corpus.ulaval.ca€¦ · Web viewEarly Developmental Trajectories of Number Knowledge and Math Achievement From 4 to 10 Years: Low-Persistent Profile and Early-Life Predictors

PREDICTORS OF LOW NUMBER KNOWLEDGE

Early Developmental Trajectories of Number Knowledge and Math Achievement From 4

to 10 Years: Low-Persistent Profile and Early-Life Predictors

Gabrielle Garon-Carrier1, Michel Boivin1,2, Jean-Pascal Lemelin3, Yulia Kovas4,5, Sophie Parent6,

Jean Séguin7,8, Frank Vitaro6, Richard E. Tremblay2,9,10, & Ginette Dionne1

1 School of Psychology, Université Laval, Canada2 Institute of Genetic, Neurobiological, and Social Foundations of Child Development, Tomsk

State University, Tomsk, Russian Federation3 Department of Psychoeducation, Université de Sherbrooke, Canada

4 Department of Psychology, University of London, Goldsmiths, England5 Laboratory for Cognitive Investigations and Behavioural Genetics, Tomsk State University,

Russian Federation6 Department of Psychoeducation, Université de Montréal, Montréal, Canada

7 Department of Psychiatry, Université de Montréal, Canada8 CHU Ste-Justine Research Center, Université de Montréal, Montréal, Canada

9 Department of Pediatrics and Psychology, Université de Montréal, Canada10 School of Public Health, Physiotherapy and Population Sciences, University College Dublin,

Ireland

Correspondence concerning this article should be sent to Michel Boivin, CRC in Child

Development, Professor, École de psychologie, Université Laval, Québec, Canada, G1K 7P4.

Email: [email protected]

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mailto:[email protected]


Research Highlights

Abstract

Little is known about the development of number knowledge (NK) and the antecedents of low-

persistent NK profiles in early childhood. We documented the developmental trajectories of NK

across the transition from preschool to elementary school, their predictive validity with respect to

Number knowledge development is not linear, and vary in onset level and rate of progression from preschool to school entry.

10% of preschool children show developmental lag in number knowledge, signaling a clear risk of math underachievement.

Poor early cognitive development, memory and visual-spatial skills uniquely predict children with low number knowledge.

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later math achievement, and the child and family early-life factors associated with low NK

profiles. Children’s NK was assessed four times at regular intervals between the ages 4 and 7

years in a large, representative population-based sample. Developmental trajectories of NK were

established for 1,597 children. These children were also assessed with respect to several features

of their family environment at 5, 17, and 29 months, as well as their cognitive skills at age 41

months. Analyses revealed a best-fitting 4-trajectory model, characterized by Low-Increasing

(10% of the children), Moderate-Increasing (39%), Moderate-Fast Increasing (32%) and High-

Increasing (19%) groups. Children of these trajectory groups differed significantly with respect

to math achievement at ages 8 and 10 years, with the Low-Increasing group persistently scoring

lower than the other groups throughout these years. Children of Low-Increasing NK group were

from household of lower income and father with low educational background, poorer early

cognitive development, and more importantly, reduced visual-spatial skills and memory span.

Children displaying reduced cognitive abilities and impoverished living conditions early in life

are at greater risk of low NK throughout late preschool and school entry, with ensuing

difficulties in math achievement. They deserve early preventive attention to help alleviate later

mathematic difficulties.

Keywords: Number knowledge, Mathematics achievement, Developmental trajectories, Early-

life predictors, Longitudinal study

Early Developmental Trajectories of Number Knowledge and Math Achievement From 4 to 10

Years: Low-Persistent Profile and Early-Life Predictors

Knowledge and skills in mathematics have been shown to predict later academic

achievement (Duncan et al., 2007; Jordan, Kaplan, Ramineni, & Locuniak, 2009; Magnuson,

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Duncan, Lee, & Metzger, 2016; National Research Council, 2009; Nguyen et al., 2016; Siegler et

al., 2012), and more generally, later educational attainment (Magnuson et al., 2016). Mathematic

abilities are paramount for college entry (Sadler & Tai, 2007) and degree completion in STEM

fields (science, technology, engineering and mathematics; Wolniak, 2016). Unfortunately, the

negative side of this predictive association is that individuals with poor mathematic abilities have

reduced educational and employment opportunities (e.g., low rates of full-time employment and

promotions, high rates of low-paying occupations; Lundetræ, Gabrielsen, & Mykletun, 2010;

Parsons & Bynner, 1997), and might even experience difficulties in common day-to-day

activities in adulthood.

