The study reported in this article had two major goals. The first goal was to investigate whether the etiology of individual differences was quantitatively similar for various aspects of mathematics. Previously, only the etiology of individual variation in a single broad measure of mathematical achievement had been investigated. The results of the present study suggest that etiologies of aspects of mathematics, such as mathematical application, understanding of numbers, computation and knowledge, mathematical interpretation, and nonnumerical processes, are similar, with heritabilities ranging from .30 to .45, shared environmental influences ranging from .08 to .30, and nonshared environmental influences ranging from .41 to .56. The only measure that seemed to stand out was Mathematical Application, in that it showed stronger shared environmental influences than other measures. However, the shared environmental estimate for Mathematical Application was not significantly different from the other four tests, indicating that more research is needed to determine whether these differences can be replicated and to identify the possible reasons for these differences.

Consistent with previous research, we found that most environmental influences on diverse aspects of mathematics were nonshared. That is, on average across the five tests, 76% of the environmental variance could be attributed to nonshared environment rather than shared environment. This means that environmental factors that shape individual differences in mathematics were not shared between the cotwins (even MZ twins) in the same family. In other words, having the same parents, being of the same socioeconomic status, and going to the same school did not contribute to the similarity between the two children beyond the similarity in their genetic make-up. To assess whether being in the same class and having the same teacher increased similarity between cotwins and affected the genetic findings, we reran our correlational analyses, splitting the data by same versus different teacher. The two groups were approximately equal in size. For example, for the Mathematical Application category, 51% of the twin pairs were in the same class (vs. 49% in different classes). The results (available upon request) showed that being in the same class and having the same teacher did not significantly increase similarity between cotwins in their performance on the Web-based battery of tests—average MZ and DZ twin correlations were .58 and .38, respectively, for twins in the same classroom and .57 and .33, respectively, for twins in different classrooms. Thus, ACE parameter estimates were also similar for twins in the same and different classrooms. We also ran analyses splitting the sample into pairs taught by the same or a different teacher when the twins were 7 and 9 years of age. The results for the two groups were also highly similar at both ages, suggesting that early shared classroom experiences did not increase similarity between cotwins on a mathematical test at 10 years of age. Overall, these findings suggest that teachers and classroom environments affect mathematical ability in different children (even twins) in different ways (and therefore these effects are subsumed under the nonshared environmental estimate).

The average correlation for MZ twins was only .58—what made them different could only be nonshared environment (and error of measurement). Nonshared environment could be explained by pre-, peri-, and postnatal factors that contribute to the differential development of mathematical ability within pairs of MZ twins. These factors could include childhood illnesses, differential parental influence, or differential effects of curricula and other school factors on children. If, as our findings suggest, three quarters of the environmental variation on mathematical ability is nonshared, it is important to identify the nonshared environmental factors that lead to differences in mathematical ability even within pairs of MZ twins growing up in the same family and attending the same school. If these nonshared environmental factors can be identified, they could lead to more individualized curricula, although much more research is necessary to clarify whether such a move toward individualization in education is necessary or feasible practically.

It should be emphasized that quantitative genetic research such as ours describes genetic and environmental influences that exist in a particular population at a particular time. Our results suggesting that most of the individual differences in mathematics performance at 10 years were due to genetic and nonshared environmental influences describe the factors that were at work at this time in our U.K. sample. Even if heritability were 1.00, new environmental interventions could have a major effect. Moreover, results could differ in different populations. For example, having a common national curriculum, as in the United Kingdom, with its guidance on the methods and the content at each age, might decrease environmental variation and thus increase the relative impact of genetic variation. In a country without a national curriculum, greater environmental variation could emerge if differences in school curriculum made a difference in children’s learning of mathematics skills. These environmental differences would be expected to be shared environmental influences to the extent that children attending the same school experienced this school-level effect similarly. Heritability would also be lower, as environmental influences would account for relatively more variance. An interesting avenue for research is to investigate whether the relative contribution of genes and environments changes as a function of changes introduced to the national curriculum.

The second goal of this study was to go beyond these univariate analyses to conduct the first multivariate genetic study into heterogeneity of mathematical ability, assessing the extent to which individual differences in diverse aspects of mathematics are influenced by the same genetic and environmental factors. Our phenotypic explorations of the data showed that correlations among the five aspects of mathematics ranged from .45 to .68. The highest correlation was between Mathematical Application and Understanding Number; the smallest correlations were between Computation and Knowledge and Non-Numerical Processes and between Computation and Knowledge and Mathematical Interpretation. These phenotypic differences in correlations seem to make sense: The highest correlation, between the Mathematical Application and Understanding Number categories, might reflect the fact that these two categories seem to be the most general, requiring some meta-understanding of mathematical information in addition to the ability to retrieve a mathematical fact from memory. The smallest correlations might reflect the relative specificity of simple computation versus spatial judgments and interpretation. However, the results of the phenotypic Cholesky analysis indicate that the best fitting model was one in which the variance in all five aspects of mathematics was due to factors shared among all of them in addition to factors unique to each aspect.

