# Math fluency is etiologically distinct from untimed math performance, decoding fluency, and untimed reading performance: evidence from a twin study.

**ABSTRACT** The authors examined whether math fluency was independent from untimed math and from reading using 314 pairs of school-aged twins drawn from the Western Reserve Reading and Math Projects. Twins were assessed through a 90-min home visit at approximately age 10 and were reassessed in their homes approximately 1 year later. Results suggested that the shared environment and genetics influenced the covariance among math fluency, untimed math measures, and reading measures. However, roughly two thirds of the variance in math fluency was independent from untimed math measures and reading, including reading fluency. The majority of this independent variance was the result of genetic factors that were longitudinally stable across two measurement occasions. These results suggest that math fluency, although related to other math measures, may also be a genetically distinct dimension of mathematics performance.

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**ABSTRACT:**In the last 20 years, longitudinal studies have demonstrated that it is important to attend to the stability of mathematical performance over time as a facet of dyscalculia, that the manifestation of mathematics difficulties changes with development, and that individual differences in cognitive profiles and learning trajectories observed in children with mathematics difficulties implicate differences between dyscalculic and non-dyscalculic subgroups. Intra-individual differences over time, and external factors related to children's learning environments, also contribute to performance trajectories; moreover, these factors may explain the inconsistent performance profiles observed among many students whose difficulty with mathematics emerges later or diminishes over time. Longitudinal studies on DD are also an important tool to elucidate why some children are more responsive to mathematics intervention than others.Trends in Neuroscience and Education. 06/2013; 2(2):65–73. - [Show abstract] [Hide abstract]

**ABSTRACT:**Working memory has been consistently associated with mathematics achievement, although the etiology of these relations remains poorly understood. The present study examined the genetic and environmental underpinnings of math story problem solving, timed calculation, and untimed calculation alongside working memory components in 12-year-old monozygotic (n = 105) and same-sex dizygotic (n = 143) twin pairs. Results indicated significant phenotypic correlation between each working memory component and all mathematics outcomes (r = 0.18–0.33). Additive genetic influences shared between the visuo-spatial sketchpad and mathematics achievement were significant, accounting for roughly 89% of the observed correlation. In addition, genetic covariance was found between the phonological loop and math story problem solving. In contrast, despite there being a significant observed relationship between phonological loop and timed and untimed calculations, there was no significant genetic or environmental covariance between the phonological loop and timed or untimed calculation skills. Further analyses indicated that genetic overlap between the visuo-spatial sketchpad and math story problem solving and math fluency was distinct from general genetic factors, whereas g, phonological loop, and mathematics shared generalist genes. Thus, although each working memory component was related to mathematics, the etiology of their relationships may be distinct.Intelligence 11/2014; 47:54–62. · 2.67 Impact Factor - SourceAvailable from: Richard K OlsonMicaela E Christopher, Jacqueline Hulslander, Brian Byrne, Stefan Samuelsson, Janice M Keenan, Bruce Pennington, John C Defries, Sally J Wadsworth, Erik G. Willcutt, Richard K Olson[Show abstract] [Hide abstract]

**ABSTRACT:**We explored the etiology of individual differences in reading development from post-kindergarten to post-4(th) grade by analyzing data from 487 twin pairs tested in Colorado. Data from three reading measures and one spelling measure were fit to biometric latent growth curve models, allowing us to extend previous behavioral genetic studies of the etiology of early reading development at specific time points. We found primarily genetic influences on individual differences at post-1(st) grade for all measures. Genetic influences on variance in growth rates were also found, with evidence of small, nonsignificant, shared environmental influences for two measures. We discuss our results, including their implications for educational policy.Scientific Studies of Reading 09/2013; 17(5):350-368. · 1.86 Impact Factor

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Math Fluency Is Etiologically Distinct From Untimed Math

Performance, Decoding Fluency, and Untimed Reading

Performance: Evidence From a Twin Study

Stephen Petrill1, Jessica Logan1, Sara Hart2, Pamela Vincent1, Lee Thompson3, Yulia

Kovas4, and Robert Plomin5

1Ohio State University, Columbus, OH, USA

2Florida State University, Tallahassee, FL, USA

3Case Western Reserve University, Cleveland, OH, USA

4Institute of Psychiatry, London, UK

5King’s College London, MRC Social, Genetic and Developmental Psychiatry Centre, Institute of

