Cognitive Predictors of the Overlap of Reading and Math in Middle School

Abstract

Math and reading skills are known to be related, and predictors of each are well researched. What is less understood is the extent to which these predictors, uniquely and collectively, overlap with one another, are differentially important for different academic skills, and account for the overlap of math and reading. We examined 20 potential predictors from four domains (working memory, processing speed, attention, and language) using latent variables and both timed and untimed achievement skill, in a sample (N=212) of at-risk middle schoolers, half of whom were English learners. The predictors accounted for about half of the overlap among achievement skills, suggesting that other factors (e.g., domain specific skills) might also be relevant for the overlap. We also found some differential prediction (language for reading, working memory for math). The present results extend and refine our understanding of the contribution of these cognitive predictors for reading and math.

Many studies focus on predictors of reading and/or math. Interest in this topic comes from their well-known relation (both high diagnostic comorbidity, and relations across the continuum of skill), from extensive work investigating their cognitive concomitants individually, and from the functional relevance of these achievement outcomes for students. However, there are gaps in this literature that if filled have potential to clarify mixed findings and thus deepen theoretical understanding, as well as inform instructional and intervention activities. The goal of the present study is therefore to evaluate four key cognitive domains (language, executive function, processing speed, and attention) as predictors of both reading and math, in terms of their: (a) shared versus unique contributions; (b) relative importance, and (c) their ability to account for the relationship between reading and math, whether timed or untimed. The context for this study is a sample of middle school students who are academically at risk.

Cognitive Predictors for Reading and Math, Considered Separately

Individual literatures of reading and math identify multiple cognitive predictors. Common components of reading include single word reading, reading fluency (at the sentence or word level), and reading comprehension. Common components of math include written computations, math fact fluency, and math applications/problem solving. The deepest literature is for single word reading and written computations, though all are regularly studied. In this study (and this review) we largely focus on word reading and written computations, and on reading fluency and math fact fluency. For reading, the strongest and most consistent predictors are language based, including phonological awareness and rapid naming (Catts et al., 2015; Melby-Lervåg et al., 2012; Oslund et al., 2017; Scarborough, 1998; Schatschneider et al., 2004), as well as more general language measures such as vocabulary and listening comprehension (Fernandes et al., 2017; Fricke et al., 2016; Gillam et al., 2023; Leppänen et al., 2008). Outside of language, other predictors of reading include executive function – particularly working memory (Cirino et al., 2018; Peng et al., 2018a; Savage et al., 2007; Siegel & Ryan, 1989); processing speed – particularly complex processing speed (Gerst et al., 2021; Shanahan et al., 2006; Willcutt et al., 2008); and a number of types of attention – from visual attention (Cirino, Barnes, Roberts, Miciak, & Gioia, 2022; Liu & Liu, 2020; Plaza & Cohen, 2007) to vigilance (Willcutt et al., 2013) to behavioral attention (Grills-Taquechel et al., 2013; Pham, 2016; Sims & Lonigan, 2013). There is support for these predictors across reading outcomes (e.g., word reading, fluency, comprehension), with an increasing role for general language processes with development, particularly for reading comprehension (Foorman et al., 2018; Gillam et al., 2023; Hoover & Gough, 1990; Ouellette & Beers, 2010). Several of these same factors (e.g., phonological awareness, rapid naming, working memory) are also relevant for English language learners (Swanson, Kudo, & Guzman-Orth, 2016), but for English reading skills, predictors measured in English are more robust than those measured in Spanish (e.g., Macdonald, Francis, Hernandez, Castilla-Earls, & Cirino, 2022).

For math, there is no analogue to the role that language plays for reading – i.e., one that dominates prediction models. Instead, math prediction is broader, with support coming from both “domain specific” factors (those with a direct and specific plausible and theoretical connection to math, such as tasks involving numbers or magnitude, and math skills lower in the math hierarchy) and “domain general” factors (those that are relevant to math, but which are less specific to it; Cirino, 2022, Cirino et al., 2022). Domain general factors as they relate to math can include executive function – particularly working memory (Child et al., 2019; Cragg et al., 2017; Friso-van den Bos et al., 2013; Peng et al., 2016; Swanson & Jerman, 2006); processing speed – particularly of the more complex variety (Andersson & Lyxell, 2007; Berg, 2008; Bull & Johnston, 1997; Child et al., 2019); attention (Cirino et al., 2007; Fuchs et al., 2006; Fuchs et al., 2010) – with most work devoted to behavioral attention (Child et al., 2019; Cirino et al., 2007; Fuchs et al., 2006; Gold et al., 2013); and spatial skill (de Hevia et al., 2008; Mix, 2019). Language skills are also relevant for math (Chu et al., 2016; Cirino et al., 2018; Koponen et al., 2007; Krajewski & Schneider, 2009; Mazzocco & Grimm, 2013). There is evidence to support the differential roles of predictors across different math outcomes. For example, language appears more important for word problems than for procedural computations (Barbu & Beal, 2010; Martiniello, 2008), and processing speed may be more important for math fluency (Fuchs et al., 2008; Petrill et al., 2012) relative to other math skills. The role for spatial skills appears to be moderated by age; that is, the strongest evidence is either early in development for pre-written arithmetic (Frick, 2019; Wei et al., 2012), or else at the high school level or beyond (Xie et al., 2020) for advanced mathematics (e.g., geometry, algebra, calculus). Of the skills described above, domain specific math predictors and spatial skills are not readily implicated for reading. There is some consistency among emergent bilinguals, where both phonological and working memory skills (to an extent in both English and Spanish) have relevance for identifying math difficulties (Swanson, Kong, & Petcu, 2022), and executive factors were not differentially related to growth in English learner versus monolingual children (Swanson, 2020).

The present study focuses on factors relevant for both reading and math, in elementary and middle school (as work in the former area is much more voluminous than in the latter, though still close in time), with attention to both timed and untimed outcomes. Given the above overview, the key candidates are language, attention, working memory, and processing speed. Below, we review the rationale and evidence for each across achievement areas, again focusing on elementary and middle school grades/ages.

Language

Vocabulary, phonological awareness (PA), and rapid automatized naming (RAN) are three language-based skills that have potential relevance for both reading and math. Vocabulary is important as a tool for drawing links between spoken words and growth in phonological awareness (Metsala & Walley, 1998), and as a general indicator of familiarity and knowledge, which are known to facilitate reading skill (Hadley & Dickinson, 2020); vocabulary is also linked to reading comprehension in prominent models (Cromley & Azevedo, 2007). For math, vocabulary is relevant for not only math word problem solving, but also specific terminology used in math; terms used for concepts (equals, more and less than, commutivity, associativity), number (ratio, proportion), and in problem solving (equality, variable) are predictive of general mathematics outcomes in younger and older students (Purpura & Reid, 2016). Vocabulary is also empirically related to both reading and math skills (Geary, 2004; Koponen et al., 2007; LeFevre et al., 2010; Sénéchal & LeFevre, 2002; Yovanoff et al., 2005).

PA has been causally and/or bidirectionally related to the development of reading skill (Castles & Coltheart, 2004; Lerner & Lonigan, 2016), as it serves to aggregate and disaggregate speech sounds (Anthony & Francis, 2005), which is particularly important when paired with the sounds made by individual letters. Perhaps surprisingly to some, PA has been implicated in the development of math fact knowledge to a particularly strong degree (e.g., Cirino, 2011; de Smedt et al., 2010). For example, knowledge of math facts requires the repeated pairing of a stem (“4 + 2”) with a response (“6”) in a well-known developmental pattern that at first relies on fingers, progressing until responses are semantically retrieved, without intervening computation (moreso for addition and multiplication than subtraction or division) (Krajewski & Schneider, 2009; Michalczyk et al., 2013). In fact, some suggest that PA may be as relevant for math as it is for reading (Amland et al., 2021; De Smedt et al., 2010).

