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British Journal of Educational Psychology (2018)
©2018 The British Psychological Society
www.wileyonlinelibrary.com
The nature of the association between number line
and mathematical performance: An international
twin study
Maria Grazia Tosto
1a
, Gabrielle Garon-Carrier
2a
, Susan Gross
3
,
Stephen A. Petrill
4
, Sergey Malykh
1,5
, Karim Malki
6
, Sara A. Hart
7
,
Lee Thompson
3
, Rezhaw L. Karadaghi
6
, Nikita Yakovlev
1
,
Tatiana Tikhomirova
5
, John E. Opfer
4
, Mich
ele M. M. Mazzocco
8
,
Ginette Dionne
2
, Mara Brendgen
9
, Frank Vitaro
10
,
Richard E. Tremblay
1,10,11
, Michel Boivin
1,2
and Yulia Kovas
1,6,12
*
1
Laboratory for Cognitive Investigations and Behavioral Genetics, Department of
Psychology, Institute of Genetic, Neurobiological, and Social Foundations of Child
Development, Tomsk State University, Tomsk, Oblast, Russia
2
School of Psychology, Universit
e Laval, Qu
ebec City, Qu
ebec, Canada
3
Department of Psychological Sciences, Case Western Reserve University, Cleveland,
Ohio, USA
4
Department of Psychology, The Ohio State University, Columbus, Ohio, USA
5
Psychological Institute, Russian Academy of Education, Moscow, Russia
6
MRC Social, Genetic and Developmental Psychiatry Centre, Institute of Psychiatry,
Psychology& Neuroscience, King’s College London, UK
7
Department of Psychology, Florida Center for Reading Research, The Florida State
University, Tallahassee, Florida, USA
8
Institute of Child Development, University of Minnesota, Minneapolis,Minnesota, USA
9
Department of Psychology, School of Psychology, Universit
eduQu
ebec
a Montr
eal,
Qu
ebec, Canada
10
Department of Psychoeducation, Department of Pediatrics and Psychology,
Universit
e de Montr
eal, Qu
ebec, Canada
11
School of Public Health, Physiotherapy and Sports Science, University College
Dublin, Belfield, Dublin 4, Ireland
12
Department of Psychology, University of London, UK
Background. The number line task assesses the ability to estimate numerical
magnitudes. People vary greatly in this ability, and this variability has been previously
*Correspondence should be addressed to Yulia Kovas, Department of Psychology Goldsmiths University of London, LondonSE14
6NW, UK (email: y.kovas@gold.ac.uk).
a
Joint first authors.
DOI:10.1111/bjep.12259
1
associated with mathematical skills. However, the sources of individual differences in
number line estimation and its association with mathematics are not fully understood.
Aims. This large-scale genetically sensitive study uses a twin design to estimate the
magnitude of the effects of genes and environments on: (1) individual variation in number
line estimation and (2) the covariation of number line estimation with mathematics.
Samples. We used over 3,000 8- to 16-year-old twins from the United States, Canada,
the United Kingdom, and Russia, and a sample of 1,456 8- to 18-year-old singleton Russian
students.
Methods. Twins were assessed on: (1) estimation of numerical magnitudes using a
number line task and (2) two mathematics components: fluency and problem-solving.
Results. Results suggest that environments largely drive individual differences in
number line estimation. Both genes and environments contribute to different extents to
the number line estimation and mathematics correlation, depending on the sample and
mathematics component.
Conclusions. Taken together, the results suggest that in more heterogeneous school
settings, environments may be more important in driving variation in number line estimation
and its association with mathematics, whereas in more homogeneous school settings, genetic
effects drive the covariation between number line estimation and mathematics. These results
are discussed in the light of development and educational settings.
Numerical competencies, such as the awareness of quantities and the understanding of
numerical magnitudes, are considered vital precursors of counting skills (Gelman &
Gallistel, 1978), formal arithmetic (Wynn, 1992) and for the development of advanced
mathematical abilities (Dehaene, 1997). One task often used to assess individuals’
numerical magnitude representation is the number line task (Siegler & Opfer, 2003).
