neuropsychological performance of each child with a premutation or a gray zone expanded FMR1 allele are presented in Figure 1. Table 4 presents descriptions of the neuropsychological phenotypes for each child.

Figure 1
Neuropsychological performance of the children with FMR1 premutation and gray zone alleles. (A) Child 1 (premutation); (B) Child 2; (C) Child 3; (D) Child 4. Deficits in individual tests were assumed when scores < or = -1.5 SD (PR6–PR7).

Table 4
Genotype and phenotype characteristics of children with expanded FMR1 alleles.

Our first question was whether the frequency of premutation or gray zone alleles was similar to that described in the literature. In samples of persons with intellectual disability, the contribution of FMR1 full mutations has been estimated to be 1–6% ^{33, 34}. The frequency of FMR1 full mutations was estimated to be approximately 1:5000 in a screening sample composed of over 36,000 newborn males ³⁵. Therefore, having an FMR1 full mutation entails a risk for intellectual disability that is 50–300 times higher than the risk in the general population. Considering the exclusion criteria implemented here and the sample size, we did not expect to identify any FMR1 full mutations in the current sample.

Our focus was on the contribution of FMR1 premutation or gray zone alleles for learning difficulties and related phenotypes. In the literature, the frequency of premutation and gray zone alleles was investigated in a sample composed of approximately 20,000 male and female adults representing the general population ³⁶. According to this study, the frequency of premutation in females was 1:148 and that in males was 1:290; the frequency of gray zone alleles (45–54 repeats) was 1:33 females and 1:62 males; and the frequency of “expanded” gray zone alleles (here defined as 41–54 CGG repeats) was 1:14 females and 1:22 males. In the current sample, the frequency of premutation, gray zone, and expanded gray zone alleles did not statistically differ from these values (results not shown).

The second question was whether children sampled from the general population and identified with premutation, gray zone, or expanded gray zone alleles also have a neuropsychological profile resembling that described for some children from families in which FXS segregates, having similar alleles ¹⁹. Two individuals were identified with performance below the PR6–PR7 in WM and arithmetic tasks. One child had a premutation and the another a gray zone allele. Both had normal intelligence and normal written language processing as assessed by word spelling, which excluded concomitant intellectual disability or a literacy acquisition disorder. In Child 1, phonological WM (Digit span backward) was below the cutoff score and spatial WM (Corsi blocks span backward) was in the lower normal range (1 SD below the average). The performance of Child 2 was at or below the cutoff in both phonological and spatial WM.

Both children also presented impairments in arithmetic abilities. Child 1 performed below the cutoff score in single-digit subtraction and more than 1 SD below the average in the other calculation tasks. Child 2 performed below the cutoff score for all single-digit operations. It is remarkable that these children presented difficulties with very basic arithmetic skills acquired in the first 3 years of schooling. It is also noteworthy that both the accuracy of nonsymbolic numerical representations (w) and number reading and writing were normal in these two children. This indicates impaired calculation together with spared number processing.

Math learning difficulties of varying degrees of severity have been reported in persons having premutations in families with FXS ¹⁹. However, clinical samples tend to include more severe cases and to overestimate both the frequency and the impact of the difficulties. A possible solution to avoid such biases is to identify cases among the school population. Two studies have investigated the frequency of FMR1 premutations and gray zone alleles in school-age children ^{37, 38}. In the study by Murray and colleagues ³⁷, boys were identified in the schools by teachers or principals due to learning disability and no specific neuropsychological evaluation was conducted. In the study by Mazzocco and colleagues ³⁸, children who were identified in pediatric or developmental neuropsychology facilities as having normal intelligence and learning disabilities were screened for FMR1 premutations and gray zone alleles. Neither of these screening studies specifically assessed WM and numerical/arithmetic abilities. Both studies may have included individuals with borderline intelligence or intellectual disability.

In both participants with expanded FMR1 alleles (Child 1 and Child 2), performance in the single-digit calculation tasks was below the PR6–PR7. Both children also presented clinical evidence of impairments in very basic math-related school abilities. Qualitatively, both children had difficulty in understanding the concepts underlying the operations and used immature strategies, such as finger counting, to solve the most basic single-digit calculations. Thus, a diagnosis of developmental dyscalculia is justifiable according to the current nosological standard ³.

Five main cognitive mechanisms have been proposed to explain difficulties in learning math in cases of developmental dyscalculia: (a) inaccurate nonsymbolic numerical magnitude representations in an approximate number system (ANS); (b) deficits in the access to nonsymbolic numerical representations from symbolic ones; (c) phonological processing deficits associated with developmental dyslexia; (d) visuospatial/visuoconstructional processing deficits associated with nonverbal learning disability; and (e) working memory/executive function impairments ⁵. These cognitive mechanisms may be interpreted as endophenotypes in different combinations in children with dyscalculia, a heterogeneous condition, often occurring with developmental dyscalculia, attention deficit disorder, and other conditions ⁵.