RAN, like PA, has been implicated for both reading and math. First, RAN is related to academic fluency, as both are timed (sharing methods variance), including reading fluency (Norton & Wolf, 2012) and math fact fluency (Koponen et al., 2013; 2017). However, beyond its obvious timing role, RAN is hypothesized to be related to both reading and math for other reasons. For example, in reading, RAN facilitates the recognition and familiarity of known associations, removing demands from word recognition (Arnell et al., 2009). RAN also serves as a proxy for retrieval of phonological representations and/or or orthographic patterns, or because of its use of domain general resources including working memory (Bowers and Wolf, 1993; Wagner & Torgesen, 1987; Papadopoulos et al., 2016), and is implicated in the double-deficit hypotheses of reading (Norton & Wolf, 2012). In math, RAN relates to untimed computational skill (Yang et al., 2022), where fluent symbol association is likely to decrease working memory demands.

Attention

Attention can be assessed with objective performance measures or behavioral ratings of attention, and each relates to both reading and math. In reading, the visual attention hypothesis (Bosse et al., 2007; Franceschini et al., 2012; Vidyasagar & Pammer, 2010) has much theoretical support, though the empirical evidence is particularly mixed (Cirino, Barnes et al., 2022; Montani et al., 2014; Tobia & Marzocchi, 2014a, 2014b). Comorbidity models often examine the overlap in the diagnostic categories of Attention Deficit Hyperactivity Disorder (ADHD) and Reading Disability (RD), which considers attention as a behavioral construct (Pennington et al., 1993; Shanahan et al., 2006; Willcutt et al., 2001; 2005). For math or Math Disability (MD), there is less detailed work on specific components of attention. Recent studies suggest that sustained attention is related to math in younger children (Barnes et al., 2020), and inhibition or executive attention is related to math in both younger children (Barnes et al., 2020), and older children even in the presence of domain-specific math factors such as approximate number sense acuity (Wilkey et al., 2020). Similarly, Anobile et al. (2013) found that sustained attention was a unique predictor of mathematics even with domain specific math factors in the same model. Math disability is also associated with deficits in both numerical processing and attention (Ashkenazi & Henik, 2010a, 2010b). Finally, there is robust evidence regarding the relation of math with behavioral attention (Cirino et al., 2007; Gold et al., 2013; Raghubar et al., 2009).

Working Memory (WM)

Beyond language, WM is the most common predictor of both reading and math (Alloway & Alloway, 2010; Child et al., 2019; Friso-van den Bos et al., 2013; McGrath et al., 2011; Peng et al., 2016; 2018; Willcutt et al., 2001). For reading, WM is important for comprehension, where increased capacity allows for more time to integrate words being read with background knowledge and earlier portions of text, consistent with many reading frameworks (Cain et al., 2004; Siegel & Ryan, 1989). However, greater WM can facilitate word reading and fluency as well (Berninger et al., 2006; Christopher et al., 2012; Cirino et al., 2019b; Knoop-van Campen et al., 2018; Peng et al., 2018a), operating at a more basic level (e.g., moving from letters to words, or word to word). WM is also relevant for math, for different reasons. For math fact fluency, WM facilitates the connection of problem stem to answer (Fuchs et al., 2010), whereas for computation (Gathercole et al., 2006; Gathercole & Pickering, 2000; Peng et al., 2018b), WM facilitates the holding of multiple necessary steps (Geary, 2004). To the extent that word/story problems require reading comprehension and the construction of a problem representation that involves coordinating both computational and linguistic knowledge, WM is also relevant (Fuchs et al., 2006; Lee et al., 2009; Peng et al., 2016).

Processing Speed (PS)

Like RAN, PS is relevant for both reading fluency and math fluency, given their timing properties (Caemmerer et al., 2018; Hudson et al., 2008; Lobier et al., 2013). However, the PS-achievement relation is stronger when the emphasis is on processing more than speed per se (when the action taken or information to be processed is complex versus simple; Gerst et al., 2021). The relation is also present for untimed skills. In reading, PS is implicated in associating letter representations with their sounds (Shanahan et al., 2006), and for computations, PS is relevant when problems have multiple steps and/or complex mental operations (Cheng et al., 2021). Processing speed may “join” with WM to facilitate both reading and math; increased familiarity of material, and increased shuttling speed of retrieval may lessen decay effects and allow WM activity to operate more efficiently.

Cognitive Predictors for Reading and Math, Considered Together

The above information is rich, but little of the work addresses the goals of the present study specifically – to examine how multiple cognitive predictors effect both reading and math, and directly address their shared contributions, relative importance, and overlap, in middle school students, while also considering automaticity/timing. This though is in the context of a vast number of studies that include measures of reading and math. Many such studies form mutually exclusive groups (RD, MD, RD+MD, and Neither), and frequently, numerous cognitive predictors are included (e.g., Cirino et al., 2015; Willcutt et al., 2013), but criteria used to form groups influences results by determining group selection/membership. It is increasingly recognized that while diagnostic decisions need to be made at a clinical level, and/or for the provision of services, examining academic variables dimensionally is a better representation of the reality of the relation between reading and math (Branum-Martin, Fletcher & Stuebing, 2013; Peters & Ansari, 2019). Although group studies are informative, a “shared” predictor in this context means that impairment in a given factor occurs in both the RD and MD subgroups, whereas “unique” means that only one of these subgroups is impaired. Moreover, in several cases such analyses examine group performance on cognitive factors separately rather than together (e.g., Andersson, 2010; Cirino et al., 2007; Moll et al., 2016; van der Sluis et al., 2005). Finally, findings are particularly mixed when comparing the comorbid group to MD and RD subgroups. A recent meta-analysis (Viesel-Nordmeyer, Reuber, Kuhn, Moll, Holling, & Dobel, 2023) examined 74 group-based studies with regard to executive function (including inhibition/shifting/updating, short-term memory/working memory, processing speed, and attention), with many fewer studies addressing processing speed or attention. They concluded that there was some evidence that the comorbid group had deficits less than the sum of the individual RD and MD groups. Like other group studies, however, results are difficult to interpret in terms of what accounts for the overlap of math and reading.

Beyond group studies, we are unaware of individual studies that include all of the cognitive domains reviewed above, and very few explicitly address the nature or source of the overlap between reading and math. We did not identify any studies focused on elementary/middle school students that examine the set of four key cognitive predictors identified above to address the overlap in both timed and untimed reading and math outcomes. We also did not identify specific studies that address all of the above in emergent bilingual students, with or without comparison to monolingual peers. However, there is some indirect literature, and a small corpus of studies that is of most direct relevance towards developing hypotheses.

For indirect literature, the genetic meta-analysis of 38 studies by Daucourt et al. (2020) found a phenotypic relation of math and reading of r = .52, with high genetic and shared environmental correlations, but did not examine individual cognitive predictors. The large meta-analysis of Ünal et al. (2023) found a median relation of math and reading of r = .42, with a wide range (.23 to .61). Although this study included many cognitive predictors, they were only evaluated collectively as a second-order latent variable (which still left a significant residual relation of reading and math) and therefore their relative contributions could not be discerned. Finally, math and reading latent variables combined timed and untimed outcomes.