Psychiatry, De Crespigny Park, London, SE5 8AF, United Kingdom

Abstract

The authors examined whether math fluency was independent from untimed math and from

reading using 314 pairs of school-aged twins drawn from the Western Reserve Reading and Math

Projects. Twins were assessed through a 90-min home visit at approximately age 10 and were

reassessed in their homes approximately 1 year later. Results suggested that the shared

environment and genetics influenced the covariance among math fluency, untimed math measures,

and reading measures. However, roughly two thirds of the variance in math fluency was

independent from untimed math measures and reading, including reading fluency. The majority of

this independent variance was the result of genetic factors that were longitudinally stable across

two measurement occasions. These results suggest that math fluency, although related to other

math measures, may also be a genetically distinct dimension of mathematics performance.

Keywords

mathematics; genetics; reading

Typically, math fluency is assessed by measuring how quickly students are able to calculate

simple math problems. This measure has begun to receive significant attention. First,

progress monitoring in schools has led to the emergence of timed measures to quickly and

repeatedly provide teachers with academic information on large groups of students (Fletcher

& Vaughn, 2009). More generally, the importance of mathematical literacy in modern

society has led to more systematic and sustained efforts to understand mathematics ability

and disability (Butterworth, 1999; Fuchs et al., 2008; Geary, 2010; Geary & Burlingham-

Dubree, 1989; Geary & Widaman, 1992; Landerl, Bevan, & Butterworth, 2004; National

Mathematics Advisory Panel, 2008; Raghubar, Barnes, & Hecht, 2010; Siegler, 1988). As

© Hammill Institute on Disabilities 2011

Corresponding Author: Stephen Petrill, Ohio State University, 135 Campbell Hall, 1787 Neil Avenue, Columbus, OH 43210, petrill.

2@osu.edu.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

NIH Public Access

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J Learn Disabil. 2012 July ; 45(4): 371–381. doi:10.1177/0022219411407926.

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part of this effort, several recent studies have suggested that math fluency, although related

to untimed math measures, may reflect additional variance in math ability above and beyond

untimed math performance (Fuchs et al., 2008; Hart, Petrill, & Thompson, 2010; Mazzocco,

Devlin, & McKenney, 2008).

After reviewing the literature, the National Mathematics Advisory Panel (2008) suggested

that math fluency was an important aspect to many subject areas of mathematics

performance, such as number sense and whole number arithmetic. In general, research has

examined the relationship between timed fluency and untimed math performance using

samples selected for math difficulties. Poor math fluency has been suggested as a possible

indicator of mathematical learning disability (MLD; Mazzocco et al., 2008). Indeed, studies

have suggested that poor math fluency may be a unique quality of mathematical difficulty

and not a predictor of other learning problems such as reading difficulty (Jordan & Hanich,

2003). In examining the difference in math fluency outcomes between children with MLD

versus children who were low achievers in math without diagnosed disability, both groups

suggested greater total errors on a math fluency measure than their typically developing

peers (Mazzocco et al., 2008). This study also examined girls with Fragile X or Turner’s

syndrome, both of which have known math performance difficulties. Girls with Turner’s

syndrome had difficulties for timed fluency outcomes, whereas girls with Fragile X did not.

In contrast, studies involving unselected samples have eschewed measures of math fluency,

instead employing measures of general processing speed. These studies suggest that general

processing speed (e.g., speed of visual matching) is a unique indicator of computational

skills (e.g., Bull, Johnston, & Roy, 1999; Fuchs et al., 2008; Hecht, Torgesen, Wagner, &

Rashotte, 2001). Fuchs et al. (2008) proposed that processing speed may indicate how

quickly numbers can be counted in a computational sequence, with slower counting

resulting in lower success in problem solving. However, because studies have not included

both math- and non-math-based measures of fluency, it is unclear whether this effect is

specific to math fluency, a reflection of more general processing speed, or a combination of

both general and math-specific fluency.

Behavioral genetic studies have also offered initial evidence concerning the overlapping but

unique etiology of math fluency with untimed math outcomes. Identical (monozygotic; MZ)

twins share 100% of their additive genetic variance, whereas fraternal (dizygotic; DZ) twins,

on average, share 50%. Genetic influences are implied if MZ twins show greater similarity

than DZ twins. Shared environmental influences are implied if familial resemblance on a

trait is equal for MZ twins and DZ twins. Nonshared environment (including error) is

implied if MZ twins are not perfectly correlated. As described more fully in the results

section, the twin design can also be used to examine the relationships among a set of

variables (also see Plomin, DeFries, McClearn, & McGuffin, 2008).

