Understanding the Cognition Related to Mathematics Difficulties: A Meta-Analysis on the Cognitive Deficit Profiles and the Bottleneck Theory

Processing Speed

Processing speed is the speed a person can encode, transform, and retrieve information (Conway, Cowan, Bunting, Therriault, & Minkoff, 2002) and is often operationalized as the speed of responding to simple tasks in which everyone would answer correctly if there were no time limits (Salthouse, 1996). Processing speed is assumed to influence early mathematics development by facilitating simple processes, such as fluent counting for finding answers, thereby permitting associations between problems and their answers to be held in working memory before committing to long-term memory (Geary, 1993). From a developmental perspective, processing speed is the major reason for the age differences in numerical competence (e.g., Holloway & Ansari, 2008). With slower processing, the interval increases for deriving counted answers and for pairing a problem stem with its answer in working memory; this creates the possibility that “decay” sets in before completing the computational sequence. In fact, some research indicates that low processing speed is a prominent cognitive feature among children with MD (e.g., Cirino, Fuchs, Elias, Powell, & Schumacher, 2015; L. S. Fuchs et al., 2006; L. S. Fuchs et al., 2008; Jordan & Montani, 1997). By contrast, others indicate that children with MD do not show deficits in processing speed (e.g., Andersson & Lyxell, 2007; Jordan, Hanich, & Kaplan, 2003; Moll, Göbel, Gooch, Landerl, & Snowling, 2016).

Types of MD, age, and domains of task may be potential factors contributing to these mixed findings. For example, some research suggests that processing speed may be especially important for the early mathematics development (e.g., L. S. Fuchs et al., 2006), and thus processing speed deficits may be most severe among young children with MD, especially for those with calculation difficulties (e.g., Bull & Johnston, 1997; L. S. Fuchs et al., 2006). Others suggest that slow processing may be related to word problem–solving difficulties, such that children with word problem–solving difficulties tend to use more time-consuming, inefficient word problem–solving strategies (Geary, 1993). Moreover, some research shows that children with MD only show slow numerical information processing, suggesting that processing speed deficits in MD may be numerical-specific (Geary, 1993).

Phonological Processing

Phonological processing refers to the ability to manipulate phonological representations (e.g., sounds) and efficiently retrieve phonological representations from long-term memory (Wagner, Torgesen, & Rashotte, 1999). According to the phonological representation hypothesis, solving calculation problems requires encoding and maintaining phonological representations in short-term memory as well as the retrieval of phonological codes from long-term memory (Hecht, Torgesen, Wagner, & Rashotte, 2001; Simmons & Singleton, 2008; Vukovic & Lesaux, 2013). For example, to solve the equation, (4 + 5)/3 =, one needs to encode this equation phonologically as “the sum of four plus five divided by three,” retrieve “9” from long-term memory for “4 + 5,” and hold the phonological information in short-term memory to solve the problem. Poor phonological processing skills may cause poor representation of numerical codes in both short-term and long-term memory, which may impede mathematics performance. This hypothesis explains the high prevalence of comorbidity between RD and MD in early childhood such that young children with RD with poor phonological processing skills often have MD (e.g., Simmons & Singleton, 2008). However, some research also documents that young children with MD, nonetheless, have good phonological processing skills (e.g., L. S. Fuchs et al., 2005; Landerl, Fussenegger, Moll, & Willburger, 2009; Simmons & Singleton, 2008), suggesting that phonological processing may not be an influential factor in early mathematics competence.

These mixed findings should be interpreted with respect to types of MD and how phonological processing was measured. For example, due to phonological processing deficits being closely related to RD, it is likely that MDRD, but not MD-only, is related to deficits in both manipulating and efficiently retrieving phonological representations (Simmons & Singleton, 2008). Previous research also shows that the manipulation of phonological representations may indirectly influence mathematical achievement via learning number words and the number-word sequence (Krajewski & Schneider, 2009), and thus deficits in phonological manipulation may be particularly related to MD identified with calculation difficulties early on (Hecht et al., 2001). Moreover, some suggest that the deficits in efficient retrieval of numerical information from long-term memory are distinct deficits in MD (Geary, 1993), and thus phonological retrieval deficits, compared with phonological manipulation, is more important to MD, regardless of comorbidity and age.

Visuospatial Skills

Visuospatial skills are the abilities to represent, transform, generate, and recall symbolic, nonlinguistic information (Halpern, 2004; Linn & Petersen, 1985). Previous research suggests that visuospatial skills are linked to mathematics achievement, and training visuospatial skills may improve mathematics performance (Cheng & Mix, 2014). However, whether deficits in visuospatial skills contribute to MD is unclear (e.g., Badian, 1999; Landerl, Bevan, & Butterworth, 2004; Swanson, 2012). Types of visuospatial skills and types of MD should be considered for the visuospatial deficit profiles of MD. Visuospatial skills are a comprehensive construct that includes (but are not limited to) several skills, such as visuospatial perception (i.e., determine visuospatial relationships with respect to the orientation of their own bodies, in spite of distracting information), spatial visualization (i.e., complicated and multistep manipulations of visuospatial information), and mental rotation (i.e., mentally rotate one stimulus to align it with a comparison stimulus; Linn & Petersen, 1985). While visuospatial perception is suggested to be related to early numerical skills, such as understanding place value, transferring and laying out numbers, and executing operations in calculations (Geary, Hoard, Byrd□Craven, Nugent, & Numtee, 2007; Mazzocco, Singh Bhatia, & Lesniak-Karpiak, 2006; Venneri, Cornoldi, & Garuti, 2003), mental rotation and spatial visualization are shown to be related to more complex and advanced mathematics skills, such as constructing schematic diagrams in word problem solving (Edens & Potter, 2008; Hegarty, Mayer, & Monk, 1995). Thus, it is expected that different types of MD may demonstrate deficits in different types of visuospatial skills, at least to varying degrees.

Memory

We focused on short-term memory and working memory. Short-term memory refers to the temporary storage of information, which is the base for early mathematics skills, such as counting, number sense, and simple arithmetic (e.g., single-digit addition and subtraction; Bull & Johnston, 1997; Hitch & McAuley, 1991; Noël, Désert, Aubrun, & Seron, 2001; Passolunghi & Siegel, 2001). Working memory refers to the ability to simultaneously store and process information, which is critical for the performance on all types of mathematics tasks (Peng, Namkung, Barnes, & Sun, 2016). Although previous research and reviews demonstrate that children with MD have deficits in both short-term memory and working memory, one question remains to be answered: Are deficits in the memory system of MD, especially working memory, domain-general or domain-specific?

The literature is mixed on this point. Previous reviews on the memory deficits among MD (Peng & Fuchs, 2016; Swanson & Jerman, 2006) indicate that individuals with MD have memory deficits across numerical, visuospatial, and verbal domains. However, Swanson and Jerman (2006) suggested that children with MD may have distinctive visuospatial working memory deficits, while Peng and Fuchs (2016) suggested that children with MD may have distinctive numerical working memory deficits. One explanation for these mixed findings may be the small and different body of studies included in those reviews. Swanson and Jerman (2006) included 28 studies before 2004 that tapped working memory measures across verbal, numerical, and visuospatial domains. Peng and Fuchs (2016) included 29 studies before 2014 and only focused on working memory tasks in the verbal or numerical domains.

Types of MD may also affect the domain specificity of the memory deficit profiles of MD. Some research shows that numerical and visuospatial processing skills are closely related to calculations, while verbal skills are more closely related to word problem solving (e.g., Raghubar, Barnes, & Hecht, 2010; Peng et al., 2016). Thus, it is expected that MD identified with calculation difficulties are likely to show more severe numerical and visuospatial memory deficits, whereas MD identified with word problem–solving difficulties may demonstrate more severe verbal memory deficits. Yet individuals with both calculation and word problem–solving difficulties may have memory deficits across verbal, numerical, and visuospatial domains to similar degrees.