Trials of the task usually show a horizontal line representing a range of numerical values,
indicated by the numbers marking the edges of the line. Participants are asked to estimate
the position of target numerals, within the range, along the line. Scores indicate
participants’ accuracy based on how close their estimation is to the correct number
position. The mainstream literature suggests that numerical thinking underlying this task
relates to how individuals mentally represent quantities, how these representations are
tagged by number symbols, and how numbers are related to each other. Recent empirical
evidence suggests that number line estimation tasks entail judgements of proportions
(Barth & Paladino, 2011; Slusser & Barth, 2017).
Irrespectively of the processes involved in number line estimation, greater accuracy in
this task predicts greater achievement in mathematics (Booth & Siegler, 2008; Fuchs et al.,
2010; Geary, 2011; Sasanguie, G€
obel, Moll, Smets, & Reynvoet, 2013; Vukovic et al., 2014).
This finding has been reliably replicated across samples in different countries, Sweden
(Tr€
aff, 2013), China (Muldoon, Simms, Towse, Menzies, & Yue, 2011; Siegler & Mu, 2008),
the Amazonian tribe of Munduruku (Pica, Lemer, Izard, & Dehaene, 2004), Denmark
(Sasanguie et al., 2013), and the United States (Booth & Siegler, 2008), and across ages
ranging from preschool (4- to 5-year-olds) (Kolkman, Kroesbergen, & Leseman, 2013) to
school-age children (~12-year-olds; Lyons, Price, Vaessen, Blomert, & Ansari, 2014).
Indeed, accuracy in estimation of numerical magnitude improves with age (Laski &
Siegler, 2007; Siegler & Opfer, 2003) and children are less accurate than adults in number
line task performance (Siegler & Opfer, 2003). For example, representation of numbers 1–
10 is generally more accurate among 6- to 7-year-olds than representation of numbers 1–
100 in the same children (Laski & Siegler, 2007). This evidence suggests the involvement
of age maturation processes in the development of number line estimation skills. Further,
2Maria Grazia Tosto et al.
a male advantage in number line estimation was found in previous studies (Gunderson,
Ramirez, Beilock, & Levine, 2012; LeFevre et al., 2010; Thompson & Opfer, 2008),
indicating that sex differences may also be a contributing factor in such variations.
Experience and practice with numbers have been found to play a role in number line
estimation accuracy (Moeller, Pixner, Kaufmann, & Nuerk, 2009; Pica et al., 2004).
Modest differences in number line performance were also detected in some cross-
cultural studies. For example, 7-year-old Italian students showed better performance in
number line estimation as they committed on average less error (17.78) than their age-
matched Austrian, German-speaking peers (21.06) (Helmreich et al., 2011). Chinese 5- to
6-year-old children showed a superior number line performance compared to age-
matched children from the United States (Siegler & Mu, 2008). Conversely, no differences
in number line performance were observed between Chinese and Scottish 4- to 6-year-old
children, although the two samples differed in mathematical performance (Muldoon
et al., 2011). Considering that the mentioned studies used relatively small samples, and
more likely not representative of their populations, some of the observed cross-cultural
differences in number line performance may stem from sample bias. However, differences
may also stem from environmental differences as social context/culture (e.g., educational
norms, social constructs, or linguistic features) have also been found associated with
number line performance (Ito & Hatta, 2004; Ramscar, Dye, Popick, & O’Donnell-
McCarthy, 2011; Shaki & Fischer, 2008; Wagner, Kimura, Cheung, & Barner, 2015).
Beyond environmental differences that may underlie some of the observed cross-
cultural differences in number line performance, average differences in the frequency of
particular genetic variants across populations may contribute to the observed differences
in number line across cultures. Indeed, genes and cultures are not independent as they are
likely to co-evolve (gene–culture co-evolution, Laland, Odling-Smee, & Myles, 2010).
Genes and culture are two interacting forms of inheritance, with offspring acquiring both
a genetic and a cultural legacy from their ancestors. Genetic propensities expressed
throughout development influence what cultural organisms learn, while culturally
transmitted information expressed in behaviour and artefacts spreads through popula-
tions and may influence how genes are selected and expressed (Laland et al., 2010).
Research so far suggests that both genetic and environmental mechanisms play a role
in the development of number line estimation skills. In order to disentangle the
contribution of genetic and environmental effects, genetically sensitive studies are
required. In this study, we use a twin design with samples from four countries to estimate
the relative genetic and environmental contribution to number line task performance.