To date, there have been two behavioral genetic studies involving measures of math fluency.

Hart, Petrill, Thompson, and Plomin (2009) examined the relationship among general

cognitive ability, reading, and math outcomes, including math fluency. Untimed measures of

math performance (e.g., problem solving) were influenced primarily by shared

environmental effects. In contrast, timed measure of math fluency indicated significant

moderate to high genetic influences. This study further suggested that there was significant

genetic overlap between math fluency and rapid automatized naming (RAN), but there was

no significant relationship between math fluency and untimed measures of reading. These

results suggest that a significant portion of the variance in math fluency is heritable and that

these genetic influences are associated with measures of naming fluency. These overlapping

genetic effects are consistent with earlier work suggesting that math fluency may involve

basic mechanisms of learning that affect multiple domains, such as speed of long-term

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memory retrieval (Geary & Widaman, 1987, 1992). However, there were also significant

independent genetic influences on math fluency above and beyond RAN, suggesting that

there may be unique genetic effects on math fluency.

To further explore this result, using the same sample, Hart et al. (2010) examined the

relationship among latent factors of untimed reading, decoding fluency, reading

comprehension, and math, including math fluency. Confirmatory factor analyses indicated

that math fluency was best associated with other untimed math measures instead of decoding

fluency measures. This would suggest that despite the general genetic association between

measures of fluency described in Hart et al. (2009), when subjected to a measurement

model, math fluency was more highly associated with untimed math outcomes than timed

reading outcomes. Moreover, the genetic covariance between the math factor (including

untimed math and math fluency) and the decoding fluency factor was statistically

significant. However, both decoding fluency and math factors also demonstrated significant

independent genetic effects, again suggesting a distinct genetic etiology for fluency.

In sum, behavioral genetic studies are consistent with the notion established in the cognitive

and educational psychology literatures that math fluency, although correlated with untimed

math measures, may also constitute an additional dimension of math performance (e.g.,

Fuchs et al., 2008; Mazzocco et al., 2008; Tolar, Lederberg, & Fletcher, 2009). These

studies also suggest that genetic effects for timed measures may be both general to reading

and math as well as specific to math.

Given these findings, three important questions need to be addressed. First, are independent

genetic influences on math fluency present above and beyond their association with untimed

math measures? Hart et al. (2010) found that math fluency loaded on a general math factor,

but 64% of the variance in math fluency was not accounted for by that general math factor.

It is unknown as to whether the remaining residual variance is attributable to random error

or whether it indicates unique variance related specifically to math fluency. Furthermore,

previous work suggests that it is important to understand the possible genetic etiology of this

residual variance. This analysis has not been conducted.

A second important question is how math fluency relates to timed and untimed measures of

reading ability. Previous work suggests significant common genetic influences between

math fluency and naming fluency, but not between math fluency and untimed measures of

reading (Hart et al., 2009). We hypothesize that fluency in reading and math may be tapping

similar constructs; however, the amount of overlap between reading and math fluency is

unclear. Therefore, it is also important to ascertain whether math fluency and reading

fluency are influenced by a common set of genetic factors or if math fluency is genetically

independent from reading fluency.

Finally, it is well established that mathematical performance is highly stable across

measurement occasions (Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; DiPerna, Lei, &

Reid, 2007; Jordan & Hanich, 2003), but the role of math fluency in the stability of math

performance is poorly understood. At present, the only study of untimed math development

found that genetic factors were important for the stability of math across measurement

occasions (Kovas, Haworth, Dale, & Plomin, 2007). Importantly, Kovas et al. (2007) also

found substantial evidence for unique genetic effects on math separate from general

intelligence and reading skills. Thus, we expect that the longitudinal stability of math

fluency will be influenced by common genetic factors. However, it is unclear whether this

stability will be specific to math fluency, shared with untimed math, and/or shared with

timed measures of reading.

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Thus, we hypothesize that math fluency will be associated with untimed math performance.

Unknown is whether math fluency will show independent genetic effects above and beyond

untimed math measures. Furthermore, if independent genetic effects on math fluency are

significant, we will examine whether they are correlated with and/or independent from

reading fluency. Finally, we examine whether the stability in math fluency across

measurement occasions is independent from or correlated with untimed math measures and

reading fluency.