The early signs of these difficulties are quite prevalent. For instance, between 6% and

10% of children suffer from learning disabilities in mathematics (Barbaresi, Katusic, Colligan,

Weaver, & Jacobsen, 2005; Shalev, Auerbach, Manor, & Gross-Tsur, 2000), and many more

struggle with mathematics without a formal diagnosis. Children showing persistent difficulties in

math may never catch up to their grade-level peers. There is also good evidence that these

difficulties emerge quite early, even before school entry (Mazzocco & Thompson, 2005; von

Aster & Shalev, 2007).

Early Development of Mathematic Skills

Children develop a broad array of early mathematic skills well before school entry

(Libertus, Feigenson, & Halberda, 2011). As early as 6 months, they possess an intuitive

impression of numbers – a number sense – that allows them to approximate magnitude difference

between small set of objects (e.g., 4 vs. 8; Feigenson, Dehaene, & Spelke, 2004; Hyde & Spelke,

2011; Lipton & Spelke, 2003; McCrink & Wynn, 2007; vanMarle, 2013; Xu, Spelke, &

Goddard, 2005). Number sense is assumed to provide the basic meaning of number and quantity

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in infancy (von Aster & Shalev, 2007). From these initial number sense skills, children learn

counting through linking numbers to objects. Children’s numerical development is thus

characterized by an increasing capacity to solve mathematic problems using tangible materials

(e.g., blocks). Around age 3 years, children typically learn number words, counting principles

(e.g., one-to-one counting, cardinality, ordinal number, numerical identification; Bermejo, 1996),

and to carry out simple operations (e.g., number combinations, see Geary, 2004, for a review).

This early knowledge about numbers is key to later mathematical development (Duncan et al.,

2007; Göbel, Watson, Lervåg, & Hulme, 2014; National Research Council, 2009; Nguyen et al.,

2016; Watts, Duncan, Siegler, & Davis-Kean, 2014).

Number knowledge (NK), that is, the conceptual and procedural understanding of whole

numbers (Okamoto & Case, 1996), has been posited to develop steadily and gradually

throughout early childhood (Piaget, 1977), and to lead to more sophisticated mathematic abilities

(Duncan et al., 2007; Göbel et al., 2014; National Research Council, 2009; Nguyen et al., 2016;

Watts et al., 2014). Specifically, it has been suggested that early NK partially originates from an

integration of number sense skills and the symbolic numerical system taught at home or in

school (Feigenson, Libertus, & Halberda, 2013; Libertus et al., 2011). Number sense skills, such

as subitizing and approximating would prepare children to associate quantities with Arabic digits

(i.e., numeral symbols, such as 0, 1, 2 or 3; von Aster & Shalev, 2007). This, in turns, leads to

hierarchically order numbers, a stepping-stone in children’s mastering of numbers and growing

math abilities (Friso-van den Bos et al., 2015; Siegler & Booth, 2004).

Thus, there is both theoretical and empirical support for the view that early NK and the

ensuing mathematic skills are developmentally interlocked (Duncan et al., 2007; Göbel et al.,

2014; National Research Council, 2009; Nguyen et al., 2016; Piaget, 1977; Watts et al., 2014).

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However, very little is known about individual differences in the development of these skills. If

the development from NK to mathematics generally follows age and grade levels, the rate at

which children transit through this period may differ significantly; some children quickly master

mathematic concepts and operations while others struggle.

Documenting early NK skills is especially crucial during the transition from late

preschool to school entry, as this period is characterized by substantial developmental changes.