Understanding the contribution of genes and environment on number line estimation
performance is particularly relevant because of its association with mathematics. The few
genetically sensitive studies that investigated the sources of variation in different aspects
of mathematical abilities suggest the influences of genetic (heritability) and environmen-
tal factors. For example, one study conducted with 10-year-old US twins found moderate
heritability for math calculation (35%) and math fluency (34%), and slightly higher
heritability for applied problems (41%) and quantitative concepts (49%) (Hart, Petrill, &
Thompson, 2010; Hart, Petrill, Thompson, & Plomin, 2009). Similarly, a study conducted
with 10-year-old UK twins assessed on three mathematical subtests, understanding of
algebraic, understanding of non-numerical processes, and computational knowledge,
found heritability estimates between 32% and 45%, non-shared environmental influences
between 42% and 48%, and almost non-existent shared environmental contribution
(Kovas, Haworth, Petrill, & Plomin, 2007). These studies also suggested similar aetiology
among those mathematic components (Hart et al., 2009, 2010), indicated by large genetic
Number line and mathematics 3
overlaps, that is largely the same genes being involved across school achievement and test
assessing understanding of algebraic and understanding of non-numerical processes and
computational knowledge (Kovas, Haworth, Dale, et al., 2007).
Although the nature of the association between mathematics and number line
estimation is unclear, previous research suggests that common genetic factors are mainly
responsible for the covariation between mathematics and other abilities. For example,
genes in common between reading and mathematics explain between 57% and 98% of
their observed association (e.g., Haworth et al., 2009; Thompson, Detterman, & Plomin,
1991), while common genetic factors explain ~70% of the covariation between general
intelligence and mathematics (67%, Kovas, Harlaar, Petrill, & Plomin, 2005; 73%,
Trzaskowski et al., 2013), and ~60% between spatial abilities and mathematics (Tosto,
Hanscombe, et al., 2014).
The present study
The current study uses a genetically informative twin design to explore: (1) the relative
contribution of genetic and environmental factors to individual differences in number line
estimation skills. Given the phenotypic sex differences in number line, we also investigate
the contribution of genetic and environmental effects on male and female performance in
number line as well as mathematics; and (2) the extent to which genetic and
environmental factors drive the covariation between number line estimation and
mathematics assessed with two components: fluency and problem-solving. Using twins
from different countries will allow to uncover the effects of genetic and environmental
factors on the measures in different cultural–educational settings. A sample of singleton
students was also included to further understand the generalizability of the developmental
pattern observed in number line estimation, from twins to the general population
(Plomin, DeFries, Knopik, & Neiderheiser, 2013).
Methods
Participants
The four twin samples are drawn from four ongoing longitudinal twin studies in the
United States (US), Canada (CA), the United Kingdom (UK), and Russia (RU). From
each study, the samples are selected at the age of the data collection of number line
and mathematics as follows: 492 English-speaking twins (246 pairs) from the US-based
‘Western Reserve Reading and Math Project’ (WRRMP; Hart et al., 2009) with age
range between age 8 and 15 years (M=12.27, SD =1.20); 674 (mostly French-
speaking) twins (337 pairs) from the Canadian ‘Quebec Newborn Twin Study’ (QNTS;
Boivin et al., 2013) at age 15 (M=15.17; SD =0.29); 5,100 English-speaking twins
(2,550 pairs) from the UK representative ‘Twins Early Development Study’ (TEDS;
Tosto et al., 2017) at age 16 (M=16.48; SD =0.27); 178 Russian-speaking twin pairs
from the ‘Russian School Twin Registry’ (RSTR; Kovas et al., 2013) at age 16
(M=16.44; SD =0.91); and 1,456 Russian singletons with age ranging from 7.51 to
18.85 years (M=12.30; SD =3.16).
Given the wide age range of the Russian singletons and the observed age-related
developmental changes in number line performance, the singletons’ results are presented
on the sample divided by age in two groups: younger (age <15.99; M=11.42; SD =2.59)
and older (age >16; M=17.14; SD =0.74). This makes the age range of the two
4Maria Grazia Tosto et al.
singletons’ groups closer to the twin samples’ age range. Further details of samples and
their recruitment can be found in Supplementary Online Material.
Measures and procedures
The number line and mathematics tests were embedded in large cognitive and
behavioural test batteries administered to participants as part of the data collections in
the longitudinal studies. The UK, Canadian, and Russian twins completed their
assessment online. US twins were assessed in person and completed the tests in pen
and paper format. No mathematical data were available for the Russian singletons, but
they completed the same online number line task as UK, Canadian, and Russian twins.