Turning now to the most relevant studies, a series by Koponen and colleagues (Koponen et al., 2013; 2016; 2019; Korpipää et al., 2017) focused on several longitudinal samples in Finland. One of these (Korpipää et al., 2017) identified language factors as prominent in accounting for the overlap of timed/fluency math and reading at Grade 1, and nonverbal reasoning at Grade 7; the other three studies emphasized the contribution more specifically of RAN and oral counting in predicting each of timed math and reading from Grade K through Grade 4, even in the context of other cognitive skills such as working memory. In these models, prediction was stronger for reading fluency than for math fluency, and in each case, there remained a residual relation of reading fluency with math fluency. These models did not consider processing speed or attention, did not address untimed outcomes, and in only one case did the analyses extend to middle school.

Four other studies are most closely relevant. For example, Fuchs et al. (2016) evaluated 747 children in grades 1 to 3 with an array of predictors; there were unique direct effects of language, phonology, and rapid naming for single word reading, which were different from those of calculations (behavioral attention, reasoning, working memory). Early reading was directly related to both later achievement skills, and several domain general predictors (attentive behavior, reasoning, visuospatial memory, and rapid naming) all had indirect effects on each outcome, mediated by addition retrieval strategy. This study included only untimed outcomes, and examined math and reading separately, making it difficult to directly address shared contributions and overlap.

Peterson et al. (2017), with a large community-based sample aged 8 to 16, considered (untimed) reading, math, and attention as outcomes, with cognitive predictors of verbal comprehension, verbal WM, PS, inhibition, PA, and RAN. The authors’ hypothesized model included paths for only two cognitive domains (verbal comprehension and PS) to predict both reading (i.e., recognition and oral fluency) and math (i.e., computations), and both were uniquely predictive. Strikingly, the latent correlation of reading and math was r = 0.77, but in the final model including all cognitive predictors, the relation between reading and math was not significant (residual r = −0.06), and further model testing found verbal comprehension to be a larger contributor to the overlap than processing speed. Results did not vary as a function of age.

Cirino et al. (2018) evaluated 193 students followed from Kindergarten to Grade 1 with a multivariate and partial correlation approach; predictors included counting, symbolic knowledge, PA, RAN, vocabulary, WM, PS, nonverbal reasoning, and behavioral inattention. The correlation between (observed, untimed) reading and math (word reading and computations) was r = .67. All predictors but PS were related to both reading and math at the zero-order level, and three showed differential prediction (counting knowledge more for math, PA and RAN more for reading). Partialling all cognitive domains reduced the math and reading relation (to r = 0.33), but it was still significant. A similar effect (residual r = 0.34) was obtained partialling only variables that showed differential prediction. Similar patterns were evident with other reading and math pairings (e.g., reading and math fluency; reading comprehension and math problem solving).

Finally, Child et al. (2019) used regression, canonical, and partial correlational approaches with a sample of 233 2^nd graders. In this study the (observed) correlation of (untimed) reading and math (word reading and computations) was r = 0.41, and PA, WM, PS, and numerosity, were tested as predictors. Both PA and WM (but not the others) were uniquely predictive of both outcomes, and each individually reduced but did not eliminate the correlation between reading and math (residual r ~ 0.27); however, their combined contribution was not examined. For timed outcomes, the initial correlation was weaker (r = .29), and so was more easily reduced to non-significance with the partialling of cognitive factors.

Summary and Present Study

As should now be clear, surprisingly few studies address the relation of multiple cognitive predictors for reading and math, their shared versus unique contributions to outcomes, how they directly account for the overlap among reading and math, and their relative importance. Beyond correlations, shared versus unique prediction requires a consideration of whether a given predictor relates to an outcome considering other predictors, and whether this is similar across outcomes (e.g., the extent to which WM is uniquely predictive of math, versus uniquely predictive of reading, considering all other predictors). Overlap requires a consideration of which predictors account for the relation of reading and math, which necessitates an evaluation of whether the relation among outcomes decreases in the presence of one or more predictors. Finally, relative importance requires a comparison of the significance and strength of the differential relation of a cognitive predictor across more than one outcome (e.g., WM predicting math versus predicting reading). The few studies that do so are diverse in their approaches and findings, so there is a strong need for further study, as clarification has implications for learning. For example, if all predictors account for robust amounts of unique overlap between reading and math, the implication is that addressing them (directly or indirectly) may lead to academic improvement, which would be most likely to impact students who struggle in both academic areas. Similarly, the array of predictors may not be all that shared, or that what is shared is dominated by a particular predictor. Each of these possibilities has corresponding implications. The present study builds on this small literature generally, but also in two specific ways, by: (a) including a number of plausible cognitive candidates, and in particular by considering a range of attentional processes; and (b) evaluating both timed and untimed achievement outcomes. Particularly with regard to the latter, the extant literature is not entirely clear on whether differential cognitive prediction might occur, or whether overlap between timed versus untimed outcomes would differ, though this is mainly because so few studies have compared them directly.

The context for this study is middle school. This time frame is relevant as it occurs during a time when cognitive functions are undergoing rapid change and differentiation. For purposes of a larger project, the sample also includes many English learners, many of whom are struggling in academic areas. The hypothesized models are not dependent on these particular age or sociodemographic characteristics; that is, we do not have reason to expect that the relations between reading and math, or their relations with cognitive processes, would be fundamentally different in students with experience in multiple languages. However, the sample does offer a unique and important context for the issues addressed in this work.

We make hypotheses based on literature review above, but especially with regard to those studies that examine the impact of numerous cognitive predictors for reading and math, considered together and in continuous fashion, with students in elementary and middle school. Some hypotheses are tentative, given that this specific literature is small, mixed, and rarely considers both timed and untimed measures. Regardless of results, the present study can begin to solidify knowledge in all of these areas, Our general hypothesis is that all four general domains are relevant to the overlap of reading and math. We expect to fit a measurement model with good fit that delineates the four cognitive domains (Hypothesis 1). We also expect reading and math relationships to be robust for timed and untimed outcomes (Hypothesis 2). We further expect (a) all four domains to have significant relations to both reading and math, whether timed or untimed, both at the individual (correlational) and collective (considering other predictors) levels (Hypothesis 3a). We also expect that (b) when considered together, all four domains will reduce the residual relationship between reading and math to non-significance (Hypothesis 3b), given the comprehensiveness of key factors covered; we expect language-related factors to have the largest contribution to this overlap.

Finally, we expect some differential prediction when considering outcomes simultaneously (Hypothesis 4). We expect that: (a) language should be more relevant for reading than for math (both timed and untimed); (b) given the larger literature on visual attention and reading, relative to math, attention may be more relevant for reading than for math (timed and untimed); (c) given that comprehension is not assessed, WM should be more relevant for math than for reading (timed and untimed); and (d) PS should be more relevant for timed measures than for untimed measures (both reading and math).

Methods

Participants

The present study included 212 Grade 6 and 7 students from two sites and three districts in the Southern US. They were part of a larger project with multiple components, some of which overlapped with the sample and measures of the present study. Table 1 displays demographics. As shown, participants were all Hispanic, and most were economically disadvantaged. Half of the sample were currently or formerly classified as limited English proficient by their school, which means that the language spoken at home and by the student is a language other than English (but may also include English), and the student falls below proficiency on a state sanctioned test involving components of listening, speaking, reading, and writing. The stated language of instruction received by the sample at these schools was English, which was also the language in which all assessments were administered (and also the language of the state-administered end-of-year benchmark tests given to these students). Approximately half the sample were identified as struggling readers and were randomized to a reading treatment or control condition (Capin et al., 2023). However, the measures used here were collected towards the beginning of the school year rather than the end, and so there was little risk that the treatment (for those students who obtained it) would conflate relationships among variables. Moreover, as discussed later (see also Supplement), eleven covariates (primarily sociodemographic characteristics, including language status) were considered, and did not lead to substantively different results.

Table 1.