Method

Participants

Participants of the present study are enrolled in the ongoing Western Reserve Reading and

Math Projects, a longitudinal twin study in Ohio. Recruiting was conducted through Ohio

state birth records, school nomination, and media announcements. We have the participation

of 293 schools throughout the state, which were asked to send information to parents with

twins. We also hired a social worker with longstanding ties to the community to assist in the

recruitment of underrepresented groups via face-to-face meetings at places of worship,

community centers, and other service organizations.

The project is ongoing (see Petrill, Deater-Deckard, Thompson, DeThorne, &

Schatschneider, 2006). Home visits began when children entered school. Waves 1, 2, and 3

focused on early reading skills. Wave 4 examined math skills. The present study is based on

home visit Waves 5 and 6, which are the first to focus on both reading and mathematics

skills simultaneously. At Wave 5, children were approximately 10 years old (age M = 9.83

years, SD = 0.97 years, range = 7.42–12.75 years). At Wave 6, children were approximately

11 years old (age M = 10.99, SD = 0.89, range = 8.42–13.42 years).

The final sample size of same-sex twin pairs with known zygosity and analyzable data was

260 pairs of MZ (n = 108) and DZ (n = 152) twins. Zygosity was determined using DNA

analysis via a cheek swab. For the cases where parents did not consent to genotyping (n =

76), zygosity was determined using a parent questionnaire about the twins similarity

(Goldsmith, 1991). Although somewhat positively skewed (skew = .04), parent education

levels varied widely and were similar for fathers and mothers: 12% had a high school

education or less, 18% had attended some college, 30% had a bachelor’s degree, 24% had

some postgraduate training or a degree, and 5% did not specify. Most families were two-

parent households (92%), and nearly all were White (92% of mothers, 94% of fathers).

Procedure and Measures

Reading and math from Wave 5 were examined. We also included math data from Wave 6

so that we could test whether the overlapping and independent effects for math fluency

identified at Wave 5 were longitudinally stable. All test sessions were conducted in the

twins’ homes in separate rooms by separate examiners, and the total time to complete all

testing was approximately 60 to 90 min per child.

Reading—Three untimed measures of reading skill were examined in the present study.

The outcome of interest in each of these measures is the number of items correctly selected

prior to reaching a ceiling. First, the Peabody Individual Achievement Test (PIAT;

Markwardt, 1997) requires participants to read a sentence and then select a picture from four

choices that best represents the meaning of the sentence. Published test–retest reliability for

10-year-old children is .93. A second measure of reading comprehension was also used, the

Woodcock Reading Mastery Test (WRMT; Woodcock, 1987). This test requires children to

read a short passage and identify a missing keyword to complete a sentence at the end of the

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passage. The published median reliability for the test is .83. The final untimed measure was

the Word Identification subtest of the WRMT. This test requires students to recognize and

read real words aloud. Published split-half reliability for this subtest is .95 (Woodcock,

1987).

Finally, the Test of Word Reading Efficiency (TOWRE; Wagner, Torgesen, & Rashotte,

1999) represents the timed, fluency aspect of reading. In the present study, a composite

score of two subtests was used. First, Sight Word Efficiency tests the number of real words

read correctly in 45 s. Second, Phonemic Decoding Efficiency tests the number of non-

words read correctly in 45 s. Test–retest reliability for both subtests of the TOWRE for

children aged 10–18 years is .88.

Math—Four untimed measures of math skill were used in the present study. For each of

these measures, the score represents the number of correct responses given prior to reaching

a ceiling. First, the Calculation subtest of the Woodcock–Johnson III (WJ-III; Woodcock,

McGraw, & Mather, 2001, 2007) measures a child’s ability to perform addition, subtraction,

multiplication, and division of positive and negative whole numbers, fractions, percentages,

and decimals. The published median reliability of the test is .85. Second, the Applied

Problems subtest of the WJ-III requires children to read word problems containing critical

and extraneous information, determine which mathematical operation to use to solve the

problem, and complete simple calculations. Published median reliability of this subtest is .

92.

The third measure used to assess untimed math skill was the Quantitative Concepts portion

of the WJ-III. Quantitative Concepts consists of two subtests: Concepts and Number Series.