Attention

Attention refers to the appropriate allocation of cognitive resources to relevant stimuli (Posner & Petersen, 1990). Because attention is often considered as the hub of cognition, children who have attention problems often demonstrate learning difficulties (e.g., Willcutt, Pennington, Olson, Chaabildas, & Hulslander, 2005). However, whether MD is necessarily associated with attention problems remains unclear (Busch, Schmidt, & Grube, 2015; Cirino, Fletcher, Ewing-Cobbs, Barnes, & Fuchs, 2007; Gold et al., 2013; McCall, 1999; Reimann, Gut, Frischknecht, & Grob, 2013; Swanson, 2012; Willcutt et al., 2013). How attention was measured may be connected to the relation between attention problems and MD. For example, studies that reported severe attention problems among MD mostly used subjective attention rating scale/checklist, such as Strengths and Weaknesses of Attention-Deficit/Hyperactivity-symptoms and Normal-behaviors (e.g., Cirino et al., 2007) and Diagnostic and Statistical Manual of Mental Disorders (e.g., Willcutt et al., 2013), whereas less severe attention problems among MD were often reported from studies using objective attention measures (e.g., Gold et al., 2013).

Age may also affect the relation between attention problems and MD. Research on children with attention-deficit hyperactivity disorder (ADHD) suggests that the negative effects of their attention deficits in mathematics are mostly due to their failure in automatized math facts retrieval. That is, the lack of automatized arithmetic facts retrieval consumes limited attention resources for children with ADHD, which hinders their mathematics performance and development (e.g., Zentall, 1990). Following this logic, if MD is related to attention difficulties, this relation may be stronger in the early learning stages, when children are actively engaged in learning and automatizing arithmetic facts. This age effect should be strong when one also considers that attention span is smaller in younger children (Ruff & Lawson, 1990). An alternative hypothesis is that the relation between attention difficulties and MD may be stronger among older individuals with MD. This may be because older individuals with MD also often struggle with automatized arithmetic facts retrieval (Geary, 1993), and the amount of attention needed in mathematics tasks may increase with more complex and advanced mathematics skills.

Executive Functions

Executive functions refer to a set of cognitive skills required to direct a behavior toward the attainment of a goal (Jacob & Parkinson, 2015). Executive functions usually include inhibition, updating, and switching (Miyake, Emerson, & Friedman, 2000). Inhibition is the ability to deliberately inhibit a dominant, automatic response in favor of a subdominant response (e.g., Passolunghi & Cornoldi, 2000; Peng, Congying, Beilei, & Sha, 2012; Swanson, 2006). Updating is the process of encoding and evaluating incoming information for relevance to the task at hand and subsequent revision of the information held in memory. Switching is the ability to shift attention or to shift between strategies or response sets. Theoretically, executive functions are critical for many mathematics tasks that involve multiple steps, sequential thinking, and complex strategies (e.g., multidigit computation, mental arithmetic performance, and problem solving; Peng et al., 2012; Raghubar et al., 2010).

Deficits in executive functions are assumed to be closely associated with poor mathematics performance, but findings on the deficit profiles of these executive functions in the MD literature are mixed (e.g., Andersson, 2008; Andersson & Lyxell, 2007; Hitch & McAuley, 1991; Passolunghi & Siegel, 2004; Peng et al., 2012; Peng et al., 2013; van der Sluis, de Jong, & van der Leij, 2004). Domains of task, types of MD, and age should be considered regarding the relation between executive function deficits and MD. For example, some research suggests that children with MD demonstrate executive function deficits across domains, but the deficits in the numerical domain are distinct (Peng et al., 2012). Some studies report that deficits in inhibition, updating, and switching are all related to calculations and word problem–solving difficulties (L. S. Fuchs et al., 2005; Passolunghi & Siegel, 2001; Passolunghi, Cornoldi, & De Liberto, 1999), while others suggest that inhibition deficits may be particularly related to calculation difficulties (Toll, Van der Ven, Kroesbergen, & Van Luit, 2011) but not to word problem–solving difficulties (Swanson & Beebe-Frankenberger, 2004). Moreover, research shows that the construct of executive function system may be different between older children/adults and younger children, such that the inhibition, updating, and switching in younger children (e.g., preschoolers) are less differentiated from one another, representing a unitary construct (Bull & Lee, 2014). Given this developmental feature of executive functions, if executive function deficits are related to MD, young children with MD should demonstrate comprehensive executive function deficits, while older children and adults with MD may show selective executive function deficits depending on types of MD (L. S. Fuchs et al., 2005; L. S. Fuchs et al., 2006; Passolunghi & Siegel, 2001; Swanson & Beebe-Frankenberger, 2004).

Types of MD

MD is often considered as a heterogeneous group, and the present study focused on several most studied subgroup comparisons in previous MD profiling research: (a) comorbidity (individual with MDRD vs. individuals with MD-only [with normal reading skills]) and (b) MD identified with calculation difficulties (MD identified with calculation measures; L. S. Fuchs et al., 2006) versus MD identified with word problem–solving difficulties (MD identified with word problem–solving measures; Fuchs, Fuchs, & Prentice, 2004) versus MD identified with comprehensive MD (identified with mathematics measures that tap calculations, problem solving, and/or other mathematics skills). Specifically, previous research suggests that different MD groups may demonstrate different cognitive deficit profiles. For example, some research suggests that compared with MD-only, MDRD show more comprehensive and severe deficits in cognitive skills (e.g., Cirino et al., 2015; Jordan & Montani, 1997; Peng et al., 2012; van der Sluis et al., 2004). In contrast, the cognitive deficit profiles among the subgroups within MD are less clear. For example, some research suggests that MD identified with calculation difficulties are mostly related to deficits in low-level cognitive skills, such as phonological processing (L. S. Fuchs et al., 2008; Swanson, 2006), processing speed (L. S. Fuchs et al., 2006; L. S. Fuchs et al., 2008; Geary et al., 2007), short-term memory (McCall, 1999), and visuospatial skills (e.g., Rovet, Szekely, & Hockenberry, 1994). By contrast, MD identified with word problem–solving difficulties are more closely associated with deficits in high-level cognitive skills, such as working memory and executive functions (L. S. Fuchs et al., 2006; L. S. Fuchs et al., 2008; Swanson, 2006; Swanson & Beebe-Frankenberger, 2004). MD identified with comprehensive MD seem to have more comprehensive and severe cognitive deficits (e.g., Cirino et al., 2015; Peng et al., 2012). Yet there are other studies suggesting that the deficit profiles among MD screened by different mathematics measures overlap (e.g., Gold et al., 2013; Landerl et al., 2004; Peng et al., 2012).

Severity of MD

Low-achievement identification is a widely adopted approach in the MD research (e.g., Andersson & Lyxell, 2007; Geary et al., 2007; Murphy, Mazzocco, Hanich, & Early, 2007; Passolunghi & Siegel, 2004; Siegel & Ryan, 1989). In accordance with low-achievement identification, individuals with MD are identified if their IQs are in a normal range but their performance is below a cutoff point on mathematics screening measures (Peng & Fuchs, 2016). However, one major problem associated with this approach is the inconsistent cutoff criterion on the mathematics screening measures. The cutoff criterion used to establish MD varies from the 35th percentile to less than the 5th percentile across studies (e.g., Landerl et al., 2004; Peng et al., 2012).

Different cutoff scores lead to different degrees of severity in MD (e.g., Murphy et al., 2007), which may affect cognitive profiles of MD. For example, Murphy et al. (2007) studied the relation between the cognitive profiles of children with MD and the mathematics achievement cutoff criteria. Murphy et al. (2007) found qualitative group differences in the profiles of mathematics related cognitive skills across groups defined by different cutoff scores (below the 10th percentile vs. between the 11th and 25th percentiles on mathematics measures). Children with less severe MD appeared to show less severe cognitive deficits than children with more severe MD. These results pose a theoretical question on whether MD is a continuous or a discrete construct, and highlight the potential effects of severity of MD on the cognitive deficits profiles. That is, if MD is a discrete construct as suggested by some research (e.g., Landerl et al., 2004), the severity of MD should not influence the deficit profiles. On the other hand, if MD is not a discrete construct, the severity of cognitive deficits in MD may change as a function of MD severity (e.g., Branum-Martin, Fletcher, & Stuebing, 2013; Murphy et al., 2007).

