Mathematical learning may not be an easy task for some children, in particular those with neurodevelopmental disorders such as attention deficit/hyperactivity disorder (ADHD). A study found that in a sample of 9- to 13-year-old children (n = 304), about 8.6% children (n = 26) had the diagnosis of ADHD. Besides, about 38% children with ADHD and 12.9% gender-, age-, grade-, and family social economic status-matched typically developing (TD) children (n = 31) had scores below the 10th percentile in school mathematic performance [1]. A population-based birth cohort study on the prevalence of learning disorders in school-aged children also reported that children with ADHD symptoms (measured by the Strengths and Difficulties Questionnaire) had higher risk for mathematic difficulties. While children with borderline ADHD symptoms had an odd ratio 2.82 [95% CI (1.58, 5.07)], children with high ADHD symptoms had an odd ratio 2.55 [95% CI (1.52, 4.26)], after adjusting for gender, age, study center, and parental age and education [2]. Results of these two studies indicate that children with ADHD had higher risk of mathematic difficulties than children without ADHD. A line of research examined the types of mathematic difficulties encountered by children with ADHD, evidence showed that children with ADHD performed worse than their peers without ADHD in basic number skills (e.g., counting, number comparison, dot enumeration, etc.) [3, 4], arithmetic factual knowledge and calculation [3, 5–7], arithmetic fluency [4, 8], applied math problems [5, 6, 9, 10], and mathematic achievement test [11, 12]. Collectively, children with ADHD had experienced difficulties in different mathematic skills. Evidence demonstrates that mathematics is closely associated with executive functions such as attention [11, 13] and working memory [14, 15]. However, the contributions of processing speed (PS) to mathematic performance are relatively understudied, particularly in children with impaired PS like those with ADHD [16, 17].
Processing speed (PS) could be defined as the time it takes a child to execute a cognitive task, it is related to the speed in which the child understands and reacts to the information he/she receives [18]. Processing speed implies a greater ability to perform a mental task, in other words, the faster the processing speed, the more efficient a child is able to think or learn. PS could be measured in many different ways. For examples, the Stop-signal task or the Go/no-go task were commonly used to measure motor PS in young children [19]; the Symbol Search subtest of the Wechsler Intelligence Scale for Children, 4th edition (WISC-IV) was employed to measure children’s perceptual PS [20]; and the Coding/Coding Copying subtests of the WISC-IV were used to measure children’s graphomotor-cognitive PS [16]. In addition to these three types (i.e., motor, perceptual, and graphomotor-cognitive) of PS, other studies also classified PS into general (including perceptual PS and decision PS) and specific (including reading and arithmetic fluency) processing speed [21].
Deficits in PS were found in children with ADHD. Evidence shows that children with ADHD had slower motor PS in a simple choice reaction time task than their peers without ADHD symptoms [4]. While graphomotor-cognitive PS has been shown to be one of the best predictors for inattention symptoms in children with ADHD [22], studies also found that children with ADHD showed deficits in graphomotor-cognitive PS as measured by the Coding subtest of the WISC and the Trail Making A + B tests [17]. Kibby and colleagues [16] measured children’s motor, perceptual, decision, and graphomotor-cognitive PS. They reported that there was no significant difference in motor PS between children with and without ADHD. Besides, in the ADHD group, subtypes analyses on all PS measures did not reach significance, indicating that children with different ADHD subtypes performed similarly on various types of PS. However, compared to children without ADHD, children with ADHD had slower perceptual (measured by the Symbol Search subtest of the WISC-III) and graphomotor-cognitive (measured by the Coding subtest of the WISC and the Trail Making A + B tests) PS. Moreover, children with Predominantly Inattention (ADHD-PI) subtype also had slower decision speed than controls [16]. Findings of this study suggest that children with ADHD might not have impaired PS if the task is simple (i.e., a simple motor reaction task), but as task demands increase (i.e., Coding subtest and Trail Making A + B tests) their PS becomes less efficient than peers without ADHD.