Concepts measures ability to count, identify numbers, shapes, and number sequences, and

knowledge of mathematical terms, but no calculation is required. Number Series measures

the ability to provide a missing number that will continue a presented series. The median

reliability for the Quantitative concepts portion of the WJ-III is .90. Finally, the Math

portion of the Wide Range Achievement Test (WRAT; Wilkinson, 1993) requires students

to name presented number symbols, solve orally presented problems, and do some

computations. The published reliability of the WRAT is .89.

The timed measure of math skill was the Fluency subtest of the WJ-III. This test measures a

child’s ability to answer addition, subtraction, multiplication, and division problems in a

limited amount of time (3 min). Twins were told to solve a series of calculation problems as

quickly as they could. Published median reliability of the Fluency subtest is .89 for children.

Results

Descriptive Statistics

Table 1 presents descriptive information for outcomes at Waves 5 and 6. Mean reading and

math scores were slightly above average and variance was somewhat restricted relative to

norming populations. There was a range of scores at or near 2 standard deviations above and

below the mean for all variables. To maintain consistency with our prior publications, for all

subsequent analyses we employed raw reading and math variables residualized for age, age

squared, and gender. Also in keeping with our prior publications, math variables were

further residualized for school months and school months squared (see Hart et al., 2009).

Intraclass correlations are also presented to provide a descriptive picture of genetic and

environmental influences related to reading and math measures. Briefly, if additive genetic

effects (h2) are significant, MZ correlations should be exactly 2 × DZ correlations, by virtue

of the fact that MZ twins share 2 times as much additive genetic variance than fraternal

twins. Shared environmental effects are indicated to the extent that MZ-DZ twin

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resemblance is similar in magnitude. Nonshared environment is suggested to the extent that

MZ twin correlations are not equal to 1.0 (which also includes error). MZ intraclass

correlations were higher than DZ intraclass correlations for all variables, suggesting genetic

influences. Shared environmental effects were also indicated, particularly for untimed math

outcomes, as evidenced by DZ intraclass correlations that were greater than expected by

genetics alone. Some evidence for dominance genetic effects in reading was suggested

because MZ twins are more than 2 × DZ correlations. However, genetic dominance was

nonsignificant for all reading measures when estimated in a structural equation modeling

framework (available from the first author on request).

Next we examined the phenotypic correlations among reading and math outcomes (see

Table 2). Statistically significant correlations were found for all bivariate comparisons,

ranging from r = .16 between Wave 5 PIAT Reading Comprehension and Wave 6 TOWRE

to r = .84 between Wave 5 and Wave 6 WJ Fluency. Most of the remaining comparisons

ranged between r = .40 and r = .60.

Quantitative Genetic Modeling

The primary goal of this study was to examine whether math fluency was etiologically

distinct for untimed math measures and untimed reading measures and, if so, whether math

fluency was correlated with, or independent from, reading fluency. A related question was

whether math fluency was longitudinally stable beyond untimed math skills and reading

fluency. Figure 1 presents a model that parameterized three latent factors: Reading Skills at

Wave 5, Math Skills at Wave 5, and Math Skills at Wave 6. This model was fit to the data

using Mx (Neale, Boker, Xie, & Maes, 2006). The overall fit of this model was −2log

likelihood = 11770.20, df = 5737, Bayesian information criterion (BIC) = −10065.72, where

negative values indicate better fit (Schwarz, 1978). Significance of individual parameters

was determined using 95% confidence intervals. The variances of the latent Reading and

Math factors were constrained to 1.0 to yield standardized estimates. Standardized factor

loadings were significant, ranging from .64 between WJ Calculation Fluency and Wave 5

Math factor and .91 between WJ Word Identification and the Wave 5 Reading Factor.

The variance and covariance among the Reading and Math factors was simultaneously

decomposed into genetic (A1, A2, A3), shared environmental (C1, C2, and C3), and

nonshared environmental (E1, E2, E3) factors. Standardized path estimates presented in

Figure 1 (with 95% confidence intervals) suggested statistically significant genetic (A1),

shared environmental (C1), and nonshared environmental (E1) influences common to

Reading and Math factors. In addition, results suggested statistically significant genetic (A2)

and nonshared environmental (E2) variance for Math factors above and beyond the Reading

factor. A2, and E2, estimates also suggest longitudinal stability between Wave 5 and Wave 6

Math factors independent from the Reading factor. There was no evidence for unique shared

environmental effects for the Wave 5 Math factor, as evidenced by nonsignificant path

estimates for C2. Furthermore, there was no evidence for independent effects for the Wave 6

Math factor, as evidenced by nonsignificant path estimates for A3, C3, and E3.