Age

Mathematics achievement is cumulative throughout and beyond the elementary school years with quantitative and qualitative changes occurring within and across grade levels. The required changes concern performance demands and the necessary prerequisite skills. Thus, performing a specific mathematics task (e.g., calculation) at different ages may involve different cognitive skills (Raghubar et al., 2010). Following this logic, a specific type of MD identified at different ages may relate to different underlying cognitive deficits. For example, some research shows that younger children (e.g., first grade) identified with calculation difficulties show deficits in visuospatial skills and working memory (e.g., Geary et al., 2007), while other studies indicate that older children (e.g., fourth grade and above) with calculation difficulties do not show deficits in these areas (e.g., McCall, 1999). This may be because young children have not established strong numerical-processing skills and tend to rely heavily on visuospatial skills and working memory to perform calculation tasks (Raghubar et al., 2010). Thus, deficits in visuospatial skills and working memory among young children may be related to their calculation difficulties. In contrast, older children are likely to use their numerical skills, not their visuospatial skills and working memory, to perform calculation tasks, and thus their calculation difficulties are less likely due to their poor visuospatial skills and working memory. These findings suggest that age, a proxy for skill level and experience with mathematics tasks, as well as the interaction between age and different MD types should be an important factor to understand the cognitive deficits profiles of MD.

Domains of Task

Information processing in different domains may play different roles in mathematics learning. Numerical and visuospatial information processing are critical in calculations (Geary, 1993; Swanson & Jerman, 2006) and verbal information processing is particularly important for word problem solving (L. S. Fuchs et al., 2006; L. S. Fuchs et al., 2008). Thus, it is reasonable to assume that domains of task may influence the cognitive deficit profiles of different types of MD. Indeed, previous reviews suggest that different types of MD may have domain-general working memory deficits, but the severity of cognitive deficits vary across domains, such that numerical and visuospatial cognitive deficits are more severe than verbal cognitive deficits (Peng & Fuchs, 2016; Swanson & Jerman, 2006). These findings, taken together, indicate that domains of task may be an important factor contributing to the working memory–deficit profiles of MD. Nevertheless, it remains unclear whether the domains of task affect the deficit profiles of other cognitive skills. This question is important for understanding the domain-general and domain-specific nature of cognitive skills in the context of MD. That is, the cognitive skills, especially those often considered as domain-general high-level cognitive skills (e.g., executive functions) may show domain-specific nature (deficits related to the core deficits of MD: numerical-processing deficits) among MD.

Literature Search

Articles for this meta-analysis were identified in three ways. First, a computer search of the Education Resources Information Center and databases of ProQuest and PsycINFO was conducted. We used the earliest possible start date (1920) till June, 2017. The following terms were used to search in full text: (math*AND difficult* OR disabilit*) OR discalcul*. The terms math*, difficult*, disabilit*, discalcul* allow for inclusion of mathematics, difficulties/difficulty/disability/disabilities, dyscalculia/dyscalculic, and so forth. Second, we searched for unpublished literature through Dissertation and Masters Abstract indexes in ProQuest, Cochrane Database of Systematic Reviews, and relevant conference programs (e.g., Conference of Society for Research in Child Development; National Council of Teachers of Mathematics Annual Meeting; Psychology of Mathematics Education Annual Conference). Third, previous relevant reviews (Geary, 1993; Gersten et al., 2005; Johnson et al., 2010; Peng & Fuchs, 2016; Swanson & Jerman, 2006) were reviewed to include extra articles not identified in the first two search steps. The initial search yielded 908 studies. Two authors of this study then reviewed all studies by titles and abstracts. After excluding the duplicate 26 articles, the remaining 882 articles were closely reviewed using specific criteria. First, the study had to include a group of individuals with MD and a group of age-matched TD individuals. That is, the study had to report information showing that the MD group scored at least below the 35th percentile or below one standard deviation on mathematics screening measures, which is a common learning disability identification criterion (Swanson & Harris, 2013). Also, the study had to provide information showing that the IQ (nonverbal IQ, verbal IQ, or combination of nonverbal and verbal IQ) of individuals with MD were in the normal range (e.g., standardized score 80–120). We excluded 499 studies that did not meet these criteria. Then, we reviewed the remaining 383 studies to select studies that compared the MD with TD group on measures that tap at least one of the following cognitive skills: phonological processing, processing speed, short-term memory, working memory, attention, visuospatial skills, and executive functions. We excluded another 308 studies that did not meet this criterion. The final sample included 75 studies, including 4 dissertations, 1 book chapter, and 70 peer-reviewed articles. Table 1 shows the definition and examples of task for each cognitive skill.

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Table 1

Description of cognitive skills and examples

Cognitive skills	Definition	Examples of tasks
Phonological processing—Manipulation	The ability to manipulate syllables, onset-rime, and phonemes	Isolate individual sounds within words; remove phonological segments from spoken words to form other words; synthesize sounds to form nonwords
Phonological processing—Retrieval	The ability to fast retrieve phonological representations from the long-term memory	Rapid letter/digit/object/picture/color naming
Processing speed	The ability to automatically and fluently perform relatively easy or overlearned cognitive tasks	Visual matching: Participants have to locate and circle fast two identical numbers that appear in a row of six numbers within a limited amount of time.
Short-term memory	The ability to temporarily store verbal, numerical, or visuospatial information	Digit span, in which participants are orally presented with a series of digits (e.g., “8, 2, 4”) and immediately repeat them back in the same order
Working memory	The ability to simultaneously process and store information	Listening recall, in which the tester says series of short sentences, only some of which make sense. The participants indicate whether each sentence is true or false. After hearing all sentences in a trial (i.e., 1–6 sentences) and determining them to be true or false, the participant recalls the final word of each sentence, in the order presented.
Attention	The ability to appropriate allocate cognitive resources to relevant stimuli	Subjective measures such as SWAN attention rating scale; objective measures such as code transmission in which participants listen to a long, monotone series of numbers for two “5s” in a row. When this pattern is noted, the child must state the number presented prior the target “5s.” Sky search, in which participants are instructed to find target spaceships on a large sheet filled with similar distracter spaceship.
Visuospatial skills—Visuospatial perception	The ability to determine visuospatial relationships with respect to the orientation of their own bodies, in spite of distracting information	Water-level test, in which participants need to draw or identify a horizontal line in a tilted bottle
Visuospatial skills—Spatial visualization	The processes of completing complicated and multistep manipulations of visuospatial information	Hidden figures, in which participants find the simple shape shown on the top in the complex shape shown on the bottom; paper-folding test, in which participants indicate how the paper would look when unfolded
Visuospatial skills—Mental rotation	The ability to mentally rotate one stimulus to align it with a comparison stimulus	Two objects are rotated relative to each other and participants need to rotate the objects along a smooth angular trajectory into congruence with one another
Executive function—Inhibition	The ability to deliberately inhibit dominant, automatic, or prepotent responses when necessary	Stroop color word in which participants are required to quickly and accurately name the name of a color (e.g., “blue,” “green,” or “red”) that is printed in a color not denoted by the name (e.g., the word “red” printed in blue ink instead of red ink)
Executive function—Updating	The ability of encoding and evaluating incoming information for relevance to the task at hand and subsequent revision of the information held in memory	N-back task, in which a single-digit number is displayed for 2 seconds every 2 seconds on a computer screen and participants have to make a judgment whether the number in the current step is the number presented n steps previously
Executive function—Switching	The ability shift attention or to shift between strategies or response sets	Wisconsin Card Sorting Test, in which a number of stimulus cards are presented to the participant and the participant is told to match the cards, but not how to match; however, the participant is told whether a particular match is right or wrong.

Note. SWAN = Strengths and Weaknesses of Attention-Deficit/Hyperactivity-symptoms and Normal-behaviors rating scale.