Demographic Characteristics (N = 212)

		N (%)
Gender	Female	102 (48.11)
Site	Site 1	97 (45.75)
	Site 2	115 (54.25)
Ethnicity	Hispanic	212 (100.00)
Special Education	No	193 (91.04)
Economic Disadvantage	Yes	149 (70.28)
	No	10 (4.72)
Limited English Proficiency	Yes	106 (50.00)
Grade	6	126 (59.43)
	7	86 (40.57)
Treatment Condition	BAU	46 (21.70)
	INT	64 (30.19)
	Not Randomized	102 (48.11)
	Mean	Standard Deviation
WASI-2 Matrix Reasoning	45.53	8.01
Age (years)	12.37	0.73

Note. Within each bolded section, when only one line is presented, the remainder are the other designation (e.g., if “no” is 91.04%, then “yes” is 8.96%). For economic disadvantage, there were missing data for 53 students (25%). WASI-2: Wechsler Abbreviated Scales of Intelligence, 2^nd Edition. Treatment refers to randomized condition for struggling readers according to the larger parent study (BAU = Business as Usual control; INT = Intervention).

Measures

We administered measures of language, attention, working memory, and processing speed, described below under their hypothesized domain. For purposes of the larger project, additional measures were administered, but are not analyzed here as the hypotheses did not invoke those measures or constructs. Measures were given early in the school year. Table 2 contains descriptive data on the 20 indicator and 4 outcome measures described below (means and standard deviations, as well as their reliabilities).

Table 2.

Descriptive Statistics for Measures

Variable	N	Raw Mean (SD)	SS Mean (SD)	Reliability	Skewness	Kurtosis
LANGUAGE
Elision	212	23.47 (6.52)	7.45 (3.04)	0.918	−0.484	−0.900
Blending Words	212	23.12 (4.80)	9.03 (3.57)	0.856	−0.484	−0.503
RAN Letters	212	18.77 (5.35)	7.12 (2.74)	0.732*	1.533	2.858
RAN Digits	212	16.39 (3.86)	7.73 (2.53)	0.732*	1.463	3.540
Vocabulary	212	20.33 (3.40)	79.31 (9.58)	0.781	−0.058	0.208
ATTENTION
VAS Letters	206	42.27 (11.89)		0.856	0.489	0.410
VAS Numbers	206	49.65 (12.62)		0.858	0.046	0.980
Search Letters	211	15.58 (3.12)		0.785*	0.199	0.171
Search Numbers	211	13.53 (2.61)		0.812*	−0.190	0.190
CPT Block 1	208	3.05 (1.06)		0.900	−0.747	0.196
CPT Block 2	209	3.15 (1.12)		0.849	−0.746	−0.247
CPT Block 3	210	2.98 (1.12)		0.844	−0.714	−0.024
Inattention	208	0.06 (12.01)		0.982	−0.252	0.113
Hyperactivity	208	−2.54 (10.50)		0.970	−0.437	0.840
WORKING MEMORY
Digits Forward	212	7.56 (1.75)	7.63 (2.44)	0.694	0.222	0.802
Digits Backward	212	8.00 (1.95)		0.664	0.031	−0.028
Digits Sequence	212	7.10 (2.10)		0.719	−0.017	−0.382
Picture Span	212	26.70 (6.10)	8.37 (2.65)	0.784	0.080	−0.126
PROCESSING SPEED
Coding	211	46.01 (9.84)	8.31 (2.36)	0.830**	0.193	−0.410
Symbol Search	210	25.07 (6.01)	8.70 (2.65)	0.810**	−0.429	0.867
READING and MATH
Letter and Word Recognition	211	523.19 (28.61)	91.23 (16.42)	0.950	−0.675	1.416
Word Reading Fluency	210	509.20 (17.69)	88.07 (13.45)	0.870**	−0.754	1.560
Computations	211	527.41 (16.25)	92.28 (12.34)	0.929	−0.605	0.201
Math Fluency	211	516.87 (21.94)	90.50 (11.63)	0.910**	0.554	0.549

Note: RAN = Rapid Automatized Naming subtests; VAS = Visual Attention Span; CPT = Continuous Performance Task. SS = Scaled scores (M=10; SD=3) for Elision, Blending Words, and RAN subtests of the CTOPP-2, and Digit Span, Picture Span, Coding, and Symbol Search subtests of the WISC-V; Standard Scores for Vocabulary (WJ-III) and Reading and Math (KTEA-3) measures. Reliability is determined by Cronbach’s alpha except:

^{* =}alternate forms reliability

^{** =}test-retest coefficients taken from test manual.

Language.

For language, there were three measures (PA, RAN, and Vocabulary) with five indicators. PA and RAN were measured with four subtests of the Comprehensive Test of Phonological Processing-Second Edition (CTOPP-2; Wagner et al., 2013). For Elision, examinees hear a word, and are asked to remove a given sound. Blending Words involves stringing together heard phonemes into words. RAN was assessed with the Rapid Letter Naming and Rapid Digit Naming subtests. All CTOPP-2 reliabilities are strong (Dickens et al., 2015). Finally, vocabulary was measured using the Picture Vocabulary subtest, requiring examinees to point to an image of a named objects, from the Woodcock Johnson-Third Edition (WJ-III) which is widely used and has strong psychometric properties (Blackwell, 2001; Schrank et al., 2001; Woodcock et al., 2001).

Attention.

There were four measures (VAS, Visual Search, CPT, and Behavioral Attention), with nine total indicators. The VAS task was adapted from Lallier, Donnadieu, and Valdois (2013), and was also used in a non-overlapping sample (Cirino, Barnes et al., 2022). For both conditions (letter and number stimuli), participants see 5 stimuli with a very brief (200 ms) presentation time, over 20 trials, with the score based on recall of an item in its proper order after brief (range 0 to 100). Visual search is a speeded scanning task; key variables were the number of correctly identified targets in the 30-second time limit for each condition (Letters or Numbers); this task has also been used previously in a different sample (Cirino, Barnes, et al., 2022).

A third measure was a continuous performance task (CPT), often used as an index of sustained attention that is performance-based (De la Torre et al., 2015; Macdonald et al., 2021; Mirsky et al., 1999). Participants see a number of stimuli after variable interstimulus intervals (500 ms, 1500 ms, 3000 ms), and are instructed to press the spacebar after (and only after) a particular target stimuli (a Hiragana symbol), from amongst distractors of letters, numbers, and other Hiragana symbols. Targets appear on 20% of the 312 trials, which are arranged in 3 blocks. Within each block, a discriminability index (d’) is computed, representing a balance of hits and false alarms; the d’ values from the three blocks were the three indicator variables. Further details on this measure are found in Author (Cirino, Barnes et al., 2022; Cirino et al., 2023).

Finally, behavioral attention was indexed with the SWAN (Strengths and Weaknesses of ADHD-symptoms and Normal-behavior; Swanson et al., 2012), which is a rating scale completed by teachers, with scales of inattention and hyperactivity/impulsivity. This measure is extensively used and has excellent psychometric properties (Child et al., 2019; Friso-van den Bos & van de Weijer-Bergsma, 2020; Geary et al., 2021; Larsen et al., 2022). Each item’s ranking is on a 7-point Likert scale that ranges from significantly below average to significantly above average, ensuring a normal distribution of scores across the population (Polderman et al., 2007; Swanson et al., 2012), with lower scores indicating greater attention problems.

Working Memory.

This was assessed with the subtests of the Working Memory Index from the Wechsler Intelligence Scales for Children-5 (WISC-5), which have good psychometric properties (Wechsler, 2014). Digit Span involves three tasks (i.e., digits forward, backward, and sequence), assesses verbal short-term working memory, and is widely used as an indicator of WM (Coy et al., 2011; Shelton et al., 2009). Picture Span involves immediate visual memory in a proactive interference paradigm, with the size of the test arrays increasing.