This transition not only coincides with shifts in physical, cognitive, emotional, and behavioral

capacities (Blair & Raver, 2015; Sasser, Bierman, & Heinrichs, 2015; Welsh, Nix, Blair,

Bierman, & Nelson, 2010), but it also entails a modification of the learning context, as children

start to be systematically exposed to formal training in early mathematic skills. Accordingly, one

goal of the present study is to assess inter-individual variations in trends of NK development

across the period from late preschool to school entry.

Early Cognitive Correlates of Number Knowledge

An important related question to that of early individual differences in NK is the issue of

their associated cognitive factors and putative environmental determinants. Previous studies of

normative samples, as well as of low math achievers and children with mathematic learning

disability have consistently revealed that poor language, visual-spatial skills, and memory-span

are correlates, if not precursors of low math skills (Bull, Espy, & Wiebe, 2008; Dehaene, Piazza,

Pinel, & Cohen, 2003; Dehaene, Spelke, Stanescu, Pinel, & Tsivkin, 1999; Geary, 2004; Geary,

2011; Geary, Hoard, Byrd-Craven, & DeSoto, 2004; LeFevre et al., 2010; Mazzocco &

Thompson, 2005; Soto-Calvo, Simmons, Willis, & Adams, 2015). Language is involved when

manipulating information within working memory, as well as when counting forward and

backward. Difficulties in processing numbers have been associated with reading difficulties,

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language impairment, or both (Jordan, Hanich, & Kaplan, 2003). Clearly, the phonetic and

semantic systems are activated when counting, if only to connect the quantities with number

words (Vukovic & Lesaux, 2013), and to solve arithmetic problems (Jordan et al., 2003).

Deficits in these systems might result in difficulties in counting and arithmetic reasoning, as well

as in concurrent reading difficulties (Dehaene & Cohen, 1995, 1997). The visual-spatial system

is also solicited when representing conceptual knowledge; visual-spatial skills are indeed

involved in basic geometry problems, in magnitude comparisons (Dehaene et al., 1999), in

logical thinking and problem solving, as well as in estimating and mentally manipulating

numbers encoded as spatial forms (e.g., number line; see Zorzi, Priftis, & Umiltà, 2002).

Children experiencing difficulties in mathematics also fail to correctly encode and/or retrieve

basic arithmetic facts from memory. Their problem typically arises from persistent difficulties in

memorizing relevant and inhibiting irrelevant information during facts retrieval (Bull et al.,

2008; Geary, 2011; Geary et al., 2004).

Thus, several studies clearly support the association of language, visual-spatial skills, and

memory with different math components (Dehaene et al., 2003; Dehaene et al., 1999; Geary,

2011; LeFevre et al., 2010; Soto- Calvo et al., 2015). However, the prediction of NK from these

cognitive abilities very early in life has not been well documented. This question will be

investigated in the current study.

Early Family Correlates of Number Knowledge

Children’s early NK is also context-dependent. Environmental adversities, such as poor

living and educational conditions (e.g., low socioeconomic status) have been associated with low

math skills (Griffin, Case, & Siegler, 1994; Jordan & Levine, 2009; Jordan, Kaplan, Oláh, &

Locuniak, 2006; Klibanoff, Levine, Huttenlocher, Vasilyeva, & Hedges, 2006; Siegler, 2009).

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For example, Levine and colleagues (2010) have documented substantial socioeconomic status

(SES) differences in mothers’ number talk to children aged 14 and 30 months. Compared to

mothers of high SES, mothers of low SES provided more input of simple verbal counting, but

less of advanced NK skills (e.g., numerical magnitude estimation), and this difference predicted

knowledge of number words cardinality at 46 months (Levine, Suriyakham, Rowe, Huttenlocher,

& Gunderson, 2010).