Coding Procedure and Interrater Reliability

These 75 studies were coded according to the characteristics of participants, mathematics, and IQ screening tasks, tasks used to measure phonological processing, processing speed, short-term memory, working memory, visuospatial skills, attention, and executive functions. The descriptive features of individual studies and coding information are provided in Supplementary Table S1 (available in the online version of the journal).

Variables were discussed until a consensus was reached between the first and the second author. Then, both authors used the coding system to conduct the final coding of all studies. Across the total variable matrix, the mean interrater agreement (i.e., the percentage of agreement indexed by the data points coded in agreement divided by total coded data points) was .98, ranging from .93 to .99 for all moderators of interest and all cognitive skills investigated in this study. Any disagreements between the raters were resolved by consulting the original article or by discussion.

Analytic Strategies

Effect Size

Hedges’ g was used as the measure of effect sizes. We chose Hedges’ g because MD profiling research tends to have small sample sizes and Hedges’ g provides a more accurate estimate of effect sizes than Cohen’s d on small sample sizes (Hedges’ g pools using n − 1 for each sample instead of n; Grissom & Kim, 2005). For studies reporting means, standard deviations, and sample size, the following formulae were used:

g u = g ( 1 − 3 4 ( N MD + N TD − 2 ) − 1 )

with

g = X ¯ MD − X ¯ TD S and g = ( N MD − 1 ) S MD 2 + ( N TD − 1 ) S TD 2 N MD + N TD − 2

in which g u is the unbiased estimate of Hedges’ g, g is Hedges’ g as traditionally defined, N MD is the number of participants in the MD group, N TD is the number of participants in the TD group, X ¯ MD is the mean of the target skill (e.g., working memory) scores for participants in the MD group, X ¯ TD is the mean of the target skill scores for participants in the TD group, S is the pooled standard deviation of S MD 2 , which is the variance of target skill scores for the participants in the MD group, and S TD 2 is the variance of target skill scores for the participants in the TD group.

Effect sizes of deficits on all cognitive skills were estimated for MD in comparison to the TD group. Next, meta-regression analyses were used to examine the moderation effects of types of MD (i.e., comorbidity and types of MD screening), severity of MD, age, and domains of task on the deficit profile for each cognitive skill. For the moderation analyses, each moderator was examined with other moderators simultaneously controlled in one meta-regression model. For moderators that were categorical, we created different sets of dummy codes and entered them into the meta-regression model (Cohen, Cohen, West, & Aiken, 2013). To examine the “bottleneck theory,” we controlled for the group differences between MD and TD on processing speed or short-term memory for the group comparison between MD and TD on working memory, attention, or executive functions. Specifically, we first identified studies that provided correlations among at least one low-level cognitive skill and one high-level cognitive skill. Then, controlling for the group difference on the low-level cognitive skill and the correlation between low-level and high-level cognitive skills, we calculated the group difference between MD and TD on the high-level cognitive skill, which was synthesized to indicate the group differences on high-level cognitive skills, partialling out the effects of low-level cognitive skills.

The Cognitive Deficit Profiles of MD

The estimated average effect size indicating the deficits of MD as compared with TD was as follows: phonological processing: Hedges’ g = −.91, CI95 [−1.24, −0.57] (Manipulation: Hedges’ g = −1.31, CI95 [−1.82, −.79]; Retrieval: Hedges’ g = −0.61, CI95 [−0.80, −0.43]), processing speed: Hedges’ g = −0.90, CI95 [−1.08, −0.72], short-term memory: Hedges’ g = −0.56, CI95 [−0.67, −0.46], working memory: Hedges’ g = −0.76, CI95 [−0.88, −0.64], visuospatial skills: Hedges’ g = −0.43, CI95 [−0.63, −0.23], attention: Hedges’ g = −0.72, CI95 [−0.92, −0.52] (Subjective: Hedges’ g = −0.44, CI95 [−0.73, −0.14]; Objective: Hedges’ g = −0.86, CI95 [−1.09, −0.63]), and executive function: Hedges’ g = −0.50, CI95 [−0.60, −0.39] (Inhibition: Hedges’ g = −0.37, CI95 [−0.48, −0.26]; Updating: Hedges’ g = −0.76, CI95 [−1.04, −0.48]; Switching: Hedges’ g = −0.75, CI95 [−0.87, −0.63]).

Factors That Influence the Deficit Profiles of MD

Next, we examined whether comorbidity, types of MD screening, severity of MD, age, or domains of task influenced the deficit profiles of each cognitive skill of MD. Table 2 presents the effect sizes of each subgroup within each categorical moderator for each cognitive skill. Specifically, across these cognitive deficits, the MDRD group showed more severe deficits than the MD-only group; MD identified by comprehensive mathematics measures showed more severe deficits than MD identified with calculation or word problem–solving difficulties; cognitive deficits were domain-general but numerical deficits were more severe on processing speed and working memory. Manipulation deficits were more severe than retrieval deficits in phonological processing. Attention deficits were more severe if measured by subjective measures than objective measures.

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Table 2

Descriptive statistics on subgroups within different cognitive skills for the comparison between MD and TD

Subgroup	k	ES	CI95	τ2	Subgroup	k	ES	CI95	τ2
	Phonological processing					Working memory
MD-only	41	−0.36	[−0.60, −0.11]	.15	MD-only	125	−0.65	[−0.82, −0.47]	.18
MDRD	42	−0.87	[−1.22, −0.51]	.42	MDRD	106	−0.81	[−0.98, −0.65]	.20
MD identified with word problem–solving difficulties	19	−0.74	[−1.30, −0.18]	.21	MD identified with word problem–solving difficulties	86	−0.44	[−0.56, −0.32]	.05
MD identified with calculation difficulties	40	−0.63	[−0.92, −0.35]	.20	MD identified with calculation difficulties	136	−0.74	[−0.91, −0.58]	.19
MD identified with comprehensive mathematics difficulties	21	−1.89	[−3.49, 0.29]	1.09	MD identified with comprehensive mathematics difficulties	52	−0.97	[−1.27, −0.68]	.27
Manipulation	24	−1.31	[−1.82, 0.79]	.51	Verbal	92	−0.64	[−0.74, −0.54]	.08
Retrieval	84	−0.61	[−0.80, −0.43]	.23	Numerical	74	−0.89	[−1.08, −0.70]	.25
					Visuospatial	81	−0.47	[−0.58, −0.36]	.07
	Processing speed					Visuospatial skills
MD-only	17	−0.66	[−0.92, −0.41]	.13	MD-only	10	−0.23	[−0.51, 0.04]	.05
MDRD	22	−1.04	[−1.31, 0.77]	.17	MDRD	8	−0.64	[−0.99, −0.29]	.05
MD identified with word problem–solving difficulties	1	−0.82	—	—	MD identified with word problem–solving difficulties	—	—	—	—
MD identified with calculation difficulties	35	−0.80	[−1.01, −0.59]	.12	MD identified with calculation difficulties	9	−0.40	[−0.71, −0.08]	.10
MD identified with comprehensive mathematics difficulties	11	−1.13	[−1.47, −0.79]	.11	MD identified with comprehensive mathematics difficulties	9	−0.49	[−0.95, −0.04]	.07
Verbal	5	−0.98	[−2.19, 0.23]	.72	Visuospatial perception	8	−0.50	[−0.88, −0.13]	.09
Numerical	14	−0.98	[−1.28, −0.67]	.10	Spatial visualization	9	−0.43	[−0.78, −0.09]	.10
Visuospatial	18	−0.87	[−1.06, −0.67]	.08	Mental rotation	3	−0.44	[−0.82, −0.06]	0
	Short-term memory					Attention
MD-only	83	−0.31	[−0.42, −0.20]	.04	MD-only	18	−0.59	[−0.79, −0.38]	.07
MDRD	73	−0.69	[−0.81, −0.58]	.03	MDRD	17	−0.97	[−1.29, −0.64]	.23
MD identified with word problem–solving difficulties	25	−0.41	[−0.52, −0.30]	.002	MD identified with word problem–solving difficulties	1	−1.28	—	—
MD identified with calculation difficulties	69	−0.51	[−0.63, −0.39]	.05	MD identified with calculation difficulties	22	−0.69	[−0.94, −0.45]	.16
MD identified with comprehensive mathematics difficulties	71	−0.71	[−0.99, −0.43]	.29	MD identified with comprehensive mathematics difficulties	26	−0.72	[−1.12, −0.33]	.31
Verbal	67	−0.47	[−0.58, −0.35]	.05	Verbal	2	−0.31	[−0.82, −0.21]	0
Numerical	62	−0.61	[−0.76, −0.46]	.17	Numerical	2	−0.92	[−3.00, 1.16]	.04
Visuospatial	56	−0.60	[−0.70, −0.50]	.002	Visuospatial	5	−0.09	[−1.19, 1.00]	.12
					Objective measures	12	−0.44	[−0.73, −0.14]	.14
					Subjective measures	37	−0.86	[−1.09, −0.63]	.25
	Executive function
MD-only	93	−0.38	[−0.52, −0.25]	.05
MDRD	37	−0.66	[−0.87, −0.45]	.19
MD identified with word problem–solving difficulties	18	−0.39	[−0.60, −0.19]	.04
MD identified with calculation difficulties	44	−0.55	[−0.71, −0.39]	.05
MD identified with comprehensive mathematics difficulties	81	−0.46	[−0.69, −0.22]	.21
Verbal	60	−0.36	[−0.51, −0.21]	.19
Numerical	58	−0.38	[−0.53, −0.24]	.07
Visuospatial	4	−0.43	[−0.92, 0.07]	.02
Inhibition	114	−0.37	[−0.48, −0.26]	.12
Updating	11	−0.76	[−1.03, −0.48]	.05
Switching	19	−0.75	[−0.87, −0.64]	.01