Number line estimation task
Number Line estimation task was used to assess estimation of numerical magnitudes. The
version used was adapted from a description in Opfer and Siegler (2007).
Mathematical abilities
Data on mathematical ability were collected in two domains: fluency and problem-solving.
Problem Verification Task (Murphy & Mazzocco, 2008) and Understanding Numbers
test (NferNelson Publishing Co. Ltd, 1994, 1999, 2001) were used to measure,
respectively, fluency and problem-solving in the UK, Canadian, and Russian twins. US
twins’ fluency and problem-solving skills were assessed with Fluency and Applied
Problems, both subtests from Woodcock–Johnson III Tests of Achievement (WJ-III,
McGrew, Dailey, & Schrank, 2007; Woodcock, McGrew, & Mather, 2001). Details on the
measures and their reliability can be found in Supplementary Online Material.
Analyses
The twin method compares monozygotic (MZ) and dizygotic (DZ) within-pair twins’
correlations (intraclass correlations) to estimate the contributions of genes (h
2
, heritability),
shared (c
2
), and non-shared (e
2
) environments to individual differences (univariate models)
and of sex differences in traits (univariate sex-limitation models). The method also
allows to estimate the contribution of genes and environments to traits’ covariation
(multivariate genetic models). Details of the twin method can be found in Supplementary
Online Material.
Results
Prior to analyses, a log 10 transformation was applied to Number Line estimation to
correct for non-normality. To control for the contribution of age and sex to the twin
correlations (Eaves, Eysenck, & Martin, 1989), twin analyses were conducted on age
and sex residualized and standardized variables (mean of 0 and standard deviation of
1.0). As analyses were conducted in each sample separately, we treated outliers in
accordance with data preparation practices of each study for consistency with other
publications reporting on the variables analysed in the current study (see
Supplementary Online Material for details of previous publications using the same
measures). Scores outside 3 standard deviations were excluded as outliers from
Number line and mathematics 5
analyses of UK twins and Russian singletons; data were winsorized to 3 standard
deviations in US and Russian twins and winsorized to the 97th percentile for the
Canadian twins.
The correlation between Problem Verification (fluency) and Understanding Numbers
(problem-solving) in CA, the UK and RU were respectively: r=.57, 95% confidence
interval (CI) (.49; .64); r=.67 (.63; .68); and r=.64 (.53; .72). In the US twins, the
correlation between WJ-III subtests Fluency and Applied Problems was r=.47 (.37; .57).
Therefore, fluency and problem-solving measures were combined into a single score in
each sample (Mathematics Composite) by averaging the standardized means. Results are
reported for the single measures and the Mathematics Composite.
Phenotypic analyses
The correlation between Number Line and the Mathematics Composite was strikingly
similar in the four samples (average correlation =.43, range =.40 and .45). When
the correlation of Number Line was carried out with the mathematics components of
fluency and problem-solving respectively, some differences emerged across the samples.
The three twin samples that used the same tests (CA, the UK, and RU in Figure 1) showed
very similar correlations between Number Line estimation and Problem Verification
(fluency) (average correlation =.42, range =.38 and .44) and between Number
Line estimation and Understanding Numbers (problem-solving) (average .37, range
.36 and .38). The 95% CIs were largely overlapping, suggesting that the correlations
may not be significantly different in these three samples. In the US twins, the correlation
WRRMP
QNTS
TEDS
RSTR
Twin samples
Fluency Problem-solving Mathematics composite
enilrebmunhtiwserusaemscita
meh
tamnoitalerroC
–.10
–.15
–.20
–.25
–.30
–.35
–.40
–.45
–.50
–.55
–.60
–.20
–.44
–.38
–.43
–.49
–.3 7 –.38
–.36
–.4 5
–.40
–.42
–.45
Figure 1. Phenotypic correlations. Magnitudes of the phenotypic correlation between Number Line
and the mathematics measures are marked on the Y-axis. The whiskers represent their 95% confidence
intervals (CIs). Mathematics measures are marked on the X-axis: fluency (Problem Verification in the
United Kingdom, Canada and Russia; Fluency WJ-III in the United States), problem-solving (Understand-
ing Numbers in the United Kingdom, Canada and Russia; Applied Problems WJ-III in the United States),
and Mathematics Composite. See Table S7 for the exact values of correlations and CIs.