Processing Speed.

This was assessed with the subtests of the Processing Speed Index of the (WISC-5), also with good psychometric properties (Wechsler, 2014). Coding involves rapidly copying novel symbols associated with numbers based on a key, while Symbol Search requires the examinee to decide whether a target is present among a group of symbols. Total correct responses within a time limit are the dependent variables for each. Both have excellent psychometric properties (Wechsler, 2014).

Achievement.

Four subtests of the Kaufman Test of Educational Achievement, Third Edition (KTEA-3) were utilized. Letter and Word Recognition is an untimed measure of single word reading, with the examinee pronouncing words that gradually increase in difficulty; students in older grades enter the list at a higher level (e.g., skipping letters and short, frequent words). Word Recognition Fluency has the examinee reads isolated words out loud rapidly over two fifteen second trials; words here also increase in difficulty, but all students read the list from the beginning, with an emphasis on speed. Math Computation is an untimed measure, requiring a variety of basic arithmetic (intermixing the four operations, and complexity; e.g., single-digit, multidigit) and other calculations (e.g., fractions, algebra). Math Fluency is a timed measure of math facts (with most initial problems of single-digit addition and subtraction – with minuends always larger than subtrahends, and later, multiplication), over 1 minute.

The KTEA-3 is a very widely used measure of academic achievement in the US, and subtests have well-known and excellent psychometric properties, and are normed on a large-scale Census-matched sample (Kaufman & Kaufman, 2014). Growth scale values (similar to Rasch-based values on a true interval scale) were used as the measure for each of the four achievement outcomes. Measures are arranged in in order of known difficulty, with established basal and ceiling rules, established through extensive normative studies.

Procedure

All procedures were reviewed and approved by the Institutional Review Board at our institution. Appropriate permissions were also obtained from the school district, schools, parents (permission), and children (assent). Students were seen at their school at times appropriate to their schedules. Assessments took about two hours (in one or two sessions), and students were provided gift cards as a token of appreciation; all assessments were administered in English. Approximately half the sample received the battery in reverse to address order effects. All examinees were trained with intense supervisory experience and passed formal “check out” procedures. Onsite supervision was present, and numerous quality checks were implemented both onsite, as well as back at the office, prior to data sheets being optically scanned, where further checks were employed. The study was originally designed to be a two-cohort cumulative study (and thus twice the N), but the second cohort was not ascertained due to Covid-19.

Analysis

Although the sample size was smaller than anticipated, some of these latent variables were used in a prior study on the structure of attention (Cirino et al., 2023); that study included multiple variables not used here, and this study included language and achievement measures not evaluated there. After examining distributional assumptions, and quality control metrics for the variables, primary hypotheses were addressed in a structural modeling framework, using MPLUS (Muthén & Muthén, 2017). Overall model fit indices included AIC, BIC, CFI, RMSEA, SRMR (Gunzler & Morris, 2015; Kline, 2015; Wang & Wang, 2019).

First, we evaluated a measurement model (Hypothesis 1) with nine latent variables, including three of language (PA, RAN, vocabulary), one each of WM and PS, and four of attention. We then added the four manifest outcome variables to this model, to examine their interrelations (Hypothesis 2). Within this model, we examined the relation of each of the four domains with the achievement variables (separately via correlations and collectively via regression, within the structural model; Hypothesis 3a). This same model was then used to test the overall reduction in reading and math correlations (Hypothesis 3b). Finally, to test differential relations of domains to outcomes (Hypothesis 4), we added to our base model a series of constraints in which paths are evaluated (e.g., testing whether the path of WM predicting untimed reading is equal to the path of WM predicting untimed math) using the model constraint command in MPLUS; collective differences (e.g., WM for untimed measures, WM for timed measures) were tested with the model test command, yielding a Wald test (based on the Chi-square distribution). The size of beta values determined the direction of effect, per hypotheses (e.g., if effects of WM on math versus reading were different, we expect the beta value for math to be higher, as the hypothesis was that WM should have a stronger effect on math than reading).

Finally, we evaluated our final model with relevant covariates (age, gender, nonverbal reasoning, special education status, limited English status, school district, and treatment condition), to determine whether they impacted substantive results. Only struggling readers were randomized (to either treatment or control), and therefore randomization can be taken as a proxy for reading level. We first derived the most parsimonious set of relevant covariates (given their relation to one another, and the strain on model fitting due to the small ratio of estimated parameters versus observations), and then considered their effect on substantive conclusions.

Results

Hypotheses 1 and 2

We expected a good fit for a measurement model that delineates the four domains (language, attention, WM, PS) with nine latent variables and 20 indicators. Vocabulary was artificially set to be a perfect indicator for parsimony (all other latent variables were defined by at least two variables). The only modification made to the model was for behavioral attention, where a small negative residual was observed for one of its two indicators; fixing this residual to zero did not appreciably change any local or global fit indices. Overall, this model showed a strong fit to the data, χ²(136) = 180.55, p = .006; CFI = .973; RMSEA = .039; SRMR = .049. Hypothesis 2 added timed and untimed measures of reading and math, and maintained a strong fit to the data, χ²(180) = 233.29, p = .006; CFI = .975; RMSEA = .037; SRMR = .046. Table 3 contains all manifest variable correlations, and Table 4 contains latent correlations. Relations of the reading and math variables were significant; specifically, from Table 4, untimed reading and math correlated r = .37, and reading and math fluency also correlated r = .37.

Table 3.

Indicator Measure Correlations

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20	21	22	23
1. Elision	--
2. Blending Words	.41**	--
3. RAN Letters	-.30**	-.08	--
4. RAN Digits	-.30**	-.11	.73**	--
5. Vocabulary	.31**	.27**	-.29**	-.26**	--
6. VAS Letters	.29**	.08	-.29**	-.34**	.18*	--
7. VAS Numbers	.20*	<.01	-.22*	-.27**	.07	.66**	--
8. Search Letters	.21*	.11	-.28**	-.27**	.14*	.18*	.16*	--
9. Search Numbers	.19*	.20*	-.24**	-.25**	.19*	.12	.11	.71**	--
10. CPT Block 1	.10	.22*	<.01	.01	.08	.21*	.18*	.10	.13	--
11. CPT Block 2	.13	.14*	-.02	-.02	.14*	.23*	.29**	.18*	.15*	.73**	--
12. CPT Block 3	.10	.12	-.02	-.03	.14*	.27**	.26**	.16*	.17*	.74**	.82**	--
13. Inattention	-.20*	.02	.14*	.16*	-.11	-.21*	-.22*	-.18*	-.10	-.11	-.10	-.15*	--
14. Hyperactivity	-.14*	.05	.06	.01	-.08	-.16*	-.14*	-.09	-.01	-.06	-.05	-.10	.85**	--
15. Digits Forward	.31**	.30**	-.21*	-.23**	.23**	.14*	.03	.17*	.17*	.08	.03	-.02	.02	<.01	--
16. Digits Backward	.36**	.16*	-.29**	-.28**	.08	.26**	.23**	.13	.18*	.18*	.20*	.13	-.17*	-.16*	.28**	--
17. Digits Sequence	.35**	.20*	-.29**	-.35**	.25**	.22*	.18*	.08	.08	.07	.10	.05	-.12	-.10	.34**	.48**	--
18. Picture Span	.26**	.20*	-.25**	-.24**	.17*	.12	.15*	.04	.01	.29**	.24**	.29**	-.20*	-.17*	.19*	.30**	.33**	--
19. Coding	-.05	.02	-.19*	-.28**	-.04	.14*	.21*	.43**	.33**	.18*	.21*	.18*	-.23**	-.21*	.04	.17*	.13	.12	--
20. Symbol Search	.02	-.03	-.13	-.22*	.14*	.08	.06	.41**	.34**	-.01	.12	.10	-.13	-.10	.03	.05	.08	-.01	.48**	--
21. Letter Word Recognition	.56**	.19*	-.42**	-.44**	.35**	.29**	.26**	.24**	.21*	.02	.14*	.16*	-.27**	-.22*	.23**	.22*	.26**	.28**	<.01	.16*	--
22. Word Reading Fluency	.51**	.18*	-.42**	-.49**	.40**	.41**	.31**	.31**	.31**	.03	.15*	.15*	-.30**	-.21*	.24**	.24**	.30**	.23**	.10	.20*	.75**	--
23. Math Computation	.31**	.08	-.23**	-.23**	.21*	.19*	.26**	.08	.13	.16*	.16*	.10	-.43**	-.36**	.23**	.41**	.43**	.29**	.26**	.17*	.36**	.37**	--
24. Math Fluency	.26**	.08	-.23**	-.30**	.17*	.19*	.24**	.21*	.26**	.01	.02	.03	-.19*	-.13*	.17*	.27**	.34**	.12	.25**	.23**	.29**	.34**	.46**