However, research linking SES and NK is limited with respect to the key factors

underlying this association. For instance, SES subsumes income, a family-level characteristic,

and educational levels, which may vary within family. These dimensions could have different

contributions to NK (Melhuish et al., 2008), and point to the specific role of mother and father in

the development of NK (McBride, Dyer, Liu, Brown, & Hong, 2009; Viljaranta, Lazarides,

Aunola, Räikkönen, & Nurmi, 2015). Furthermore, parents’ outcome expectancies (Bandura,

1997; Benasich & Brooks- Gunn, 1996; Bornstein, 2002; Kouimtzi & Stogiannidou, 2009; Parke

& Buriel, 1998), that is, the belief that their parenting behaviors are important for their child’s

development could account for providing a stimulating environment, such as exposing and

engaging young children in numerical activities (e.g., cooking, board games, moving along

number paths, Laski & Siegler, 2014; LeFevre et al., 2009; Levine et al., 2010; Ramani, Siegler,

& Hitti, 2012; Siegler & Ramani, 2009). This should be investigated more systematically.

The Present Study

Few studies have longitudinally documented individual variations in the development of

NK across the late preschool and school transition periods. Previous studies have provided

evidence for the relevance of mathematical skills at school entry for academic outcomes up to

age 9 and 10 (Aunola, Leskinen, Lerkkanen, & Nurmi, 2004; Duncan et al., 2007; Jordan et al.,

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2009; Jordan, Glutting, & Ramineni, 2010; Nguyen et al., 2016). However, few studies have

been conducted on NK as early as age 4 years. NK has only been studied from age 5-6 years

and/or only at single time-points (Aunola et al., 2004; Jordan et al., 2009; Jordan et al., 2010;

Nguyen et al., 2016), providing little information on its early development and validity in

predicting later math achievement and difficulties.

In the present study, we argue for the identification of subsets of developmental

trajectories based on repeated measures of evolving NK during the preschool and early school

years. To reflect the evolving and discrete nature of NK, we used the Number Knowledge Test

(Okamoto & Case, 1996), which provides a four-level assessment of several aspects of numerical

competence (see Method). Such an approach posits heterogeneity in the developmental

trajectories, and thus identifies homogeneous subgroups with respect to the evolution of NK over

the targeted period of time, including children that show developmental lags in NK. Indeed, as

some children may experience learning difficulties at various times in their NK development, it

is therefore important to distinguish children with persistent difficulties from those with

normative transient difficulties. Previous studies have often used arbitrary cut-offs to establish

groups or patterns of NK and mathematic skills (e.g., above 10% vs. over 90%; 2 SD above and

over the mean). For example, one study classified children with mathematic scores under the 15th

percentile as children with mathematic learning disability, and those scoring between the 15th and

30th percentiles as math low achievers (Geary, Bailey, & Hoard, 2009). Here, we used a

clustering procedure based on semi-parametric modeling (Nagin, 1999). This quantitative

approach offers several advantages over uniform longitudinal analysis, such as growth curve

analysis (Jordan et al., 2003; Jordan et al., 2006; Jordan, Kaplan, Locuniak, & Ramineni, 2007).

First, it considers individual differences in trajectories to determine the optimal number of

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groups needed to describe different patterns of change over time. Second, it does not make

strong assumptions about the population distribution of the putative developmental trajectories.

Third, the model defines the form of the trajectory for each potential cluster of trajectories,

which allows for the possibility that subgroups of children show distinct developmental trends in

NK, thus providing a more nuanced view of NK development, and one that could reveal non-

linear changes in development (see Leblanc, Boivin, Dionne, Tremblay, & Pérusse, 2008, for a

more extensive discussion of these points).

Furthermore, from a preventive perspective, not only is it essential to document the

various early developmental pathways in NK, but it is also crucial to (a) estimate from a person-

centered perspective (Laursen & Hoff, 2006) whether and how some of these developmental

profiles foresee later difficulties in mathematics, (b) identify early risk factors that predict low

and enduring profiles of NK over time, and (c) forecast a possible evolution into mathematic

difficulties. Identifying specific predictors of low NK may help preschool and childcare

practitioners to better understand the cognitive and family profile of children with low NK, with

the hope of guiding preventive efforts aimed at children at risk of failure in school.