Note. CI = confidence interval; ES = effect size; TD = typically developing; MD-only = individuals with only mathematics difficulties; MDRD = individuals with mathematics difficulties and reading difficulties; — = no results due to no/insufficient data points. The second group in each group comparison variable is the reference group (e.g., in MD-only vs. MDRD, MDRD is the reference group in the dummy coding of comorbidity).

Next, we used meta-regression models with all moderators in one model to examine the group comparison between MD and TD on each cognitive skill (see Table 3). Regarding the phonological processing deficit profiles of MD, age, comorbidity, types of MD screening, and type of phonological processing were significant moderators. Specifically, younger individuals with MD showed more severe phonological processing deficits than older individuals with MD, β = −0.15, p = .03; the MDRD group showed more severe phonological deficits than the MD-only group, β = 0.49, p = .03; MD identified with word problem–solving difficulties showed less severe phonological processing deficits than those identified with calculation difficulties, β = 0.76, p = .001. Manipulation deficits were more severe than retrieval deficits, β = −0.45, p = .02.

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Table 3

Meta-regression of the moderation analysis for the comparison between MD and TD on different cognitive skills

	β	SE	t	CI95	p
Phonological processing
Publication type (peer review vs. nonpeer review)	.87	.34	2.55	[0.11, 1.64]	.03
Age	−.15	.06	−2.60	[−0.28, −0.02]	.03
Comorbidity (MD-only vs. MDRD)	.49	.20	2.45	[0.04, 0.94]	.03
Severity of MD	.05	.13	0.42	[−0.23, 0.34]	.69
Type of MD screening
Word problem solving vs. calculation	.76	.17	4.54	[0.38, 1.13]	.001
Comprehensive vs. calculation	.15	.31	0.47	[−0.55, 0.84]	.65
Word problem solving vs. comprehensive	.61	.30	2.07	[−0.05, 1.27]	.07
Manipulation vs. retrieval	−.45	.16	−2.85	[−0.80, −0.10]	.02
Processing speed
Publication type (peer review vs. nonpeer review)	−.44	.22	−2.02	[−0.94, 0.05]	.07
Age	.04	.06	0.78	[−0.08, 0.17]	.46
Comorbidity (MD-only vs. MDRD)	.48	.18	2.71	[0.08, 0.89]	.02
Severity of MD	−.26	.09	−2.80	[−0.48, −0.05]	.02
Type of MD screening
Word problem solving vs. calculation	—	—	—	—	—
Comprehensive vs. calculation	−.14	.16	−0.83	[−0.51, 0.24]	.43
Word problem solving vs. comprehensive	—	—	—	—	—
Domain of tasks
Verbal vs. numerical	.27	.31	0.86	[−0.43, 0.97]	.41
Verbal vs. visuospatial	−.39	.48	−0.83	[−1.47, 0.68]	.43
Visuospatial vs. numerical	.66	.23	2.90	[0.15, 1.18]	.02
Short-term memory
Publication type (peer review vs. nonpeer review)	−.05	.18	−0.29	[−0.42, 0.32]	.78
Age	−.002	.01	−0.15	[−0.02, 0.02]	.89
Comorbidity (MD-only vs. MDRD)	.37	.10	3.57	[0.16, 0.58]	.001
Severity of MD	−.01	.03	−0.33	[−0.07, 0.05]	.74
Type of MD screening
Word problem solving vs. calculation	.06	.16	0.36	[−0.27, 0.38]	.72
Comprehensive vs. calculation	.08	.17	0.49	[−0.27, -0.43]	.63
Word problem solving vs. comprehensive	.08	.17	0.49	[−0.27, 0.43]	.63
Domain of tasks
Verbal vs. numerical	.09	.08	1.14	[−0.07, 0.24]	.26
Verbal vs. visuospatial	.04	.10	0.42	[−0.16, 0.25]	.68
Visuospatial vs. numerical	.04	.10	0.46	[−0.15, 0.24]	.65
Visuospatial skills
Publication type (peer review vs. nonpeer review)	−.12	.34	−0.36	[−1.06, 0.82]	.74
Age	−.08	.04	−1.96	[−0.19, 0.03]	.12
Comorbidity (MD-only vs. MDRD)	.46	.16	2.89	[0.02, 0.89]	.04
Severity of MD	−.33	.12	−2.74	[−0.67, 0.004]	.05
Type of MD screening
Word problem solving vs. calculation				—
Comprehensive vs. calculation	.42	.41	1.04	[−0.71, 1.56]	.36
Word problem solving vs. comprehensive				—
Visuospatial perception vs. spatial visualization	−.07	.18	−0.40	[−0.56, 0.42]	.71
Visuospatial perception vs. mental rotation	.65	.33	1.96	[−0.27, 1.58]	.12
Mental rotation vs. spatial visualization	.78	.45	1.71	[−0.48, 2.04]	.16
Working memory
Publication type (peer review vs. nonpeer review)	−.03	.25	−0.11	[−0.53, 0.47]	.91
Age	−.04	.02	−1.90	[−0.09, 0.002]	.06
Comorbidity (MD-only vs. MDRD)	.21	.12	1.70	[−0.04, 0.46]	.10
Severity of MD	−.17	.13	−1.30	[−0.43, 0.09]	.20
Type of MD screening
Word problem solving vs. calculation	.37	.16	2.27	[0.04, 0.69]	.03
Comprehensive vs. calculation	−.03	.15	−0.23	[−0.34, 0.27]	.82
Word problem solving vs. comprehensive	.40	.17	2.40	[0.07, 0.74]	.02
Domain of tasks
Verbal vs. numerical	.23	.18	1.27	[−0.13, 0.59]	.21
Verbal vs. visuospatial	−.07	.17	−0.38	[−0.41, 0.28]	.70
Visuospatial vs. numerical	.30	.12	2.52	[0.06, 0.53]	.02
Attention
Publication type (peer review vs. nonpeer review)	.10	.19	0.51	[−0.31, 0.51]	.62
Age	−.10	.02	−4.13	[−0.15, 0.05]	.001
Comorbidity (MD-only vs. MDRD)	.09	.09	0.99	[−0.10, 0.29]	.34
Severity of MD	−.12	.07	−1.60	[−0.27, 0.04]	.13
Type of MD screening
Word problem solving vs. calculation	—	—	—	—	—
Comprehensive vs. calculation	−.13	.10	−1.26	[−0.34, 0.09]	.23
Word problem solving vs. comprehensive	—	—	—	—	—
Domain of tasks
Verbal vs. numerical	—	—	—	—	—
Verbal vs. visuospatial	—	—	—	—	—
Visuospatial vs. numerical	—	—	—	—	—
Objective measures vs. subjective measures	.42	.13	3.17	[0.14, 0.70]	.01
Executive function
Publication type (peer review vs. nonpeer review)	.75	.14	5.15	[0.44, 1.05]	<.001
Age	.04	.03	1.43	[−0.02, 0.09]	.17
Comorbidity (MD-only vs. MDRD)	.21	.07	3.14	[0.07, 0.36]	.01
Severity of MD	−.01	.05	−0.17	[−0.11, 0.10]	.87
Type of MD screening
Word problem solving vs. calculation	.27	.15	1.83	[−0.04, 0.58]	.08
Comprehensive vs. calculation	–.27	.10	−2.79	[−0.47, −0.07]	.01
Word problem solving vs. comprehensive	.54	.18	2.98	[0.16, 0.92]	.01
Inhibition vs. updating	.48	.18	2.61	[0.09, 0.87]	.02
Switching vs. updating	.29	.18	1.63	[−0.08, 0.67]	.12
Switching vs. inhibition	−.19	.11	−1.77	[−0.41, 0.04]	.09