6Maria Grazia Tosto et al.
between Number Line estimation and Fluency (WJ-III) was weaker (.20) and the
correlation between Number Line estimation and Applied Problems (problem-solving;
WJ-III) was stronger (.49) compared to the correlation of Number Line and the same
mathematical domains in the other three samples. Summary of phenotypic correlations
between number line estimation and mathematics measures is presented in Figure 1 and
Table S7.
Descriptive statistics for each sample are presented in Table S1. The oldest samples
were the most accurate in Number Line estimation, and the youngest samples were the
least accurate. Number Line estimation scores were smaller (less error, hence more
accurate) in the older groups (Canadian, UK and Russian twins, and older Russian
singletons) than in the two youngest groups (US twins and the younger Russian
singletons). In the UK sample, the largest and the most age-homogeneous, participants
were the most accurate on average. The median of the Number Line task in the six groups
of twins and singletons shows a pattern consistent with the previously reported increase
estimation accuracy with age (Table S1).
Males were on average more accurate than females in Number Line and mathematics
measures, except for the Russian twins where females performed slightly better on the
mathematical tests. However, these sex differences were negligible, ranging between
0.0% and 3.1% of variance across samples.
Twin analyses
Univariate models
Twin genetic analyses were conducted only in the United States, Canada, and the United
Kingdom because the Russian sample is smaller than the others and currently
underpowered for these analyses. Given the differences in age and sample size, these
analyses were conducted separately in each sample.
MZ intraclass twin correlations (ICCs) were greater than DZ correlations for all measures,
suggesting some genetic influences on Number Line estimation and mathematics measures
(see Figure 2 and Table S3). Heritability estimates for Number Line estimation were modest:
.16 in the United States; .35 in Canada; .27 in the United Kingdom. The contribution of shared
environmental factors was: .34 in United States; .08 in Canada; and virtuallyzero in the United
Kingdom. Non-shared environmental factors were the most important component in all
samples: .50 in the United States; .57 in Canada; and .73 in the United Kingdom. Variance in
the mathematics measures was explained by genetic, non-shared, and shared environmental
factors in the United States, whereas in Canada and the United Kingdom, variance in
mathematics was largely explained by genetic and non-shared environmental factors. For
example, heritability of the Mathematics Composite was .46 in the United States, .59 in
Canada, and .64 in the United Kingdom; shared environmental sources of variance were
significant only in the United States (.40); non-shared environments were .14 in United States,
.41 in Canada, and .31 in United Kingdom (see Table S4).
Univariate sex-limitation models were fitted only to UK data, as the sample has the
adequate size to be divided into 5 sex-by-zygosity groups required for these analyses (Eley,
2005). ICCs for the 5 sex-by-zygosity groups are presented in Table S3. Sex-limitation
model-fitting analyses revealed no qualitative or quantitative sex differences in Number
Line estimation, Problem Verification (fluency), and Mathematics Composite (Table S5).
This suggests that, for these measures, the same genetic and environmental factors
explain individual variation to the same extent in males and females. For Understanding
Number line and mathematics 7
Numbers (problem-solving), small quantitative sex differences were detected, indicating
significantly higher heritability for females. It should be noted that aetiological sex
differences might not necessarily give rise to phenotypic sex differences (Kovas,
Haworth, Dale, et al., 2007). Small but significant differences in variances between males
and females were also observed for this measure (see footnote Table S5).
Multivariate models
In Canada and the United Kingdom, the cross-trait cross-twin correlations were greater in
MZ than DZ on all measures, suggesting genetic effects in the covariation of all measures.
In the United States, the MZ cross-trait correlations were equal or smaller than the DZ
correlations, suggesting greater environmental contribution in the covariation of the
measures (Table S6). Figure 3 depicts the extent to which genetic and environmental
factors account for the phenotypic (observed) correlations of Number Line estimation
with fluency and problem-solving measures and Mathematics Composite in the three
samples. For example, 82% of the phenotypic correlation between Number Line
estimation and Problem Verification (fluency) in the UK sample (r=.38) is explained by
common genetic factors. Non-shared environmental factors explain the remaining 18% of
the correlation.
In the US sample, a significant genetic correlation was found between Number Line
estimation and Applied Problems (problem-solving), while the association between
Number Line and the other two mathematics variables (fluency and composite) was
explained by common shared and non-shared environmental factors (see Table S7).