Note.

^*p < .05

^**p < .001

RAN = Rapid Automatized Naming subtests; VAS = Visual Attention Span; CPT = Continuous Performance Task. RAN, Elision, and Blending Words are from the CTOPP-2; Vocabulary from WJ-III; Digits measures, Picture Span, Coding, and Symbol Search are from the WISC-V; reading and math measures are from the KTEA-3.

Table 4.

Factor Correlations for Latent Variables

	1	2	3	4	5	6	7	8	9	10	11	12
1. PA	--
2. RAN	-.35**	--
3. Vocabulary	.34**	-.31**	--
4. VAS	.35**	-.45**	.22*	--
5. Visual Search	.25	-.35**	.17*	.20*	--
6. CPT	.12	-.02	.15*	.34**	.20*	--
7. Behavioral Attention	-.20*	.16*	-.11	-.28**	-.19*	-.12	--
8. WM	.57**	-.52**	.31**	.41**	.20**	.22*	-.20*	--
9. PS	-.04	-.35**	-.01	.24*	.59**	.23*	-.27**	.23*	--
10. Letter Word Recognition	.58**	-.49**	.36**	.37**	.27**	.14*	-.26**	.41**	.06	--
11. Word Reading Fluency	.55**	-.58**	.43**	.48**	.37**	.18*	-.33**	.46**	.19*	.77**	--
12. Math Computations	.32**	-.26**	.21*	.30**	.11	.15*	-.44**	.61**	.31**	.37**	.41**	--
13. Math Fluency	.27**	-.32**	.17*	.29**	.26**	.04	-.20*	.42**	.31**	.30**	.37**	.46**

Note.

^*p < .05

^**p < .001

PA = Phonological Awareness; RAN = Rapid Automatized Naming subtests; VAS = Visual Attention Span; CPT = Continuous Performance Task; WM = Working Memory; PS = Processing Speed. PA and RAN are derived from the CTOPP-2; Vocabulary from WJ-III; WM and PS are derived from the WISC-V; reading and math measures are from the KTEA-3.

Hypothesis 3

There were two parts. The first was that each of the four domains (language, attention, WM, PS) would relate to each academic outcome. From Table 4, at the zero-order level, most latent variables correlated significantly with all four outcomes (r > .14). For untimed (single word) reading, the range was |r| = .06 to .58; for untimed math (computations) |r| range = .11 to .61; for timed (single word fluency) reading |r| range = .18 to .58; and for timed math (fact fluency), |r| range = .04 to .42; sustained attention had the weakest relations across outcomes, |r| range = .04 to .18.

Next, all predictors were examined together to examine unique and shared relations; factor loadings and significant beta weights appear in Figure 1. For untimed reading (overall R² = 47.8%), two language latent variables (PA, p < .001; RAN, p < .001) and one of four attention latent variables (behavioral attention, p = .021) were uniquely predictive, but neither WM nor PS was (both p > .05). For untimed math (overall R² = 51.8%), language was unrelated, but behavioral attention, p < .001, and both WM, p < .001, and PS, p = .034, were. For timed reading (overall R² = 56.8%), all three language variables (PA, p = .007; RAN, p < .001; Vocabulary, p = .006), and behavioral attention, p = .003, were predictive, but neither WM nor PS was (both p > .05). Finally, for timed math (overall R² = 26.1%), only WM (p < .012) was uniquely predictive. Although predictor latent variables were not multicollinear, the difference in significance between zero-order and regression-based results may be due to shared variance of predictors (median r = .23; range |r| = .00 to .59).

Figure 1.

Measurement Model.

Note: Non-significant paths are not included. Paths significant at p < 0.05 Numbers pointing toward academic measures are fully standardized path estimates (of latent variables onto manifest outcomes). Numbers are factor loadings of manifest variables onto latent variables.

The second part of hypothesis 3 was whether predictors account for the math and reading relationship, and this was only partially supported. Considering all predictors, the residual untimed reading and math correlation was now r = .25, and for reading and math fluency was r = .15. While both are substantially reduced from their zero-order relations (by 42% and 63% respectively), both remained significant (p = .002 and p = .037, respectively).

Hypothesis 4

Hypothesis 4 concerned the differential relation of the four predictor domains to achievement. The final unconstrained model was compared to one that constrained the path from each (e.g., language) to be the same for reading and for math, or for type of outcome (timed versus untimed). Wald tests and model constraints are presented in Table 5.

Table 5.

Model Constraint Testing

Factor	Outcome	Value	df	p
LANGUAGE	UNTIMED	31.69	3	.005
LANGUAGE	TIMED	11.93	3	.008
LANGUAGE	ALL	33.99	6	<.001
ATTENTION	UNTIMED	7.10	4	.131
ATTENTION	TIMED	6.56	4	.161
ATTENTION	ALL	11.99	8	.151
WM	ALL	9.96	2	.007
PS	ALL	1.30	2	.521
Factor	Outcome	Estimate	S.E.	t = Estimate / S.E.	p
PA	UNTIMED	10.97	3.88	2.830	.005
RAN	UNTIMED	-12.38	2.80	-4.427	.000
Vocabulary	UNTIMED	0.41	0.63	0.652	.515
VAS	UNTIMED	0.44	2.42	0.180	.857
Visual Search	UNTIMED	5.13	3.07	1.671	.095
CPT	UNTIMED	3.33	2.09	1.593	.111
Behavioral Attention	UNTIMED	1.09	1.85	0.592	.554
WM	UNTIMED	-11.08	3.80	-2.919	.004
PA	TIMED	3.69	2.74	1.346	.178
RAN	TIMED	-6.85	2.55	-2.689	.007
Vocabulary	TIMED	0.58	0.57	1.020	.308
VAS	TIMED	-0.14	2.28	-0.062	.951
Visual Search	TIMED	1.58	2.71	0.584	.559
CPT	TIMED	4.07	1.94	2.094	.036
Behavioral Attention	TIMED	-1.98	1.70	-1.167	.243
WM	TIMED	-7.17	3.06	-2.341	.019
PS	READING	-2.74	2.46	-1.111	.266
PS	MATH	-0.77	2.66	-0.288	.773

Note: UNTIMED = untimed achievement measures (KTEA-3 Letter Word Recognition and Math Computations); TIMED = timed achievement measures (KTEA-3 Word Reading Fluency and Math Fluency); ALL = both timed and untimed achievement measures; READ = reading achievement measures (KTEA-3 Letter Word Recognition and Word Reading Fluency); MATH = math achievement measures (KTEA-3 Math Computations and Math Fluency). Top half of Table: Wald tests constraining all indicators of a specific cognitive domain to be the same across outcomes; for example, the first line (LANGUAGE, UNTIMED) tests whether the four language measures collectively can be constrained to be equal across the two untimed achievement measures (they cannot without negatively effecting model fit). Bottom half of Table: individual model constraints evaluating whether pairs of estimates (one across two outcomes) are different; for example, the first line (PA, UNTIMED) tests whether the PA factor can be constrained to have equal loadings across the two untimed achievement measures (they cannot without negatively effecting model fit).