The study also provided a unique opportunity to test for possible sex differences in

number knowledge and achievement in mathematics. Previous studies found differences favoring

boys in number knowledge. Jordan et al. (2006) found small but significant sex effects on

calculation with objects and on numerical estimation in kindergarten (mean age = 5.8 years);

boys had an edge over girls even when income level, age, and reading ability were controlled for

in the analyses, and more boys than girls were classified in the highest performing group.

However, several studies found no such differences in later math performance (Lachance &

Mazzocco, 2006; Levine, Jordan, & Huttenlocher, 1992). To date, only a few longitudinal

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studies have tested for the possible sex difference in both number knowledge and to its

prediction to math achievement.

Accordingly, the following research questions were addressed in the present study: (1)

Using a group-based modeling approach, are there discrete patterns of developmental trajectories

of NK? At least three putative trajectories of NK were expected: a small group of children with

high level of NK, a larger group, likely the majority of children, with an average/moderate level

of NK, and a small but significant group of children with low NK profile (Jordan et al., 2006).

All these groups were expected to gain in NK over time, but we did not have a specific

hypothesis regarding the rate of their relative gains. Based on Jordan et al. (2006), we also

expected more boys than girls to fall on a high-level trajectory of NK.

(2) Does early NK predict later math achievement? To answer this question, we

compared school-based mathematic achievement in 2nd and 4th grade (age 8 and 10) according to

NK trajectories and sex. This period is a time when mathematics becomes more complex and

differentiated. We expected that children falling on the lowest trajectory would show the lowest

math achievement scores at both ages, whereas those of the highest NK trajectory-group would

display higher scores, again, boys having higher scores than girls.

(3) To what extent can we predict a child’s trajectory of low NK from specific early

cognitive and family predictors? Cognitive factors, specifically receptive vocabulary, visual-

spatial skills, and memory-span were considered central in the model. Also included was an

assessment of early general cognitive development at 29 months to account for initial

developmental gains in cognitive skills, and thus more precisely estimate the unique

contributions of these specific skills to NK. Because family factors may also play a role in the

development of both the specific cognitive factors and NK, they were assessed early in

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development. Here we distinguished between family income and parent’s education, and

considered early parenting outcome expectancies (i.e., the parents’ beliefs that their parenting

behaviors are important for their developing child). We expected both lower cognitive skills and

poorer family background factors to independently predict membership in a trajectory of low

NK.

Method

Participants and Setting

Participants were from the Quebec Longitudinal Study of Child Development (QLSCD),

a longitudinal population-based study aimed at understanding the role of early childhood in later

school adjustment and academic achievement. Participating families were recruited through the

Quebec Master Birth Registry of the Ministry of Health and Social Services to be representative

of children born in 1997-98 in the province of Quebec, Canada. For practical reasons, data could

not be collected on children living on Cree or Inuit territories, in Indian reserves, and in northern

Quebec, and thus, were excluded from the study. A stratified three-stage sampling design based

on living area and birth rate was used. The territory covered by the survey was first divided into

primary sampling units (regions), which were divided into second-stage units composed of one

or two county regional municipalities, and then further divided in third-stage units according to

the number of births registered in 1996. All selected infants had been born after October 1, 1997

to ensure that they would enter school the same school year. Families were excluded if the

mother could not speak French or English, and if babies were born before 24 weeks or after 42

weeks of gestation. Those with a gestation period of less than 24 weeks could not be retained

because they had a higher risk of mortality between entry in the register and the conducting of

the survey. Similarly, births occurring after 42 weeks of gestation had to be excluded because the

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delay in selection would have meant waiting until they became available for the sampling frame.

These two a priori exclusions represented approximately 0.1% of registered births of the date

collection.

A sample of 2,940 families with newborns was initially identified with the goal of

obtaining reliable longitudinal statistics within the budget. Selected families that could be located

(N = 2,675) were approached by mail and by phone. Of those, 2,223 families were first visited

when the child was 5 months old (83%), and 2,120 of them (79% response rate) were followed

longitudinally and regularly assessed on various child and family characteristics (Jetté & Des

Groseillers, 2000a, 2000b).