Conversely, genetic influences were mainly responsible for the covariation between
Number Line estimation and mathematics in Canada and the United Kingdom (Figure 3);
in these samples, almost all shared environmental correlations were non-significant while
all genetic correlations were significant (Table S7).
Figure 2. Univariate heritability estimates. The bars represent the variance of each measure
decomposed in genetic (h
2
, red portion), shared (c
2
, green portion), and non-shared environmental
(e
2
, blue portion) variances. The first block of bars represents heritability and environmental estimates for
number Line Estimation in the United States, Canada, and the United Kingdom, respectively. The second
block shows heritability for fluency (Fluency WJ-III in the United States, Problem Verification in Canada
and the United Kingdom). The third block shows heritability for the problem-solving component (Applied
Problems WJ-III in the United States, Understanding Numbers test in Canada and the United Kingdom).
The last block shows heritability for the Mathematics Composite. All the univariate heritability estimates
are statistically significant; details can be found in Table S4.
8Maria Grazia Tosto et al.
Discussion
This study investigated the genetic and environmental sources of individual differences in
estimation of numerical magnitudes on a number line task, and of the covariation between
number line estimation and mathematics measures of fluency and problem-solving. It
further explored genetic and environmental contribution to sex differences in estimation
of numerical magnitudes and in mathematics.
The results showed that sources of individual differences in Number Line estimation
and mathematics measures differed across populations. We found larger contribution of
shared environmental factors in the United States and greater influences of non-shared
environmental factors in Canada and the United Kingdom. However, the nature of
individual differences in Number Line was the same for males and females, suggesting that
Figure 3. Bivariate heritability estimates. The length of each bar equals the magnitude of the phenotypic
correlation between Number Line estimation and the mathematics variables. The portions in each bar
represent the contribution of genetic (h
2
-bivariate, red), shared environmental (c
2
-bivariate, green), and
non-shared environmental (e
2
-bivariate, blue) influences to the phenotypic correlation between Number
Line and the relevant mathematics measure. The bars are organized in three blocks showing genetic,
shared, and non-shared environmental contribution to the phenotypic correlation between: Number
Line and fluency (Fluency WJ-III in United States, Problem Verification in Canada and the United
Kingdom), Number Line and problem-solving (Applied Problems in the United States, Understanding
Numbers in Canada and the United Kingdom), and Number Line and the Mathematics Composites. The
asterisk (*) indicates when the estimate of the bivariate heritability is non-significant. Estimates of the
bivariate heritability depicted in this figure can be found in Table S7.
Number line and mathematics 9
any sex differences observed in mathematical ability (e.g., Reilly, Neumann, & Andrews,
2015) are unlikely to be related to Number Line estimation skills. The covariation between
Number Line estimation and mathematical performance was largely driven by shared
environmental component in the United States but was mainly driven by genetic factors in
Canada and the United Kingdom.
Data from all countries supported the typical developmental trend whereby younger
students were less accurate than older ones in number line performance. The same
patterns of results observed in twins were replicated in non-twin participants. This
suggests that results derived from twins in this study can be extended to the general
population.
Which factors contribute to individual differences in number line?
The genetic analyses conducted in the samples from the United Kingdom, Canada and the
United States showed that individual differences in number line estimation are largely
driven by individual-specific environments and are only modestly associated with genetic
factors. As reported by previous studies, estimation of numerical magnitudes on a number
line improves with practice, feedback, or relevant experiences (Siegler & Booth, 2004;
Siegler & Mu, 2008). Although some of these factors may be thought of as shared
environments (e.g., quality and quantity of feedback provided by the teacher), they may
act as individual-specific environments by interacting with individual characteristics.
Examples of such interaction may include perceptions and motivation associated with
engaging and practicing number line estimation skills.
Developmental factors may also be responsible for individual variation in number line
estimation. Discrepancies in the magnitude of genetic effects in Number Line
performance of the younger US twins and the genetic effects of the relatively older
Canadian and UK twins may stem from biological or maturational changes across
development. For example, previous research suggests that heritability of general
intelligence increases with age (Davis, Haworth, & Plomin, 2009; Plomin & Deary, 2015).
Some indication of developmental effects was provided by exploratory analyses
suggesting that age, rather than country or culture per se, is the main factor explaining
mean differences in Number Line estimation across samples (details in Table S2).