We first evaluated language. Results demonstrated that the value across the constrained paths where the three language parameters were fixed to be the same for untimed reading and math was significant (p = .005; line one of the top of Table 5), with more specific model constraints (lines 1–3 of the bottom of Table 5) indicating that PA (p = .005) and RAN (p < .001) were stronger for untimed reading than untimed math; vocabulary was not different (p = .515). Similarly, for timed outcomes, language was significant (p = .008; line 2 of the top of Table 5) overall, but only RAN (p = .007; line 5 of the bottom of Table 5) was stronger for timed reading than timed math; neither PA (p = .178) nor vocabulary (p = .308) were significant.

When attention was examined, the collective paths from attention to reading and math outcomes were similar, whether untimed (p = .131) or timed (p = .161). That is, there was no difference in fit between these models when all four attention variables were constrained to have similar paths to reading and to math. The only individual model constraint comparison that was significant was for sustained attention to have a greater impact for timed math than timed reading (p = .036); it should be noted that in this context, sustained attention did not have a significant unique effect on either timed math (p = .075) or timed reading (p = .394).

For WM, the effects on reading and math were different, whether untimed (p = .004) or timed (p = .019) outcomes were considered. With only a single (latent) WM, the model constraint results were the same; in both cases, results showed that WM was more important for math than reading, whether untimed or timed.

Finally, for PS, the comparison hypotheses were not about reading versus math, but rather untimed versus timed content. Like WM, there is only a single (latent) PS predictor, so Wald tests and model constraints were identical. Results showed that the effect of PS on untimed versus timed outcomes were similar, within reading (p = .266) and within math (p = .773).

Supplements

Sensitivity analyses to evaluate the robustness and extension of the results (including covariates) were generally supportive of the above results. Details can be found in the Supplement 1. MPlus Syntax for primary analyses appear in Supplement 2.

Discussion

The goal of the present study was to evaluate the unique and shared cognitive predictors of achievement and its overlap. Results were partially supportive of hypotheses. A measurement model of latent variables for language, attention, working memory, and processing speed showed good fit to the data. Math and reading were related to one another, to a moderate degree. All cognitive domains were at least in part related to both academic outcomes when considered separately. With all predictors considered together, for untimed achievement, aspects of language (PA and RAN) were uniquely related to reading (but not math), WM and PS were uniquely related to math (but not reading), and behavioral attention was shared across both reading and math. For timed achievement, language was unique to reading (but not math), and WM (but not PS) was unique to math (but not reading); there were no shared predictors. When differential effects were tested directly, language was more important for reading than math, and WM was more important for math than reading; these results held whether achievement was timed or untimed. Attention was not differentially predictive of math versus reading, and PS was not differentially predictive of timed versus untimed outcomes. Predictor domains only partially accounted for the relation between reading and math (timed or untimed).

The novelty of the present study is in the comprehensiveness of predictor domains considered, its consideration of both untimed and timed achievement, and its direct testing of the general, relative, and shared contributions of predictors, and the extent to which they account for achievement overlap, simultaneously in a single model. To reach similar conclusions from prior work would require pulling from different papers and populations and making high-level inferences rather than testing hypotheses based on direct testing as in this study. The fact that the empirical results were not fully supportive of hypotheses spurs questions for future work.

The present study is also highly relevant to individual differences research. All of the variables included (the cognitive, the achievement, and the sociodemographic) are individual differences across which children vary. The present work addresses the overlap among two highly important academic individual differences (in reading and math) and does so in different ways (via timed and untimed measures), while using an array of key cognitive individual differences. The sensitivity analyses explore additional individual differences variables (sociodemographic ones) – though their inclusion did not substantively change the conclusions of the primary analyses/hypotheses. Nonetheless, from among these variables, language status appears particularly relevant – it would be interesting to contrast the present results in samples of monolingual and emergent bilingual populations to understand whether and how a variety of cognitive factors are unique versus shared and/or account for the overlap of reading and math.

Reading and Math Relation

Reading and math did not correlate as strongly we had anticipated (r = .37). However, closer inspection of prior work though shows a wide range of relations of (at least for untimed) reading and math (Child et al., 2019, r = .41; Cirino et al., 2018, r = .67; Fuchs et al., 2016, r = .54; Peterson et al., 2017, r = .77) with similar or larger sized samples; Korpipää et al. (2017), in a large sample but using only timed outcomes, found math and reading correlations of r = .44 in Grade 1, and r = .36 in Grade 7. Therefore, the size of the relations in the present study (.37 for both untimed and timed reading and math) are perhaps more in line with prior work than they may at first appear. For example, the recent meta-analysis of Ünal et al. (2023) found that the median correlation between math and reading measures was .42 (range .23 to .61).

Relation of Predictors to Achievement

As hypothesized, all four predictor domains related to all achievement outcomes, and reflected differences between manifest and latent correlations. For reading, manifest language variables correlated median r = .40, and manifest visual attention span variables correlated median r = .30, whereas their latent counterparts correlated .54 and .41, respectively. These ranges are consistent with prior work (Cirino, Barnes et al., 2022; Peng et al., 2022; van den Boer & de Jong, 2018), and corroborate the relation of phonology with math (median r = .23 for manifest variables, and r = .30 for latent variables) (Yang et al., 2022). Manifest (median |r| = .25) and latent (range |r| = .26 to .43) variable correlations of behavioral attention with reading and math appeared somewhat lower than in prior work (Fuchs et al., 2016; Jiang & Farquharson, 2018; Miller et al., 2014), though consistent with a recent meta-analysis (Gioia, 2023). Sustained attention, however, did not relate well to achievement, which was unexpected. It is possible that the students’ overall good performance on the CPT (the Table 2 discriminability scores indicate that students in the sample made relative few errors of omission or commission over the 306 trials and 12 minutes of this task) may have limited their relation to other measures, even though there were not obvious distributional anomalies on this task.

Manifest WM variables correlated median r = .23 with reading and median r = .28 with math, and their latent counterparts correlated .43 and .51, which are in line with most meta-analytic findings (Peng et al., 2016; 2018b; 2022; Spiegel et al., 2022). PS results (even of the complex variety), even as a latent variable, were stronger for math (r = .30) than for reading (r = .07 and .19). These relations with reading were smaller than expected given some samples (Gerst et al., 2021; Peterson et al., 2017). However, the general pattern in meta-analytic findings is more balanced; for example, Zaboski, Kranzler, and Gage (2018) found that speed related r = .20 with basic reading, and r = .23 with basic math.

Across all measures, the sample as a whole (see Table 1) had normative performance scores that were near the 15^th-25^th %iles on most measures, and there was some slight restriction of range (e.g., SD of 10–12 versus the expectation of 15) for some (not all) achievement and predictor variables (e.g., math, WM, PS, vocabulary). However, nothing about the overall level of performance automatically implies that relations among predictor variables and achievement measures should be different. The overarching consistency with prior work supports the generalization of that work to the present sample with demographic characteristics that are less widely represented in the literature (i.e., Hispanic middle school students with a Spanish language background, some of whom were English learners and/or struggling readers). When sociodemographic characteristics were included as covariates, however, results were generally substantively the same.