The QLSCD children were of Canadian, European, British or French origin in 82.5% of

the cases; 12.1% were of American, African, Haitian or another ancestry, and 0.5% were of an

unidentified ethnic origin. At 5 months, 31.4% reported an income lower than 30,000 Canadian

Dollar, 17.9% of mothers and 17.6% of fathers had no high school diploma, and 17% were

reconstituted or one-parent families. French was the most frequent language spoken at home

(75.2%), followed by English (10.1%). The remaining families spoke other languages or a

combination of languages (Jetté & Des Groseillers, 2000a, 2000b). Other details of the cohort

study can be found at: http://www.iamillbe.stat.gouv.qc.ca/publications/baby_no1.pdf

The present study focused on data relevant to the study aims and collected in infancy and

the early school period. The study covers the preschool and early school periods, and uses data

collected when children were ages 5 months (N = 2,120; age: M = 5.0 months, SD = .48), 17

months (N = 2,045; age: M = 17.1 months, SD = .49), 29 months (N = 1,997; age: M = 29.1

months, SD = .47), 41 months (N = 1,950; age: M = 41.1 months, SD = .52), and then at ages 4

years (N = 1,944; age: M = 50.0 months, SD = 3.09), 5 years (N = 1,759; age: M = 61.9 months,

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http://www.iamillbe.stat.gouv.qc.ca/publications/baby_no1.pdf


SD = 3.09), and 6 years (N = 1,492; age: M = 73.8 months, SD = 3.00), and in the following

school years: grade 1 (N = 1,528; age: M = 7.15 years, SD = 3.06), grade 2 (N = 1,451; age: M =

8.10 years, SD = .26), and grade 4 (N = 1,334; age: M = 10.14 years, SD = .26).

At age 4 years, a minority (16.7%, N = 324) of the participants attended a nursery school,

a preschool, or educational activities offered by a municipality or a community or recreational

center on a regular basis (M = 8.59 hours/week; SD = 8.88). At age 5 years, 47.5% (N = 835)

attended daycare, and 37.5% (N = 659) attended junior or senior kindergarten at a full day or

half-day frequency.

Starting at age 4, all data collections were done in the winter and spring of the year. The

average attrition rate from ages 5 months to 10 years was 4.0% per year, although it varied

slightly across measures.

Measures

Number knowledge. NK was measured through an adapted version of Okamoto and

Case (1996) Number Knowledge Test (NKT; Côté et al., 2013; Duncan et al., 2007; Romano,

Babchishin, Pagani, & Kohen, 2010). The NKT was developed with the goal of documenting

children’s understanding of whole numbers and basic operations, and as a tool for teachers to

identify children with mathematic difficulties (Gersten, Clarke, & Jordan, 2007). The NKT has

four levels of complexity (from 0 to 3). Each level of the test reflects a current developmental

stage of children’s NK conceptions (Gersten et al., 2007; Gersten, Jordan, & Flojo, 2005). The

baseline level (level 0, around ages 3-4) requires children to count small sets of tangible objects

(e.g., The experimenter shows five unordered chips to the child, and then asks him to count the

chips; The experimenter shows image with two piles of chips, and then asks the child which pile

has more chips). The first level (around age 6) measures uni-dimensional mental representations

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of numbers, i.e., numerical comparison, with items that were designed to probe for the “mental

number line” structure (e.g., Which number is closer to 5: 6 or 2?). The second level (about age

8) reflects bi-dimensional representations, i.e., understanding numerical “difference” (e.g., How

many numbers are there in between 7 and 9?), and the base-ten system with double-digit

numbers (e.g., What number comes 5 numbers after 49?). The third level (age 10) reflects

integrated bi-dimensional mental representations of numbers, i.e. constructing and comparing

two sums or differences rather than just one (e.g., which difference is smaller: the difference

between 48 and 36 or the difference between 84 and 73?). At this level, children also have to

manage triple-digit numbers and/or to solve more complex problems involving double-digit

numbers.