Note. MD-only = individuals with only mathematics difficulties; MDRD = individuals with mathematics difficulties and reading difficulties; TD = typically developing); — = no results due to no/insufficient data points; Comprehensive = MD identified with comprehensive mathematics difficulties; Word problem solving = MD identified with word problem–solving difficulties; Calculation = MD identified with calculation difficulties; CI = confidence interval. All moderators were entered in one model. Several models were run for thorough subgroup comparisons among moderators with more than 2 categories. For the convenience of presentation, subgroup comparisons within domains of tasks, types of MD screening, types of visuospatial skills, and types of executive functions are all listed in the model. The second group in each group comparison variable is the reference group (e.g., in MD-only vs. MDRD, MDRD is the reference group in the dummy coding of comorbidity). The values in bold indicate statistically significant moderators.

*

p < .05. **p < .01. ***p < .001.

Next, we examined moderators for the manipulation and retrieval deficit profiles of MD, respectively. For the manipulation deficit profiles, with all moderators (i.e., publication type, age, MD severity, comorbidity, and types of MD screening) in one model, only age and comorbidity were significant moderators such that younger individuals with MD and individuals with MDRD have more severe manipulation deficits than older individuals with MD and individuals with MD-only, β = 0.67/0.70, ps < .01. For the retrieval deficit profiles, with all moderators (i.e., publication type, age, MD severity, comorbidity, and domains of retrieval: numerical vs. nonnumerical, and types of MD screening) in one model, only types of MD screening was the significant moderator such that MD identified with calculation difficulties showed more severe retrieval deficits than MD identified with word problem–solving difficulties or comprehensive MD, β = 0.67/0.70, p = .01/.03. We also examined moderators for MD defined by different screening measures. Results showed that there were no significant moderators for the phonological processing deficit profiles among MD identified with calculation difficulties or MD identified with word problem–solving difficulties, ps > .05. However, for MD identified with comprehensive MD, manipulation deficits were more severe than retrieval deficits, β = 0.53, p = .01. Moreover, we examined moderators among the MD-only group and the MDRD group, respectively, and we did not find significant moderators, ps > .08.

Regarding the processing speed deficit profiles of MD, comorbidity, MD severity, and domains of task were significant moderators. That is, the MDRD group showed more severe processing speed deficits than the MD-only group, β = 0.48, p = .02. The processing speed deficits were more severe among those with more severe MD, β = −0.26, p = .02. Deficits in numerical-processing speed were more severe than deficits in visuospatial processing speed, β = 0.66, p = .02. We also examined moderators among MD identified with calculation difficulties, and we did not find significant moderators, ps > .06.

With respect to the deficit profiles of short-term memory and visuospatial skills, only comorbidity affected the group differences such that the MDRD group showed more severe short-term memory and visuospatial processing deficits than that of the MD-only group, β = 0.37/0.46, p = .001/.04.

With respect to the working memory deficit profiles of MD, types of MD screening and domains of task were significant moderators. That is, MD identified with word problem–solving difficulties showed less severe working memory deficits than those identified with calculation or comprehensive MD, β = 0.37/0.40, p = .03/.02. The deficits in numerical working memory were more severe than the deficits in visuospatial working memory, β = 0.30, p = .02. Next, we examined moderators for the working memory deficit profiles among MD defined by different screening measures. Results showed that only domains of task was a significant moderator for MD identified with comprehensive MD such that the visuospatial working memory deficits were less severe than the verbal and numerical working memory deficits, β = 0.35/0.49, ps < .05, with the numerical working memory deficits being the most severe. Moreover, we examined moderators for the working memory deficit profiles among the MD-only group and the MDRD group, respectively. For the MD-only group, we did not find any significant moderators, ps > .05. For the MDRD group, only types of MD screening was significant such that those screened with word problem–solving difficulties have less severe working memory deficits than those identified with calculation or comprehensive MD, β = 0.55/0.64, ps < .05.

Regarding the attention deficit profiles of MD, age and type of attention measure were the only significant moderators. That is, younger individuals with MD showed more severe attention problems than older individuals with MD, β = −0.10, p = .001; attention problems seemed more severe when attention was measured by subjective measures than objective measures, β = 0.42, p = .01.

With respect to the executive function deficit profiles of MD, comorbidity and types of MD screening were significant moderators such that the MDRD group showed more severe executive function deficits than the MD-only group, β = 0.21, p = .01; MD identified with comprehensive MD showed more severe executive function deficits than those identified with calculation difficulties or word problem–solving difficulties, β = 0.27/0.54, ps = .01. We did not examine the effects of domains of task because we did not have data on domains for switching (all switching tasks used materials across domains). That said, among inhibition and updating, we did not find that domains of task was a significant moderator, ps > .30. Next, we examined whether type of executive functions and age affected the executive function deficit profiles among MD defined by different screening measures. Results showed that type of executive functions did not influence executive deficits for MD identified with calculation difficulties or comprehensive MD, ps > .31. However, for MD identified with word problem–solving difficulties, updating deficits were more severe than inhibition deficits, β = 0.44, p = .03 (not enough effect sizes to make comparison with switching). Age did not explain the executive function deficit profiles of MD identified by different screening measures, ps > .31.

Testing the Bottleneck Theory

Next, we examined whether the group differences between MD and TD on high-level cognitive skills (i.e., working memory, attention, and executive function) were (partially) explained by group differences on low-level cognitive skills (i.e., short-term memory and processing speed) as suggested by the bottleneck theory of cognition. There were 12 studies that provided correlation tables to calculate 221 effect sizes on group comparisons on working memory, attention, or executive functions, partialling out the effects of short-term memory or processing speed.

Among these 12 studies, the mean group differences between MD and TD on working memory, attention, and executive function was Hedges’ g = −0.75, CI95 [−0.90, −0.60]. When the effects of short-term memory or processing speed were controlled for, the mean group differences between MD and TD on working memory, attention, and executive function decreased, Hedges’ g = −0.50, CI95 [−0.61, −0.38]. t-Test showed that there was a significant difference between these two effect sizes (i.e., with vs. without partialling out short-term memory or processing speed). This finding indicates that low-level cognitive skills significantly accounted for some variance (Hedges’ g = .25) in the group differences between MD and TD on high-level cognitive skills. However, the low-level cognitive skills could not completely explain the group differences between MD and TD on high-level cognitive skills.