The link between PS and mathematic performance in children has been demonstrated in previous studies. Clark and coworkers [23] examined the relation between PS and mathematic abilities in preschoolers. The Visual Matching subtest of the Woodcock-Johnson III Tests of Cognitive Abilities (WJ III Visual Matching) was administered to measure children’s perceptual PS, and the Test of Early Mathematic Ability − 3 and the Applied Problems subtest from the Woodcock-Johnson III Tests of Achievement III were used to measure children’s mathematic performance. Results showed that perceptual PS was significantly correlated with early mathematic abilities [23]. A similar study examined the relationship between perceptual PS and mathematic cognition (i.e., number sense, counting skills, arithmetic etc.) in children aged 4 to 6 years. PS was measured by a timed figure matching task. After controlling for age, perceptual PS was found to be a significant predictor for mathematic cognition [24]. Taken together, evidence showed that early mathematic abilities were predicted by perceptual PS in preschool-aged children. In school-aged children, a study found that perceptual PS, measured by a timed number matching task, had both direct and indirect (via both reading and arithmetic accuracy) effects on arithmetic fluency in first graders [25]. Another study measured perceptual PS using the WJ III Visual Matching test and examined the relationships between PS and untimed (arithmetic computation) and timed (addition and subtraction fluency) math abilities in 2nd graders. Results showed that perceptual PS was significantly correlated with both untimed and timed math abilities. However, perceptual PS predicted children’s timed math abilities only [20]. Fuchs and colleagues [18] also used the WJ III Visual Matching test to measure PS in a sample of 3rd graders. Results showed that perceptual PS was a significant predictor for single-digit addition and subtraction fluency tests but not for double-digit fluency tests or arithmetic word problems [18]. A study examined the association between graphomotor-cognitive PS and math abilities in a sample of children and adolescents aged 6–20 who had sluggish cognitive tempo symptoms. PS was measured using the Coding subtest of the age-appropriate version of the Wechsler scales. Untimed math computation was measured by the relevant subtests from either the Woodcock-Johnson Test of Academic Achievement III, or the KTEA-3, while timed math abilities were measured by the Math Fluency subtest of the Woodcock-Johnson Test of Academic Achievement III, the Math Fluency Addition subtest of the Wechsler Individual Achievement Test, third edition, or the Math Fluency subtest of the KTEA-3. Results showed that timed math abilities (Math Fluency) were predicted by graphomotor-cognitive speed and the Slow subscale of Sluggish Cognitive Tempo Scale [26]. Collectively, findings of the above-mentioned studies demonstrate that PS (perceptual and graphomotor-cognitive) better predicted math abilities which demand speed (math fluency) in school-aged children. However, due to the lack of studies, it is unclear if PS also predicts math fluency in children with ADHD. Given that children with ADHD would become less efficient in processing information when the task demand increases [16], it is possible that PS measured by tasks of different complexities and cognitive loads might have different contributions to timed mathematic performance in this clinical population. The current study would investigate the contributions of perceptual (simple) and graphomotor-cognitive (more complex) PS to math fluency (MF) in children with ADHD.
Although the link between PS and MF was demonstrated in previous studies, it is not fully clear what cognitive processes underlie the contributions of PS to MF. There is evidence to show that PS and working memory (WM) are related. A study has manipulated PS within a WM paradigm in order to examine the relationship between PS and WM in school-aged children with and without ADHD. Findings showed that children with ADHD had worse WM than children without ADHD. Besides, slowing PS led to a decrease in WM capacity due to an increase in cognitive load. Children with ADHD were also found to experience higher cognitive load than peers without ADHD when performing the tasks. As a whole, results of this study suggested that slow PS was a plausible cause of WM deficits in ADHD [27]. Another study has examined if math difficulties (MD) associated with executive functions (i.e., working memory, inhibition, cognitive flexibility, and PS) deficits in 4th to 6th graders. The authors reported that children with MD had deficits in both WM and PS compared to children without MD. When covarying the effect of PS, group difference in WM was attenuated, suggesting children with MD experienced WM deficit which could be explained by deficits in PS [28]. Collectively, results of these two studies point to the suggestion that the relationship between PS, WM, and mathematics in children might follow the direction PS ◊ WM ◊ mathematic performance. In fact, evidence shows that PS (indexed by the composite score of both perceptual and graphomotor-cognitive PS) had both direct and indirect effects via working memory on numerical operations in typically developing (TD) children [29] and on mathematic achievement in 8-year-old children with surgically-repaired dextro-transportation of the great arteries (d-TGA) [30], after controlling for attention and general cognitive abilities. Deficits in PS and WM are commonly co-existed in children with ADHD [31–33], and a significant number of children with ADHD also experienced mathematic difficulties [1]. It is likely that the relationship between PS and mathematic performance might be mediated by working memory in children with ADHD. The current study would further investigate the relationships between PS and MF by examining if processing speed (PS) had both direct and indirect effects via working memory (WM) on math fluency (MF) in children with ADHD.