In this present study, neuropsychological results excluded impairments in written language and phonological processing in the two children with expanded FMR1 alleles and dyscalculia. None of these two children presented difficulties with the accuracy of or access to nonsymbolic numerical representations. These results exclude a role for phonological and nonsymbolic numerical processing in their difficulties. Their difficulties were otherwise associated with impairments of both verbal and visuospatial WM.

The literature on females with FXS has disclosed an uneven cognitive profile of assets (verbal memory and analytic visual perception) and deficits (visuospatial and executive function) ³⁹. Other evidence indicates spared number reading/writing and rote counting abilities with deficits in magnitude judgments, mental number line judgments, understanding of counting principles, and basic addition in FXS girls ^{40, 41}.

There are reports of individuals with FMR1 premutation presenting deficits in cognitive functions such as WM, executive function, visuospatial perception, phonological processing, and reaction time ^{16, 42, 43}. Females with a premutation allele may present difficulties in basic numerical comprehension and numerical transcoding of mathematical questions and calculations ^{6, 18, 19}. It is remarkable, however, that the impairments described in premutation alleles are usually of lesser severity than required for the diagnosis of developmental dyscalculia.

Considering this literature, our neuropsychological results suggest that (a) WM impairments in individuals having FMR1 expansions may play a role in difficulties learning math, as it has been previously reported; (b) ­accuracy of nonsymbolic numerical representations is normal in at least a subgroup of individuals with FMR1 mutations; and (c) neuropsychological impairments are observed not only in FXS and premutation alleles but also, as described here, in individuals with gray zone alleles.

Our results must be cautiously interpreted. One limitation is the number of individuals having FMR1 abnormal alleles detected. From the 2,195 children taking part in the initial population screening phase, 328 concluded the second phase of the study and, among these, only 4 had FMR1 alleles in the abnormal range. These numbers seem small, reflecting the complexities underlying a population design necessary to detect FMR1 expansions (due to their low frequency) and dyscalculia (due to the wide range of tests needed to diagnose it). However, it is important to highlight that the neuropsychological profile presented by these two children is typically seen in association with FMR1 premutations, suggesting that the presence of developmental dyscalculia is not a fortuitous finding. The exclusion of low and borderline intelligence may have removed children with full mutations from the sample. However, the focus here was actually to ascertain children with normal intelligence and MD.

Child 1, a girl with a 57 CGG premutation, was the single individual identified with an FMR1 premutation in our sample and she also presented developmental dyscalculia. According to the literature, these math difficulties should be expected only in individuals with more than 100 CGG repeats ^{11, 18, 44}. ­However, ­Lachiewicz and colleagues ⁶ reported that girls with premutation alleles around 80–88 CGG repeats have lower arithmetic performance than women with alleles greater than 100 CGG repeats. The findings from ­Lachiewicz and coworkers and this study suggest that the typical neuropsychological profile may be present in the premutation range, independent of the number of CGG ­repeats.

Child 2, a boy with a 46 CGG gray zone allele was identified as having developmental dyscalculia. The other two children with gray zone alleles, as well as the four children with alleles in the 41–44 CGG range, had typical neuropsychological performance. Although other studies have investigated the contribution of FMR1 full mutation and premutation alleles to MD, this is the first study to investigate their contribution in a sample of school-age children undergoing a detailed neuropsychological and cognitive-numerical assessment. We provided evidence that gray zone alleles may occur in individuals with normal neuropsychological profiles; whereas other individuals with gray zone or premutation alleles display specific neuropsychological deficits, characterized by low performance in WM and low math achievement, corresponding to a diagnosis of developmental dyscalculia. This suggests that other factors, in addition to the gray zone alleles, must be present to cause the developmental difficulties in learning math for these children.

The finding that two out of four children with an FMR1 premutation and gray zone allele presented developmental dyscalculia associated with WM impairments suggests this is an important phenotype in FMR1 expansions. In contrast, the frequency observed in this study, two children with abnormal FMR1 alleles out of 121 children with MD, suggests that this gene makes an important contribution to MD and developmental dyscalculia. As impairments in WM were the most salient finding, it also suggests that WM impairment is an important endophenotype of developmental dyscalculia. Indeed, FMR1 provides one of the larger contributions of a single gene to the dyscalculia phenotype reported so far. Similar findings have been previously reported ³⁸. A genetic contribution to MD has also been reported for 22q11.2 deletion syndromes ^{45, 46}. Most importantly, our data suggest that even minor alterations of the FMR1 gene may contribute to developmental dyscalculia.

This is the second study from our group to investigate the genetic basis of developmental dyscalculia. In the same sample, a child has already been identified with a 22q11.2(LCR4–LCR5) deletion syndrome ⁴⁶. Therefore, we found three children with a genetic condition out of 121 children with MD. None of these children had severe congenital malformations or even a funny face that would refer them for a genetic investigation. These findings are preliminary, but raise the question whether and when a genetic investigation should be considered in otherwise healthy children with MD.

Correspondence: Maria Raquel Santos Carvalho; Emails: ma.raquel.carvalho@gmail.com; mraquel-carvalho@ufmg.br