These path estimates can also be used to estimate overall heritability (h2), shared

environment (c2), and nonshared environment (e2) for the Wave 5 Reading factor, Wave 5

Math factor, and Wave 6 Math factor (see Table 3). These estimates are calculated by

squaring the A, C, and E matrices presented in Figure 1 (see Neale & Cardon, 1992).

Heritability estimates for reading and math factors were statistically significant, with a trend

for higher heritability in Wave 5 Reading (h2 = .77) versus the Math factors at Wave 5 (h2

= .41) and Wave 6 (h2 = .34). Importantly, shared environmental estimates were also

statistically significant for all three factors. Shared environmental estimates were statistically

significantly lower for the Wave 5 Reading factor (c2 = .12) compared to the Wave 5 Math

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factor (c2 = .52), as evidenced by nonoverlapping confidence intervals. Shared

environmental estimates for Wave 5 Reading was also lower than Wave 6 Math factor (c2

= .52), but this comparison was not statistically significant. Child specific nonshared

environmental (e2) estimates were statistically significant for all factors. Importantly,

because these are based on latent factors, they do not include random error.

Decomposition of Residuals

The central question of this study is whether unique genetic and/or environmental influences

are significant for Math Fluency above and beyond latent Reading and Math factors. The

model presented in Figure 1 tests for this explicitly by decomposing the residual variance for

each measure into genetic (a), shared environmental (c), and nonshared environmental (e)

sources of variance. For presentational purposes, only statistically significant residual

estimates are presented in Figure 1, but confidence intervals for residual estimates and factor

loadings are presented in Table 4. Nonshared environmental estimates were statistically

significant for all residuals. Unlike the latent factors, nonshared environmental influences on

the residuals include measurement error. Genetic sources of variance were statistically

significant for Wave 5 TOWRE, Wave 5 WJ Calculation Fluency, and Wave 6 WJ

Calculation Fluency, suggesting independent genetic influences for fluency residuals after

accounting for the latent factors. Shared environmental influences (c) on residuals were not

significant.

We also examined whether the residual genetic effects for Reading and Math Fluency were

correlated and/or independent. Genetic variance for Math Fluency was correlated with

Reading Fluency both concurrently and longitudinally (indicated by significant a1 loadings

in Figure 1 and Table 4). There was also evidence for independent genetic effects for Math

Fluency that was stable across measurement occasions (as evidenced by a2 loadings). There

was no evidence for independent genetic effects for Math Fluency at Wave 6 above and

beyond Wave 5 measures, as evidenced by a loading of 0.00 on a3. Dropping the genetic

overlap among residuals led to a significant decrease in fit (χ2change = 269.448, dfchange = 3,

BICchange = 126.38), providing further evidence that genetic overlap among residuals is

necessary to model fit.

Summary of Latent and Residual Effects on Reading and Math Fluency

Taken together, the a1 and a2 path estimates described in Figure 1 suggest that there is

significant genetic overlap among Reading Fluency and Math Fluency as well as significant

independent genetic variance for Math Fluency. Both a1 and a2 appear to be longitudinally

stable as evidenced by significant loadings with Wave 6 Math Fluency. To provide a better

measure of the magnitude of these independent effects, we present the influences of these

various pathways on the total variance for Math Fluency. The total variance for Math

Fluency at Waves 5 and 6 is divided into genetic (h2), shared environmental (c2), and

nonshared environmental (e2) sources of variance that emanate from the latent factors plus

variance from residual factors. For example, using the data from Figure 1, the heritability of

Wave 5 Math Fluency was

where

(A1)

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(A2)

(a1)

(a2)

Figure 2 presents the proportion of h2, c2, and e2 of Wave 5 Math Fluency that was

explained by these sources of variance along with tests of significance (using confidence

intervals, as indicated by an asterisk). Results suggested small but significant genetic effects

from the latent factors and reading fluency, with most of the genetic variance specific to

math fluency. In contrast, shared environment (c2 = .25) was influenced by the general

factor common to all reading and math measures (C1 in Figure 1). Nonshared environment

(e2 = .17) was attributable mainly by variance unique to Math Fluency (as well as error).