Summary of Results

To sum up, our findings show that (a) MD is related to deficits (from most severe to less severe) in phonological processing, processing speed, working memory, attention, short-term memory, executive functions, and visuospatial skills; (b) regarding the comorbidity, the MDRD group showed more severe deficits than MD-only group on phonological processing, processing speed, short-term memory, visuospatial skills, and executive functions; (c) types of MD screening affected the deficit profiles of phonological processing, working memory, and executive functions such that MD identified with word problem–solving difficulties seemed to have less severe deficits on these cognitive skills than those identified with calculation difficulties or comprehensive MD. For MD identified with comprehensive MD, they had more severe numerical working memory deficits than their visuospatial working memory deficits, and they also had more severe phonological processing-manipulation deficits than their retrieval deficits; (d) severity of MD was only related to the severity of processing speed deficits; (e) age only affected deficits in phonological processing and attention such that younger individuals with MD have more severe deficits in these skills than older individuals with MD; (f) domains of task mostly affected the deficit profiles of processing speed and working memory such that deficits on these skills among MD are most severe in the numerical domain; (g) low-level skills (short-term memory and processing speed) significantly accounted for some (not all) variance in the group differences between MD and TD on high-level skills (working memory, attention, and executive functions).

The Deficit Profile of Processing Speed and Moderators

We found that processing speed deficits were one of the most salient cognitive deficits among MD, in line with most previous research (Bull & Johnston, 1997; L. S. Fuchs et al., 2006; L. S. Fuchs et al., 2008; Geary, 1993). Some previous research suggests processing speed may be especially important for the early mathematics development (e.g., L. S. Fuchs et al., 2006), and processing speed deficits may be most severe among young children with calculation difficulties (Bull & Johnston, 1997; L. S. Fuchs et al., 2006). However, our findings showed that processing speed deficits were not affected by types of MD or age, indicating that processing speed deficits may be fundamental cognitive deficits for all types of MD (e.g., L. S. Fuchs et al., 2008; Geary, 1993) and can serve as a stable marker of MD across time (e.g., McCall, 1999; Mussolin, Martin, & Schiltz, 2011, Swanson, 2012).

In addition, Geary (1993) suggests that MD only demonstrate processing speed deficits in the numerical domain. Although our finding showed that numerical-processing speed deficits were most severe, in line with Geary (1993), we did find that MD was related to processing deficits across domains. One way to understand the relation between domain-general processing speed deficits and MD is based on the bottleneck theory. That is, low-level cognitive skills such as processing speed directly support the process of working memory, attention, and executive functions in mathematics performance (e.g., Alloway et al., 2006; Noël et al., 2001; Swanson & Sachse-Lee, 2001). Slow numerical facts processing during calculations (Cirino et al., 2015; L. S. Fuchs et al., 2006; L. S. Fuchs et al., 2008; Jordan & Montani, 1997) or use of inefficient and slow strategies during word problem solving (Geary, 1993) may create a ‘‘bottleneck’’ that constricts information flow (e.g., information may decay easily in short-term memory because of slow processing speed) to high levels of processing during mathematics performance. Indeed, our findings showed that the deficits in processing speed and short-term memory accounted for some variance in the deficits of high-level cognitive skills (i.e., working memory, attention, executive functions) among MD, partially supporting the bottleneck theory and highlighting the fundamental role of processing speed deficits in MD.

The Deficit Profile of Memory and Moderators

Regarding the memory deficits of MD, we found that MD has deficits in both short-term memory and working memory, which is consistent with previous reviews (Peng & Fuchs, 2016; Swanson & Jerman, 2006). Our findings showed that the deficits in short-term memory and working memory among MD are across verbal, numerical, and visuospatial, also consistent with previous reviews (Peng & Fuchs, 2016; Swanson & Jerman, 2006). However, numerical working memory deficits were more severe than the visuospatial working memory deficits, especially among MD identified with comprehensive MD. This finding is more in line with the findings of Peng and Fuchs (2016) and Gersten et al. (2005), indicating that numerical working memory, but not visuospatial working memory as suggested by Swanson and Jerman (2006), may be the most significant working memory deficits among MD.

We found that MD identified with word problem–solving difficulties had less severe working memory deficits than MD identified with calculation and comprehensive MD. This finding is contrary to our hypothesis that working memory deficits are closely related to word problem–solving difficulties, at least in comparison to calculation difficulties. This hypothesis was based on the assumption that solving word problems involves more complex steps, such as reading and comprehending the problem, identifying relevant/irrelevant information to set up an equation, and then solving calculations, thereby requiring more working memory. There may be two explanations for this unexpected finding. First, a recent meta-analysis on the relation between mathematics and working memory suggests that calculations and word problems actually show comparable relations with working memory (Peng et al., 2016), suggesting that word problem–solving difficulties may not necessarily be associated with more severe working memory deficits than do calculation difficulties. Second, among the reviewed studies, in which MD were identified with word problem–solving difficulties, these MD individuals were often reported to have relatively normal calculation competence whereas most studies, in which MD identified with calculation difficulties, did not report whether these MD individuals had normal word–problem solving skills. Thus, it is likely that some MD identified with calculation difficulties may also have word problem–solving difficulties. Future studies are needed to compare word problem–solving difficulties only to calculation difficulties only to further test whether these two groups differ on working memory.

The Deficit Profile of Phonological Processing and Moderators

We found that MD had deficits in phonological processing, with the MDRD group showing more severe deficits than the MD-only group. However, the phonological processing deficits of MD were not stable across time. That is, we found decreasing effects of age on the severity of phonological processing deficits among MD. Moreover, comorbidity and types of MD screening affected the phonological processing deficits profile of MD, and phonological manipulation is more severe than retrieval. These findings, taken together, suggest that phonological processing deficits are not only an important feature of RD but also an early marker for MD, regardless of comorbidity (although phonological processing deficits are worse in RDMD compared with MD-only). It appears that phonological processing plays a key role in early stages of mathematics learning, especially in calculation development (language and reasoning may be more important prerequisite skills for word problem solving as evidenced by L. S. Fuchs et al., 2006), but it may not be a distinct marker of MD among older individuals. This is in line with the phonological representation hypothesis that solving a simple calculation problem initially requires manipulating and maintaining phonological representation in short-term memory (Geary, 1993; Krajewski & Schneider, 2009; Simmons & Singleton, 2008; Vukovic & Lesaux, 2013). As children with MD build more fluency and automaticity, they rely less on phonological processing (especially phonological manipulation) to solve a mathematics problem. At the same time, phonological processing deficits are often alleviated with school instruction (Bus & Van IJzendoorn, 1999), which may reduce the relation between phonological processing deficits and MD among older population.

The Deficit Profile of Visuospatial Skills and Moderators

Regarding the relation between visuospatial deficits and MD, we were especially interested in the moderating effects of types of visuospatial skills and types of MD. However, we did not find those as significant moderators, nor did we obtain enough effect sizes to compare different types of visuospatial skills among different types of MD. Thus, we cautiously conclude that types of visuospatial skills do not affect the visuospatial deficit profiles of MD. That said, we do point out that the visuospatial deficits were the least severe cognitive deficits of MD, and the deficits in the visuospatial domain of other cognitive skills, such as attention, short-term memory, executive functions, and working memory all appeared to be nonsignificant or play a less important role in MD, compared with the deficits in the verbal or numerical domain.

There are two plausible explanations for this moderate relation between visuospatial deficits and MD. One is that the majority of MD individuals included in the present meta-analysis were elementary or middle school children (94% of included studies) and were identified as MD based on calculation or word problem–solving difficulties (77% of included studies). Prior studies suggest that visuospatial skills are more strongly related to geometry and other higher level mathematics skills (e.g., algebra) among older individuals. For example, Rohde and Thompson (2007) found that visuospatial skills significantly explained performance on college students’ mathematics scores on the Scholastic Aptitude Test. Similarly, Reuhkala (2001) found moderate correlations (r = .44–.57) among visuospatial processes and high school mathematics competence, which included items on algebra and geometry. In addition, Tolar, Lederberg, and Fletcher (2009) found visuospatial skills affected algebra achievement among college students. Another explanation is that the MD sample in this meta-analysis were individuals with MD and normal IQ. Because IQ, in our review, was often indexed via measures that assess visuospatial abilities (e.g., WASI matrix reasoning and Raven’s Progressive Matrix), IQ and visuospatial skills share significance variance and may be competing against each other in our analyses. Further studies are needed to clarify whether visuospatial deficits are related, but not a distinct marker for MD.