Rapid automatized naming (RAN) is a strong predictor for reading particularly reading fluency [34, 35]. In recent years, RAN is considered as a processing speed (phonological PS) measure by some researchers and its relationship with mathematic achievement were examined [25, 36]. RAN relates to mathematics because both skills are thought to tap the ability to access and retrieve phonological representations from long-term memory [37]. RAN refers to the ability to name a sequence of familiar visual stimuli such as colors, letters, digits, and objects as fast as possible [38]. Structural equation modelling reveals that different RAN measures (i.e., digits, vowels, consonants, dice, finger-numeral configurations, objects, and colors) support a one-dimensional RAN factor in 6- to 7-year-old TD children [39]. However, RAN may have differential contributions to different mathematic skills. A meta-analytic study reported that RAN had stronger associations with arithmetic calculation than general math achievement; it also had stronger associations with single-digit calculation than double-digit calculation; and stronger associations with math fluency than math accuracy [36]. Some studies had specifically examined the relationships between different RAN measures and children’s MF. RAN was found to be a significant correlate of arithmetic fluency in kindergarteners and the correlations did not vary as a function of the type of RAN measures (i.e., digits, letters, colors, and dice) [40]. A meta-analytic study also found that alphanumerical (digits and letters) RAN and non-alphanumerical (colors and objects) RAN had similar association strength with arithmetic fluency in children [36]. Results of these two studies suggest that RAN-Math fluency relationship is independent of the types of RAN tasks. However, there is evidence that RAN-Digits had specific contributions to MF in school-aged children. Balhinez and collaborators [41] found that RAN-Digits and WM predicted arithmetic fluency in 1st and 2nd graders and RAN was a unique predictor for MF in 3rd graders. RAN-Digits were also found to have both direct and indirect (via both reading and arithmetic accuracy) effects on arithmetic fluency in 1st graders [25].
Deficits in RAN were also found in children with ADHD. Some studies had compared the performance on RAN tasks between children with ADHD and TD controls. Tannock and coworkers [42] found that 7- to 12-year-old children with ADHD performed worse than age-matched TD controls on both RAN-Colors and RAN-Letters. The group difference in RAN-Colors remained significant after controlling for vocabulary and reading skills. In addition to deficits in RAN-Colors, children with ADHD were found to be significantly slower than controls on rapid letter naming (RAN-Letters) [43] and rapid digit naming (RAN-Digits) [44]. Measuring a single RAN task cannot tell us if children with ADHD had a general or a specific RAN deficit. Thus, several studies had examined the performance on different RAN measures in children and adolescents with ADHD. Alves and collaborators [45] measured different RAN (digits, letters, colors, and objects) tasks in 8- to 11-year-old children with and without ADHD. The authors reported that while the whole group of children with ADHD took longer time than controls without ADHD to complete RAN-Digits and RAN- Objects, children with ADHD aged 10–11 performed worse than age-matched controls in RAN- Colors and RAN -Digits. Besides, there were no significant group differences in RAN-Letters across age [45]. Ghelani and collaborators [46] also measured different RAN tasks (digits, letters, colors, and objects) in adolescents with and without ADHD. Results revealed that adolescents with ADHD took longer time to name colors and objects but not digits and letters than adolescents without ADHD [46]. Similarly, Whipple and colleagues [47] also found that adolescents’ performance on non-alphanumeric RAN (colors and objects) was significantly worse than alphanumeric RAN (digits and letters). Collectively, it is still unclear whether children with ADHD had a general or a specific RAN deficit. Yet, it appears that older children/adolescents with ADHD had experienced difficulties in naming colors.
Studies on the relationships between RAN and mathematic performance in TD children demonstrate that RAN-Digits was a significant predictor for MF in 1st − 3rd graders [25, 41]. Studies on the relationship between RAN and MF in children with ADHD are rare, as we know both studies were on adolescents. Capodieci and collaborators [8] have investigated the relationships between RAN (digits and colors) and the error types in math fluency in adolescents with ADHD aged between 14 and 17 years. The authors found that RAN-Colors but not RAN-Digits was significantly correlated with the switch errors of MF, however, it was not a significant predictor. The other study has investigated the performance of adolescents and young adults with ADHD on measures of alphanumeric (digits and letters) and non-alphanumeric (colors and objects) RAN and the relationships between RAN and academic achievement. Results revealed that alphanumeric but not non-alphanumeric RAN was a significant predictor for MF [47]. Collectively, results of these two studies are inconsistent. RAN-Digits was found to be a significant predictor for MF in TD children, due to the lack of studies, it is unclear if the same relationship could be found in children with ADHD. The current study would like to bride this research gap and investigate the contributions of RAN-Digits (phonological PS) to math fluency (MF) in children with ADHD.
The Current Study
The current study aimed to improve our knowledge about the contributions of processing speed (PS) to mathematic performance in school-aged children with ADHD. There are two objectives: (1) To investigate the contributions of three types of processing speed (i.e., perceptual, graphomotor-cognitive, and phonological PS) to a timed mathematic performance (i.e., math fluency) in children with ADHD; and (2) To investigate if PS had both direct and indirect effects via working memory on math fluency in children with ADHD. Based on aforementioned previous studies, it was hypothesized that, for objective (1), all three PS (i.e., perceptual, graphomotor-cognitive and phonological) would be significant predictors for math fluency in school-aged children with ADHD. For objective (2), it was hypothesized that perceptual and graphomotor-cognitive PS would have both direct and indirect effects via working memory on math fluency in children with ADHD.