However, our results as to the effects of development are merely suggestive, hindered by
the differences in sample sizes and other limitations.
Discrepancies in the environmental estimates of number line for the US twins
compared to the Canadian and UK twins may stem from homogeneous school
environments existing in Canada (Quebec) and the United Kingdom. In these countries,
the Government sets both the educational levels and school curricula resulting in a unified
and standardized system across the whole territory. In addition, teachers in both countries
undergo regular standardized training. At the time of this data collection, the US Federal
Government set only the most basic educational standard levels; more specific school
policies, details of public school curricula, and teaching practices were set through local
school boards. The different school policies may give rise to more homogeneous school
environments in the United Kingdom and Canada compared to the United States. In
homogeneous environments, genetic influences may be more significant in driving
individual differences in a trait compared to environmental influences. Lower genetic
influences on cognitive abilities and achievement have been reported in US twin studies
compared to non-US twin studies (Australian and Western European samples). Such
discrepancies are explained by gene–environment interaction mechanisms whereby
10 Maria Grazia Tosto et al.
genetic effects may be suppressed in conditions of socio-economic inequality (Tucker-
Drob & Bates, 2015). A similar mechanism, related to cross-cultural differences, might also
explain why the mathematical measures showed lower heritability estimates and higher
shared environmental component in the United States than in Canada and the United
Kingdom.
Despite some differences, heritability of Number Line estimation was overall modest in all
samples. Number line estimation skills are thought to be developmentally more basic than
computational skills or more advanced mathematics; number line estimation is often
categorized as core or domain-specific numerical skills (Fuchs et al., 2010). However, the
importance of a basic skill does not mean that individual differences in this skill are genetically
driven. Another measure of basic numerical skills, non-symbolic estimation, has shown
modest influences of genetic factors in normal populations (~30%) with most of the variance
explained by non-shared environmental factors (Lukowski et al., 2017; Tosto, Petrill, et al.,
2014). More genetically sensitive and cross-cultural research at different ages is needed to
investigate possible developmental or maturational changes and the role of homogeneity/
heterogeneity of the environment in the development of number line estimation. For
mathematics, the results of this study are consistent with previous findings, suggesting that
individual differences in mathematics are driven by genetic and environmental factors to
various extents, depending on the mathematics components and sample characteristics.
Which factors explain the covariation between number line and mathematics?
In the United Kingdom and Canada, the covariation between number line estimation and
mathematics was largely driven by genetic factors (85% on average) with the remaining
portion of the phenotypic correlation driven by shared and non-shared environment. In
the United States, the covariation between number line estimation and mathematics was
more strongly driven by shared environmental factors (63% on average) compared to
genetic factors (27% on average). In all the samples, non-shared, individual-specific
environments had small or non-significant influence.
It is possible that the different pattern of association between accuracy in Number Line
estimation and mathematics reflects cultural differences across Canada, the United
Kingdom, and the United States. In the presence of heterogeneous environments such as
varying school curricula (in the US sample), environmental rather than genetic factors are
the driving force shaping up individual differences in number line estimation and of its co-
variation with mathematics. In the presence of more homogeneous common environ-
ments (school), genetic factors (of modest influence) drive variations in number line
estimation in addition to the individual-specific environments (in Canada and United
Kingdom). It needs to be noted that the observed correlation between mathematics and
number line estimation was overall small to modest in all samples. Thus, even if the
genetic and environmental factors contributing to the association were completely
overlapping, Number Line and mathematics remain largely different domains.
Are there sex differences in number line and in mathematics?
Mean sex differences were negligible in all twin samples for all measures, suggesting that
the observed sex differences in mathematical ability are unlikely to be related to number
line estimation. The proportion of genetic and environmental contributions to variation in
Number Line estimation, the Mathematics Composite, and fluency (Problem Verification)
was also the same for males and females.
Number line and mathematics 11
Only the mathematical component of problem-solving (Understanding Numbers)
showed small aetiological –quantitative and variance –sex differences in the United
Kingdom. This suggests that, in this mathematical component, the same genes and
environments drive individual differences in boys and girls at age 16, but the magnitude of
their effect is different for males and females. The test Understanding Numbers is designed
according to UK school curricula and can be considered a good index of mathematical
achievement. Previous studies in the UK twins have consistently reported small mean
male advantage in mathematics, but no aetiological sex differences (Kovas, Haworth,
Dale, et al., 2007). However, age 16 marks the first time that aetiological sex differences
are detected in mathematics school achievement (General Certificate of Secondary
Education mathematics exams) in this sample (Shakeshaft et al., 2013).