Unique and Shared Prediction

A surprising aspect of the present study was the sporadic unique and shared predictive power of the cognitive domains, and how these contribute to the overlap of reading and math. Of the nine latent predictors, only RAN, PA, WM, and behavioral attention related uniquely to more than one of the four outcomes, and only behavioral attention was predictive of at least one reading and math outcome. Clearly, failure to be uniquely predictive in a regression context is not the same as concluding that the skill is unimportant or irrelevant to the outcome, especially because all of the predictor variables were selected precisely because prior literature has pointed to a consistent role for each. Rather, such results speak to the strong role of shared variance among predictors, which has been noted before in the literature (Cirino, Miciak et al., 2019; Cirino et al., 2022). Perhaps because there were few predictors that had shared relations with both reading and math, the predictors collectively reduced the reading-math correlations but did not eliminate them. There are few studies that explicitly attempt to account for the relation of reading and math by considering the differential role of multiple predictors (Child et al., 2019; Cirino et al., 2018; Peterson et al., 2017), and only Peterson et al. (2017) was able to completely account for the relation, with verbal comprehension dominating the overlap. That study was significantly larger (N = 636) and had a much wider age range (8–16, though results did not vary by age). Perhaps a greater distinction was that performance levels in Peterson et al. (2017) were rather higher (~15 standard score points) on many cognitive and academic measures; variability (SDs) of manifest measures though did not seem substantially different in that study relative the present. However, performance levels in Cirino et al., (2018) were similar to those of Peterson et al. (2017), whereas those of Child et al. (2019) were more consistent with the present study.

Differential Prediction

Despite the unexpected findings above regarding the relatively sparse unique and shared predictive power of the cognitive domains, the pattern of differential/relative prediction was consistent with expectation. The reading literature unquestionably supports a dominant role for language, and thus it makes sense for language to be more robustly related to reading versus math. This is consistent with literature that suggests that language is critically important for math as well (Amland et al., 2021; Cirino et al., 2018; Koponen et al., 2007; Krajewski & Schneider, 2009). Finding that WM is more robust for math than reading is also consistent with meta analyses (Spiegel et al., 2021) as well as with theoretical models of math (e.g., von Aster & Shalev, 2007; Geary, 2004) that reserve a special place for WM, as opposed to a number of reading models that invoke WM (typically in the context of executive function) but in a somewhat more indirect manner (Butterfuss & Kendou, 2017), and where the role of language is considered “first” (Cirino, Miciak et al., 2019). However, the present results add nuance as well. For example, it was quite clear that “language” did not operate uniformly; in the context of one another, vocabulary had a much smaller effect on even reading outcomes, relative to PA and RAN, and this was apparent at both the manifest and latent level, even in this middle school sample. Although broader language measures such as vocabulary might be expected to play a larger role for older students, perhaps the lower achievement levels of the present sample as a whole made the particular relevance of PA and RAN stand out more.

Importance and Implications

The results are novel and important new knowledge is gained, some of which confirms prior suggestions from the literature (e.g., on the relation of math and reading, about predictors of each, about predictors shared between them), but does so simultaneously and in a technically sophisticated manner, and adds nuance to the idea that math and reading have both shared and independent correlates. Although the study has limitations which can be probed in future studies (e.g., with a larger sample size, with latent achievement variables, with different populations such as younger students or in groups with different linguistic backgrounds), the sensitivity analyses suggest a general robustness to the findings.

There are also key implications of the results. First, domain general cognitive skills are unlikely to fully determine a student’s academic performance – competence and mastery of achievement skills must also invoke direct learning, and/or at least more domain specific foundational skills (e.g., numerosity skills in the case of math). This is consistent with the literature on reading development that recognizes a strong role of PA for reading, but also that print exposure is critical (Nation & Hulme, 2011), and from cognitive literature that suggests that training of domain general skills is rarely sufficient to drive distal transfer (e.g., to achievement; Melby-Lervåg & Hulme, 2013; Sala & Gobet, 2020; Schwaighofer, Fischer, & Buhner, 2015). There is recent work in this area demonstrating transfer (Fuchs et al., 2022), but the magnitude of transfer is much stronger within domain than across domain (in that case, working memory training vs. word problem solving training). Documentation of any transfer, along with the contributions of both, is also consistent with literature suggesting bidirectional influences of domain general and domain specific factors (Coolen et al., 2021; Mareva et al., 2022; Peng & Kievit, 2020). Therefore, there is a need to consider both. Even for skills that appear to be domain-specific may have cross-achievement predictive power. For example, counting is clearly a number-based function (and therefore domain specific), but inasmuch as it requires non-number features (e.g., symbolic association), it may account for a relation between reading and math, which has been found (Child et al., 2019; Cirino et al., 2018; Koponen et al., 2106; 2019).

Given the manifest differences between reading (e.g., word reading) and math (e.g., computations), it could be considered impressive that they share more than 20% of their variance. It makes sense to say that domain general skills would account for the overlap, and the literature shows convergence on which domain general skills are more or less likely to do so (e.g., the ones examined in this study). Those domain general skills did in fact account for about 50% of the variance in math and reading outcomes (examined separately), though not all of it, consistent with other reviews in this literature (Peters & Ansari, 2019). It is possible that even the addition of other domain general (and specific) factors, be they cognitive or non-cognitive (e.g., anxiety, motivation), might not account for all of the overlap between math and reading within a given sample, particularly at the manifest level, and given sociodemographic variability, method variance, and measurement error. What is less clear is whether other types of achievement measures (e.g., curriculum based measures, course grades, GPA) might show more versus less overlap that those evaluated here; those measures may also show different patterns of prediction.

Despite the inability to identify all the sources of overlap, given that the overlap is reduced does suggest that domain general skills could be leveraged to support math and reading learning (and/or difficulties in these areas). Doing so must though consider at least two things: (a) the extent to which domain specific information/instruction/intervention would need to be triggered simultaneously and/or interact with domain general factors to net improvement in each area; and (b) that domain general skills themselves overlap with each other, and/or that the same domain general skill (e.g., WM) might be used in different manners for each of math and reading. Future studies of both a correlational/descriptive nature as well as experimental approaches should be brought to bear to affect such leveraging.

Second, the present results also have relevance for understanding academic fluency. For example, non-academic processing speed was not a unique predictor of either timed achievement measure and did not show differential prediction for untimed versus timed achievement, and implies that in terms of processing speed, what is more relevant is what is being processed than speed per se (Gerst et al., 2021). This could suggest that academic fluency may be more a reflection of the strength of representation of the academic skill, rather than an independent construct. However, more work is needed to address such questions directly, as studies are variable in terms of how strongly academic fluency measures relate. For example, in Child et al. (2019), reading and math fluency measures correlated r = .29, whereas in a different study (which included only timed not untimed measures, and including individuals with spina bifida myelomeningocele over a much larger age range of children and adults), academic fluency measures were much more strongly related (r ~ .75 or more), and reaction time measures were predictive of all outcomes, albeit with some differential prediction (Cirino, Kulesz, et al., 2019).

Conclusions

The present study affirms prior work into the relations of key cognitive domains for math and reading achievement, while at the same time lending nuance to the pattern of these relations, and placing limits on their relevance. The scant basis for comparison and the variability in findings in the literature highlight the need for future work that considers math and reading concomitantly, particularly in terms of these cognitive predictors that can be considered domain general, preferably in the context of domain specific factors that are relevant to or directly involve one academic domain but not others. The present study might serve as a guide and lens through which to interpret future findings.