In the present study, the adapted NKT (Côté et al., 2013; Duncan et al., 2007; Romano et

al., 2010) was administered by a trained research assistant at ages 4 (N = 1,768), 5 (N = 989), 6

(N = 1,189) and 7 (N = 1,461) years. Five items of the baseline level (level 0) were slightly

modified to facilitate the administration of the test, but aimed for the same knowledge and skills

as the original items (see Table S1 in Supplementary materials). The items of the next levels

were identical to the original test. To reduce the length of the assessments, alleviate fatigue, and

preserve children’s motivation, the administration of the NKT slightly departed from Okamoto

and Case (1996). According to Okamoto and Case (1996), only children who correctly answered

at least 60% of the items at one level should move to the next. Here, the procedure was

systematized differently so that the same levels would be administered to all children at a given

age, thus insuring within-year standardization. Specifically, at ages 4, 5, and 6 years, only the

levels 0 and 1 were systematically administered, and the test was stopped if the child made three

consecutive errors at level 1. Thus, up to age 6, the procedure started at the baseline level and

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moved to level 1 until the child failed three consecutives items. At 7 years, the baseline level was

omitted and levels 1 and 2 were administered. The test was stopped if the child made three

consecutives errors at level 2. The score consisted of the total number of correct items across

levels. Points were automatically assigned for the baseline level items if testing began at level 1

(for children aged 7 years).

With the exception of the low reliability at age 5 (α = .55), the internal consistency of the

adapted-NKT in our sample was found to be adequate (α = .68 at age 4, .92 at age 6, and .79 at

age 7 years); and the test-retest stability was high across all time points: r = .74 between ages 4

and 5; r = .92 between ages 5 and 6; and r = .82 between ages 6 and 7. We also tested the

measurement-invariance of the adapted-NKT across time using Mplus 7.11 (Muthén & Muthén,

1998–2012). The test was invariant over time, with non-significant difference in the model fit

when comparing the non-constrain model to model with constrained to equality factor loadings

for matching items of NK, ∆χ2 (1) = .008, p = .78 (results available from the authors). Scores

were normally distributed at ages 6 and 7 years, but a small trend toward bimodal distribution

was found at both ages 4 and 5 years, which again justifies the use of a modeling approach that

does not assume normal distribution (Nagin, 1999).

Achievement in mathematics in elementary school. Achievement in mathematics in

elementary was assessed through the Mathematics subtest of the Canadian Achievement Test

(CAT), an evolving Canadian-based subtest reflecting grade appropriate math achievement in

elementary school (Canadian Test Center, 1992; Carter, Dubois, & Ramsay, 2010). Using a

multiple-choice response and a pen-and-paper formats, the CAT was administered when children

were in grade 2 (age 8) and in grade 4 (age 10). The grade 2 version of the CAT assessed

children’s capacity to perform addition (7 items), subtraction (8 items), and multiplication (6

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items) operations. The grade 4 version assessed the ability to compute slightly more complex

(compared to grade 2) addition (5 items), subtraction (5 items), multiplication (6 items)

operations, and also required to compute division (4 items) operations. Children had to select the

right answer out of four choices within a given time (between 45 and 60 seconds depending on

the item difficulty). Children had to move to the next item when the time was up. The test ended

when three successive errors were made within a type of operation (e.g. three successive errors

in subtraction operations). We computed a total raw score for each child by summing the number

of correct items across all types of operation. The internal consistency of this measure in our

sample at ages 8 and 10 was α = .76, and α = .81, respectively, and its grade 2-4 stability was r =

.49 (Garon-Carrier, Boivin, Ouellet, Tremblay, & Dionne, in preparation).

Family predictors. Data on two categories of family predictors—socio-demographic

factors and parenting outcome expectancies—were collected.