Limitations and conclusion
Diversity in size, age, and schooling guarded against a formal comparison of the UK, US,
and Canadian samples on genetic analyses. One alternative would have been using a
subset of the UK sample to match in size the other two; however, any criteria for carrying
out the matching (e.g., on the basis of socio-economic status, IQ, verbal, non-verbal
ability) would have introduced different confounding elements on the causes of
discrepancies or similarities across the samples. Further, matching would have resulted
in sample-size reduction and we preferred to carry out analyses on the largest number of
participants available to provide reliable heritability estimates (Martin, Eaves, Kearsey, &
Davies, 1978; Martin, Eaves, & Kendler, 1994).
Some discrepancies in results may have stemmed from measurement differences.
For example, the questions of the two problem-solving tests, Understanding Numbers
and Applied Problems, are designed to assess similar cognitive abilities and similar
mathematical domains, but the items are different in the two tests. In the US sample,
all tests were administered in person; in particular, the pen and paper Number Line
test had a very strict administration protocol. This may have generated more accurate
measurements compared to the online tests. Some evidence in support of accuracy
can be found in the US heritability estimates where all the measures showed the
lowest non-shared environment, which in twin model-fitting includes measurement
error. However, US non-shared environment for pen and paper administered Number
Line was very similar to that of the Canadian Number Line test administered online. In
addition, the Number Line tests yielded adequate test–retest reliability and internal
validity in all samples, although the latter varies widely across samples (from .63 to
.95). Thus, the discrepancies in the number line results are unlikely to be affected by
differences in administration mode.
As our study could control for age only to some extent, future studies should include age-
homogeneous twin sample from different countries in order to provide further support to
the explanatory role of age in the observed cross-cultural differences. Future research is also
needed to explore the role of other cultural factors such as the reading/writing systems in
number line estimation (Shaki & Fischer, 2008). In our study, all samples had the same left-
to-right writing direction and therefore were not suited to investigate these effects.
These limitations notwithstanding, this study is the first genetically sensitive
investigation into number line estimation skills and mathematics that use data from
twins from different countries. Although the results are not ready to be translated into the
real-world practice, they provide novel insights into the aetiology of individual differences
in number line performance and its covariation with mathematical ability.
12 Maria Grazia Tosto et al.
Acknowledgements
This research was supported by Grant from the Russian Science Foundation (project # 14-48-
00043) to Tomsk State University; by the Russian Science Foundation (project 15-18-30055); by
a Programme Grant (G0901245; previously G0500079) from the U.K. Medical Research
Council (MRC); by Eunice Kennedy Shriver National Institute of Child Health and Development
(HD059215; HD038075), and (HD075460); by the Qu
ebec Ministry of Health, Fonds
Qu
eb
ecois de la Recherche sur la Soci
et
e et la Culture; the Fonds de la Recherche en Sant
e
du Qu
ebec; the Social Science and Humanities Research Council of Canada; the National Health
Research Development Program; the Canadian Institutes for Health Research; and Sainte-
Justine Hospital’s Research Center. Michel Boivin is supported by the Canada Research Chair
(Tier 1) programme.
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Supporting Information
The following supporting information may be found in the online edition of the article:
Table S1. Means and standard deviation on all measures for each sample.
Table S2. ANOVAs comparison of Number Line estimation and Mathematics means in
groups with restricted age range.
Table S3. Intraclass correlation (ICC). Twins similarity in Number Line estimation and
mathematics.
Table S4. Heritability estimates from twin model-fitting for Number Line estimation
and mathematics measures.
Table S5. Sex limitation model fitting and parameter estimates for males and females
separately (UK TEDS sample only).
Table S6. Cross-trait twin correlations between Number Line estimation and
mathematics measures.
Table S7. Genetic and environmental correlations; bivariate heritabilities; phenotypic
correlations.
Table S8. Bivariate model fitting between Number Line (NL) and mathematics
measures.
Figure S1. Phenotypic correlations between Lambda scores and the absolute mean
error scores of Number Line estimation task for the twin samples.
Figure S2. Phenotypic correlations between Lambda scores and the absolute mean
error scores of Number Line estimation task for the Russian singletons.
`
Number line and mathematics 17
