Fragile X syndrome (FXS) is a neurodevelopmental disorder caused by a single-gene mutation on the X chromosome. FXS is the most common inherited cause of intellectual impairment, affecting approximately 1 in 3,600 males and 1 in 4,000 females
[1]. It is also the most common cause of non-idiopathic autism
[2, 3]. FXS results from a trinucleotide (CGG) expansion on the 5’ untranslated promoter region of the fragile X mental retardation 1 gene (FMR1), located on a distal tail of the X chromosome
[4]. The gene is normally polymorphic up to ~44 CGG repeats. The full FXS mutation is characterized by >200 repeats and renders the gene highly susceptible to DNA methylation and transcriptional silencing, which consequently results in a reduced level or complete loss of the gene’s protein product, fragile X mental retardation protein (FMRP)
[5]. FMRP is an RNA binding protein involved in regulation of translation of multiple dendritic mRNAs important to synaptic development and plasticity, suggesting that FMRP is integral to the development of neural networks and functional integration across brain regions
[6–8].

There is considerable evidence showing that cognitive deficits observed in individuals with FXS map onto abnormal processing in frontal-parietal neural networks in the brain. While the predominant end-phenotype of FXS is characterized by mild to severe intellectual impairment, individuals with FXS do not present with global deficits, but rather display selective impairments along with areas of spared skills. Individuals with FXS demonstrate marked weakness in performance on tasks of inhibitory control, selective and sustained attention, visual-spatial integration, motor coordination, and numerical processing
[9]. The current body of literature has lead some researchers to suggest that abnormal connectivity between parietal cortex and occipital regions, as well as disruptions to a purported frontoparietal attentional network, underlie the specific cognitive profile seen in individuals with FXS
[10–14]. Focusing primarily on the prominent cognitive deficit of mathematical disability observed in older and higher-functioning individuals with FXS, in this study we examine whether such impairments in the domain of numerical processing can be seen early in development.

Disruption of number processing and arithmetic abilities in higher-functioning adults and female adolescents with FXS is well established in the empirical literature
[15–19]. In illustration, Rivera *et al.*[14] tested a group of adolescent girls with FXS on an arithmetic task and showed that they performed worse than controls, they also displayed a different pattern of brain activation when viewing simple arithmetic problems. Subjects exhibited less overall activation than did unaffected individuals during both 2-operand (e.g., 2 + 1 = 3) and 3-operand (e.g., 3 + 2 – 1 = 5) trials
[14]. In response to increasing arithmetic complexity (i.e., going from 2- to 3-operand equations), unaffected subjects showed increased recruitment in parietal regions known to be associated with arithmetic processing (including left angular gyrus and intraparietal sulcus), whereas participants with FXS did not show this “ramping up” of parietal activation. Importantly, these researchers demonstrated that higher levels of serum FMRP were associated with more typical activation patterns in parietal areas known to support arithmetic processing
[14]. Given the striking genetic dose-dependent nature of mathematical impairment seen in older individuals with FXS, the domain of numerical processing provides an important area for investigation of altered development and cognitive behavioral deficits in FXS. Still, neither mathematical skills nor developmentally antecedent cognitive processes (which presumably underlie more complex numerical processing), have been investigated in toddlers with FXS.

Research in typical cognitive development strongly suggests that human infants are able to represent and discriminate number at a very young age, long before verbal counting principles develop. A growing body of evidence supports the notion that human infants, adults and non-human primates all share a non-verbal system for representing quantity, and that this system is both ontogenetically primitive and is present very early in human development
[20]. The approximate number system (ANS)
[21] is theoretically and functionally distinct from that served by the verbally-based counting principles
[22, 23], and can be thought of as a mental “number line” which allows representation of imprecise magnitudes in terms of relative spatial relationships along a continuum
[22, 24]. The ANS is thought to support representation of specifically large, approximate numbers as non-exact, or “un-counted”, quantities, serving the judgment of “generally, how much”. Research in both adults and infants suggests that there is a distinct and separate system for representing small exact number, serving the judgment of “precisely one, two, three or four”. Evidence shows that the ANS and exact number systems develop at different rates
[20, 25]. In a task of numerical discrimination, Xu
[25] demonstrated that typically developing 6-month old infants successfully discriminated between large numbers (4 *vs.* 8), but not small numbers (2 *vs.* 4), suggesting that ANS develops earlier than exact small number knowledge.

Studies of typically developing toddlers indicate that two-year-olds are able to recognize ordinal numerical relationships before they are able to use verbal counting to describe these relations, and Brannon
[26] demonstrated that typically developing infants could detect ordinal relationships between numerical magnitudes by as young as 11 months of age. Implicit in the organization of the ANS is ordinality (the basic principle of which quantity comes first, second, third, etc.); the mental number line is ordered such that magnitude increases in one direction. Ordinality is also important for more exact numerical skills, such that stable order is essential to appropriate enumeration, development of verbal counting skills, and understanding of the relations among precise cardinal quantities represented by numerical symbols. Based on the “continuity hypothesis”
[27], ontogenetically and evolutionarily primitive processes that support a general sense of quantity extend into basic cognitive representations of magnitude supported by the ANS, which developmentally precedes and underlies exact numerical representations. According to this view, perception of ordinal relationships is an essential skill for both approximate magnitude judgments and precise numerical processing and math abilities. Several researchers have proposed that perception of ordinal relations serves as the bridge across which exact numerical representations are mapped onto the more fundamental sense of approximate quantity
[28, 29]. In support of this perspective Halberda *et al.*[30], and Lyons and Beilock
[29] demonstrated that individual differences in approximate magnitude judgments predicted symbolic math performance in adolescents and adults. Lyons and Beilock further demonstrated that among college students, better ability to order symbolic precise numbers significantly predicted performance on a mental arithmetic task, supporting the link between ordinal processing and more advanced mathematical cognition
[29]. Nonetheless, these studies were conducted with relatively mature subjects and did not address developmental trajectories. It is therefore appropriate to examine ordinal processing in young individuals with FXS, who as a group, show deficits in later math ability that are considered hallmarks of the syndrome.

The goal of the current study was to examine the specific nature of mathematical deficits reported in individuals with FXS by investigating how very young children with FXS respond to ordinal relationships among numerical magnitudes. The tasks were adapted from Brannon
[26] to address the early development of numerical knowledge in FXS and to begin to identify the developmental trajectory of math deficits displayed in older children and adults with FXS. This is the first empirical study of numerical processing in young toddlers with FXS, and results could shed light on the nature of math disability in FXS, as well as contribute to the developmental debate over the continuity of the ANS and mathematical skills. For example, if toddlers with FXS demonstrate impaired perception of ordinal magnitudes (as compared to typically developing (TD) developmentally age-matched controls), this would support the hypothesis that disruptions in the mental number line are associated with, and possibly underlie, dysfunction in higher-level math abilities in FXS. However, if toddlers with FXS demonstrate intact ordinal recognition, this would suggest that later deficits in precise number representation (e.g., arithmetic calculations) stem from qualitatively distinct processes in FXS.