The Deficit Profile of Attention and Moderators

MD was also related to attention difficulties. This finding is consistent with previous research (Cirino et al., 2007; McCall, 1999; Swanson, 2012; Willcutt et al., 2013) and the view that individuals with learning disabilities (e.g., RD, MD, and MDRD), even without the diagnosis of ADHD, demonstrate attention difficulties to certain degrees (e.g., Mayes, Calhoun, & Crowell, 2000). As expected, the attention difficulties of MD were more severe if attention was measured by subjective measures than by objective measures. This suggests that different methods used to measure attention may affect the sensitivity of attention measures when screening MD. One possible explanation is that the subjective measures of attention, which mostly rely on teachers’ and parents’ ratings, may have been clouded by students’ poor mathematics performance. That is, the subjective ratings of attention may serve as a proxy for students’ mathematics competence (L. S. Fuchs et al., 2006)

Research on children with ADHD suggests the lack of automatized arithmetic facts retrieval consumes limited attention resources for children with ADHD, which hinders their mathematics performance (e.g., Zentall, 1990). Following this logic, we had two competing hypotheses with regard to the age effects on the relation between attention difficulties and MD. One was that the relation between attention problems and MD is stronger in the early learning stage when children are actively engaged in learning and automatizing math facts. The other was that the relation between attention difficulties and MD is stronger among older individuals with MD because older individuals with MD also struggle with automatized arithmetic facts retrieval to certain degrees (Geary, 1993) and the amount of attention needed in mathematics tasks may increase with more complex and advanced mathematics skills. Our finding is consistent with the former. The relation between attention problems and MD decrease with age possibly because of the increasing mathematics learning experiences and thus more fluent arithmetic facts retrieval abilities among older individuals with MD. That said, most of the MD screening measures we reviewed tap calculations or word problem solving, which may not reflect complex and advanced mathematics skills. Future studies should further investigate whether MD identified with more complex mathematics skills (e.g., algebra) reflects a different age effect on the relation between MD and attention problems.

The Deficit Profile of Executive Functions and Moderators

We found that individuals with MD have comprehensive executive function deficits, which is consistent with findings from previous research (Peng et al., 2012; van der Sluis et al., 2004). We did not find domains of task as a significant moderator, suggesting that executive function deficits among MD are domain general. These findings are interesting when compared with working memory deficit profiles of MD. Specifically, working memory deficits were more severe than the executive function deficits among MD. Also, the working memory deficits of MD showed domain specificity in the numerical domain whereas executive functions deficits of MD did not show such domain specificity. These inconsistent findings on the deficit profiles between working memory and executive functions of MD do not support the view that working memory and executive functions are the same construct (Engle, 2002). These findings, however, are in line with previous research showing that compared with executive functions, working memory seems to be a more influential component to mathematics (Friso-van den Bos, van der Ven, Kroesbergen, & van Luit, 2013; Jacob & Parkinson, 2015; Peng et al., 2016), especially among MD (Friso-van den Bos et al., 2013; Peng et al., 2016), which may be due to that working memory includes not only executive functions but also the memory component as well as the coordination between memory and executive functions (Baddeley, 1986). That said, we did not acquire sufficient data on executive functions (especially on updating and switching), nor did we have enough studies that included both working memory and other executive functions. Future studies are needed to investigate the difference on the deficit profiles between working memory and different types of executive functions among MD, ideally controlling for each other in the analysis (partialling out the common variance).

Regarding the specific components of executive functions, we found deficits in inhibition, updating, and switching were related to different types of MD, which is consistent with previous research (L. S. Fuchs et al., 2005; Passolunghi & Siegel, 2001; Swanson & Beebe-Frankenberger, 2004). This finding is consistent with the executive function framework that emphasizes the high intercorrelations among different executive functions (Miyake et al., 2000), suggesting that different executive functions may work simultaneously and collaboratively in mathematics tasks and thus deficits in executive functions are related to all types of MD. That said, we found that updating deficits were more severe for MD identified with word problem–solving difficulties. This provides further evidence that word problem solving draws heavily on updating as students continuously comprehend, integrate, and update the information read in the word problem, and thus word problem–solving difficulties are more related to updating deficits (Palladino, Cornoldi, De Beni, & Pazzaglia, 2001; Passolunghi & Pazzaglia, 2004). Moreover, based on the developmental feature of executive functions, we hypothesized that younger children with MD would demonstrate comprehensive executive function deficits, while older individuals with MD may show selective executive function deficits depending on types of MD (L. S. Fuchs et al., 2005; L. S. Fuchs et al., 2006; Passolunghi & Siegel, 2001; Swanson & Beebe-Frankenberger, 2004). However, we found that age did not explain the executive function deficit profiles of MD identified by different measures. This finding may suggest the executive function deficits are relatively stable among different types of MD across age.

Limitations

Our findings are based on the combined results of 75 studies conducted with more than 13,000 participants. Despite the scale of our literature search and sample size, we note several limitations when interpreting the findings. First, due to the small number of studies that reported comparisons between the MD and TD groups on visuospatial skills and attention, we could not run moderation analyses on types of MD screening for visuospatial skills and domains of task for attention. We may also be underpowered, as mentioned earlier, to run analyses on some moderators, such as types of visuospatial skills for the visuospatial deficit profiles. Thus, the moderation analyses for the profiles of MD on visuospatial skills and attention should be regarded as exploratory in nature and warrant further investigation.

Another limitation is that we were unable to investigate other interesting moderators due to insufficient data. For example, for attention, we could not examine the moderation effect of types of attention. Specifically, among the attention measures, we coded for alerting (becoming and staying attentive toward a specific stimulus) and orienting (directing attention to a specific stimulus) based on Posner and Petersen’s (1990) attention model, but found only less than five effect sizes that represented each of those attention components. For the same reason, for types of MD screening, we only focused on the most common MD screening measures (i.e., calculation vs. word problem solving vs. comprehensive mathematics), and did not include less common ones, such as nonsymbolic measures (e.g., Approximate Number System). As some research suggests that nonsymbolic measures may be especially related to numerical-processing and visuospatial skills (e.g., Dehaene, 2011; Halberda & Feigenson, 2008), MD children identified with these measures may show more severe deficits in visuospatial skills and cognitive skills measured in numerical materials. Moreover, we had a fair amount of IQ data missing (26% of the total effect sizes) in the original studies, so we were unable to examine whether IQ differences between MD and TD influence the cognitive deficits of MD. Future studies may further examine these potential moderators for the cognitive deficit profiles of MD.

Following the logic of the bottleneck theory, if low-level cognitive skills are foundational to high-level cognitive skills, then deficits in low-level cognitive skills may cause the Matthew effect on the deficits in high-level cognitive skills from a developmental perspective. That is, compared with the deficits of low-level cognitive skills, deficits in high-level cognitive skills among MD may become more severe as a function of age. However, our results did not support this Matthew effect, which may be because the deficits in low-level cognitive skills of MD only partially explained the deficits in high-level cognitive skills. That said, we used concurrent data from different samples to examine the age effect, which may not accurately reflect the accumulative age effects. A longitudinal design on the same sample may better investigate the Matthew effect related to the bottleneck theory.

Last, we did not use the Cohen’s benchmarks (Cohen, 1998) to indicate magnitude of effect size. Hill, Bloom, Black, and Lipsey (2008) argued that for education research, effect sizes should be interpreted with respect to empirical benchmarks that are relevant to specific population and measures. Given that there is a lack of empirical studies to establish meaningful benchmarks in MD profiling research, future studies should examine whether cognitive deficits of MD are causally related to mathematics performance and to what degrees the change of cognitive skills are associated with meaningful changes in mathematics among MD (Hill et al., 2008). This line of research can provide more refined effect size benchmarks for MD profiling research.

Implications