Next, we applied multivariate genetic analysis to determine the extent to which the five aspects of mathematics were related etiologically. The results showed that these diverse aspects of mathematics were influenced largely by the same genetic factors. On average, genetic overlap accounted for over 60% of the phenotypic correlation among the five aspects of mathematics (). Moreover, the average genetic correlation among the five aspects of mathematics was .91 (), which indicates that the same genes largely affected all five aspects of mathematics. These findings represent additional support for the generalist genes hypothesis, which proposes that genetic influences within and between cognitive abilities and disabilities largely overlap (

Plomin & Kovas, 2005).

What are the processes by which genetic influences have such broad effects? In genetics, the word

*pleiotropy* is used to refer to manifold effects of genes. It is possible that many cognitive processes might also be pleiotropic. Fundamental cognitive mechanisms—spatial attention, for example—might contribute both to calculation (attention to space on a number line) and to mental rotation. Indeed, spatial and numerical representations seem to have overlapping brain correlates (

Hubbard, Piazza, Pinel, & Dehaene, 2005;

Pinel, Piazza, Le Bihan, & Dehaene, 2004). Moreover, many behavioral protocols have shown a close connection between numbers and space, in which small numbers are represented on the left side of space and large numbers on the right. It is also possible that the adult competence for arithmetic arises from this fundamental number sense (

Hubbard et al., 2005), which might be linked to the general space sense. Both abilities also involve many nonspecific mechanisms, such as memory and speed of processing. These commonalties and many other examples of this sort might reflect pleiotropic genetic influences.

The finding that most genetic effects are general charts the course for future molecular genetic research that aims to identify genes that account for heritability. Rather than focusing on specific aspects of mathematics, attempts to identify genes would profit from focusing on what is in common among different aspects of mathematics. When such genes are found, it might be possible to use them to identify children at genetic risk for mathematical disability early enough to prevent the problem from arising (

Plomin & Walker, 2003).

The finding that genetic influences on diverse aspects of mathematics were largely general suggests that differentiation of mathematical abilities and disabilities was due to environmental rather than genetic factors. Moreover, most of this environmental heterogeneity was due to nonshared environmental influences. However, we have little idea about the sorts of mathematics-relevant environments that are nonshared and that are specific to each aspect of mathematics. A discouraging prospect is that these nonshared environmental influences might reflect many idiosyncratic experiences of very small effect, multiple gene–environment and environment–environment interactions, and other factors (e.g., in utero environment and diseases) over which we have little control. However, until research is conducted on nonshared environment specific to mathematics, it is far too early to accept such a gloomy hypothesis (

Plomin, Asbury, & Dunn, 2001).

It is possible that even more specific aspects of mathematical ability may indeed have unique genetic etiology. However, the large genetic overlap among different aspects of mathematics observed in this study speaks against this prediction. Future genetic studies using a wider range of more specific mathematical tasks can clarify this issue. Our research also suggests that if the aim of the national curriculum classification is to identify more specific strengths and weaknesses in children’s mathematical ability, this classification may need reassessment. For example, more in-depth assessment of spatial abilities may be contrasted with more specific fact retrieval ability, whereas some other distinctions might be less informative.

Another future direction for research is to investigate whether most of the variance in the diverse aspects of mathematics can be explained by factors that also influence reading and general cognitive ability. For example, one might be tempted to say that what is in common among these different aspects of mathematics is intelligence. However, our view is that this does not take us much farther in terms of understanding mechanisms, because we do not know what intelligence is any more than we know what causes the general factor that influences different aspects of mathematics. Although many brain and cognitive processes are likely to contribute to the phenotypic overlap among the subdomains of mathematics, the point of the present results is that the same set of genes is largely responsible for genetic influence in these domains (for more discussion on this issue, see

Plomin & Kovas, 2005). We have collected data on reading and general cognitive ability in addition to mathematics as part of a large Web-based battery. The next step for our research is to include these variables in multivariate genetic analyses. From previous research (e.g.,

Kovas et al., 2005) and the present study, we predict substantial overlap among genetic influences on mathematics, reading, and general cognitive ability but also some unique genetic influences on mathematics.

Finally, despite the large sample of this study, an even larger sample is needed to assess whether the small quantitative differences in etiology of the five aspects of mathematics found in this study are statistically significant. We are planning to investigate this issue when the data from the second TEDS cohort are available. Increasing the sample size will also allow us to investigate sex differences in the etiology of individual differences in mathematical ability.