Notably, the estimates of total h2, c2, and e2 for Wave 5 Math Fluency are very similar to

what would have been expected from the intraclass correlation results presented in Table 1.

Figure 3 presents estimates for Wave 6 Math Fluency. What is important to note is that the

genetic and shared environmental variance at Wave 6 was explained by measures assessed

in Wave 5. The majority of the genetic variance in Math Fluency at Wave 6 was explained

by genetic variance unique to Math Fluency at Wave 5, whereas most of the shared

environmental variance was explained through the latent factors. Nonshared environment

(and error) was influenced mostly by variance specific to Wave 6 Math Fluency.

Discussion

The purpose of this study was to examine whether math fluency was genetically and/or

environmentally distinct from other measures of math and reading performance. Latent

factor modeling suggested that two thirds of the variance in math fluency was independent

from other math measures. Most of this residual variance was affected by specific genetic

influences related to math fluency, with additional genetic variance explained by reading

fluency. A secondary purpose of the study was to examine if variance for math fluency was

longitudinally stable in late childhood or early adolescence and, if so, if this stability was

related to or independent from other math and reading measures. Independent genetic

influences for math fluency overlapped completely across two annual measurement

occasions, above and beyond reading fluency and untimed math measures. Thus, genetic

variance for math fluency was not only etiologically distinct from other math measures but

also longitudinally stable over a 1-year period.

In contrast, the majority of the shared environmental influences on math fluency were

general, associated via the latent math factor with measures of reading. As was the case for

the genetic influences on fluency, complete overlap across measurement occasions

suggested strong longitudinal stability. Taken together, genetic effects on math fluency were

largely specific, whereas shared environmental effects on math fluency were largely general.

Both genetic and shared environmental effects on math fluency were longitudinally stable.

These findings are novel in the following ways. First, this study is the first longitudinal twin

study to examine direct assessments of math and the first to examine math fluency. Kovas et

al. (2007) employed a longitudinal genetic design in twins who were roughly the same age

as our sample (7, 9, and 10 years) but employed teacher ratings. Both studies find evidence

for not only significant genetic overlap between math and reading factors but also significant

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genetic specificity for overall math, separate from reading. Both studies also suggest high

longitudinal stability. Unique to our study are significant shared environmental effects for

the math factor. Aside from differences in the measurement of math, the Kovas et al. study

employed a U.K. sample where there is a national math curriculum. The current study

employed a U.S. sample where curricula are locally administered. Because twins attend the

same schools, greater variability in the U.S. schools may be reflected in higher estimates of

shared environment (see Petrill & Plomin, 2007, for a discussion).

More central to the purpose of the study, results provide strong evidence for the specificity

of math fluency as a genetically distinct dimension of math performance. As described in the

introduction, unselected studies of math have found that general processing speed (e.g.,

speed of visual matching) independently predicts computational skills (e.g., Bull &

Johnston, 1997; Fuchs et al., 2008; Hecht et al., 2001), whereas studies of math disability

have shown that math fluency predicts independent variance in math ability (Jordan &

Hanich, 2003; Mazzocco et al., 2008). Our study replicates these findings but further

suggests that math fluency, although related to other math measures, may also be influenced

by a separate genetically stable component of math performance. As research continues to

examine the effects of working memory, general processing speed, number sense, and math

problem-solving strategies on math performance, we expect that math fluency will stand

apart from these measures. We are currently conducting such a study using a twin sample

and expect that independent genetic factors will be primarily responsible for the

discriminant validity of math fluency.

Finally, the results of the current study have important implications for progress monitoring,

education, and intervention. Timed measures are clearly an efficient means to assess

academic performance (Fletcher & Vaughn, 2009), and the literature to date suggests that

math fluency does indeed covary with other measures of math. Thus, math fluency does

appear to be a viable screening measure. Our data further suggest that the shared

environment affects math fluency through variance common to all math measures. However,

math fluency also stands apart from other math measures. Our data suggest that two thirds of

the variance is unique to math fluency, a large portion of which is affected by independent

genetic factors that are longitudinally stable. Intervention studies have shown that math

fluency is modifiable but requires different approaches compared to untimed measures

(Fuchs et al., 2010). This is not surprising. Genetic influences aside, math fluency, although

related to untimed math performance, is not synonymous with untimed math performance.

Understanding the multidimensional aspects of math performance is necessary to develop

curricula and intervention strategies that target students’ strengths and weaknesses across

these domains.

Acknowledgments

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of

this article:

This work is supported by Eunice Kennedy Shriver National Institute of Child Health and Human Development

grants HD038075 and HD059215.

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Biographies

Stephen Petrill is professor of human development and family science at the Ohio State

University. His research interests involve the application of quantitative and molecular

genetic designs to the development of math and reading ability/disability

Jessica Logan is a postdoctoral fellow at the Ohio State University. She is a methodologist

with expertise in modeling the development of academic achievement.

Sara Hart is a postdoctoral fellow at Florida State University. Her interests include the

cognitive underpinnings of mathematics ability and disability.

Pamela Vincent is a graduate fellow at the Ohio State University. Her interests include the

development of mathematics and related environmental factors.

Lee Thompson is professor of psychology at Case Western Reserve University. Her

research interests involve the genetic and environmental underpinnings of cognitive ability,

and their relationship with academic achievement.

Yulia Kovas is a lecturer at the Institute of Psychiatry in London. She has conducted

international studies on the genetic and environmental underpinnings of math performance.

Robert Plomin is professor and director of the Social, Genetic, and Developmental

Psychiatry Research Centre, at the Institute of Psychiatry. His interests involve the

application of quantitative and molecular genetic designs to understand the etiology and

development of learning disabilities.

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Figure 1.

Latent factor model with 95% confidence intervals for genetic, shared, and nonshared

environmental effects on latent reading and math factors

Note: PIAT = Peabody Individual Achievement Test; WRMT = Woodcock Reading

Mastery Test; TOWRE = Test of Word Reading Efficiency; WJ = Woodcock–Johnson;

WRAT = Wide Range Achievement Test. Confidence intervals for latent reading and math

factor loadings and residual loadings are presented in Table 4.

*p < .05.

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Figure 2.

Proportion of genetic (h2), shared environmental (c2), and nonshared environmental (e2)

variance in Wave 5 Math Fluency accounted for by Wave 5 Reading factor (Read5Factor),

Math factor (Math5Factor), Reading Fluency (Read5Fluency), and Math Fluency

(Math5Fluency), with 95% confidence intervals in braces

*p < .05.

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Figure 3.

Proportion of genetic (h2), shared environmental (c2), and nonshared environmental (e2)

variance in Wave 6 Math Fluency accounted for by Wave 5 Reading factor (Read5Factor),

Math factor (Math5Factor), Reading Fluency (Read5Fluency), and Math Fluency

(Math5Fluency), plus Wave 6 Math Fluency (Math6Fluency) residuals, with 95%

confidence intervals in braces

Note: No independent variance was attributable to Wave 6 Math factor (Math6Factor).

*p < .05.

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Table 1

Descriptive Statistics

Variable

M

SD

Min

Max

rMZ

rDZ

Wave 5

Reading PIAT Comp

106.02

11.41

77.00

144.00

.61

.21

WRMT Comp

103.49

11.09

65.00

134.00

.68

.26

WRMT Word ID

106.87

10.40

68.00

140.00

.83

.38

TOWRE

103.61

11.99

70.00

130.50

.80

.37

Math WJ Calc

106.38

13.19

71.00

152.00

.73

.57

WJ Fluency

101.37

14.63

62.00

154.00

.83

.54

WJ Applied Prob

107.85

11.48

61.00

144.00

.74

.56

WJ Quant Concept

105.55

12.92

73.00

145.00

.81

.49

WRAT Math

103.72

13.66

68.00

153.00

.77

.50

Wave 6

Math WJ Calc

106.70

13.41

69.00

151.00

.73

.62

WJ Fluency

101.15

14.38

65.00

153.00

.83

.54

WJ Applied Prob

107.05

10.93

69.00

143.00

.69

.51

WJ Quant Concept

105.56

12.55

65.00

139.00

.64

.43

WRAT Math

104.93

14.64

68.00

150.00

.66

.59

Note: rMZ = monozygotic; rDZ = dizygotic; PIAT = Peabody Individual Achievement Test; WRMT = Woodcock Reading Mastery Test; TOWRE = Test of Word Reading Efficiency; WJ = Woodcock–

Johnson; WRAT = Wide Range Achievement Test.

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