Socio-demographic factors. We collected socio-demographic data through questions

from the National Longitudinal Survey of Children and Youth (see Willms & Shields, 1996)

when the child was 5-month-old. Household income was assessed on a 9-point scale ranging

from 1 ($10,000 or less/year) to 9 ($80,000 or more/year). Mother and father school attainment

was measured as their highest diploma, as revealed through a 4-point self-report scale ranging

from 1 (no high school diploma) to 4 (undergraduate degree). These rank-ordered variables were

normally distributed, and were thus treated as continuous variables in the following analyses (see

Pasta, 2009; Powers & Xie, 2008).

Parenting outcome expectancies. Mother’s and father’s parenting outcome expectancies

were measured through a subscale (5 items) of the Parental Cognitions and Conduct Toward the

Infant Scale (PACOTIS) (Boivin et al., 2005), a self-report questionnaire designed to assess

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parenting perceptions and behavior tendencies toward a recently born infant. For most children

(i.e., 82% of them), both mothers and fathers filled out the questionnaire when their child was

aged 5 months, 17 months and 29 months. They had to indicate on a continuous 11-point scale

(i.e., from 0 to 10) to what extent each statement regarding the perceived impact of their

behavior on their developing child accurately described their actions or thoughts (e.g.,

Regardless of what I do, my baby will develop on his/her own; My behavior has little effect on

the development of emotions (happiness, fear, anger) in my baby, etc.). The item scores were

reversed to reflect perceived parenting impact/expectancies. These parenting expectancy items

presumably reflect the quality of parents’ involvement vis-à-vis their child (see Boivin et al.,

2005 for a detailed description of the construction and validity of the scale). The measure

revealed acceptable internal consistency in our sample from 5 to 29 months (α = .71–.78). For

both mother and father, we averaged the three time-specific scores into a global parenting

outcome expectancy score.

Child Cognitive Abilities. Data on four child cognitive abilities—early cognitive

development, memory span, receptive vocabulary, visual-spatial skills—were collected.

Early cognitive development. When the child was aged 29 months, the person “most

knowledgeable about the child” (the mother 99.7% of the time) reported the child’s early

cognitive development through a 3-item scale (naming four colors, counting three objects and

pronouncing partial sentence of three words or more). The three items were moderately

correlated (.26 and .31) and summed to compute a total score of early cognitive development.

This measure was also correlated to the other measures of child cognitive abilities at 41 months,

i.e., with memory-span (r = .22), receptive vocabulary (r = .33), and visual-spatial skills (r

= .20).

18


Memory-span. Children were also assessed with respect to their memory-span at age 41

months through the Visually Cued Recall task (VCR; Zelazo, Jacques, Burack, & Frye, 2002), a

reliable (α = .95 in our sample) measure of the child’s incremental capacity to encode visual

items and to recall the spatial locations of the items after a short delay. In each trial, a cardboard

with pictures of 12–18 objects was shown to the child by a research assistant who pointed to

items and asked the child to remember them. The research assistant then flipped the cardboard

for a short delay. When flipped back, the child was cued to identify the items pointed to

previously. The number of items to remember increased after each trial, up to 12 different levels

of difficulty. The test ended when the child made two errors on two subsequent levels. The score

consisted of the highest level reached by children.

Receptive vocabulary. Children’s receptive vocabulary at age 41 months was evaluated

with the Peabody Picture Vocabulary Test-Third Edition (PPVT-III; Dunn & Dunn, 1997; Dunn,

Theriault-Whalen, & Dunn, 1993), a standardized language test that assessed phonological

recognition and semantic understanding of words upon hearing them. This test has high internal

consistency (α = .93; Dunn et al., 1993), and is valid for use with French and English speakers

(Dunn et al., 1993; Flipsen, 1998). The test consisted of 170 cards each depicting four different

objects, actions, or emotions. Children had to identify the correct corresponding image to the

word said by the experimenter. One point was allowed for each correct answer. The test stopped

when children made six errors within a sequence of eight cards.

Visual-spatial skills. The visual-spatial skills were measured at 41 months through the

Block Design subtest of the Wechsler Preschool and Primary Scale of Intelligence–Revised

(WPPSI-R; Wechsler, 1989). This subtest was composed of 14 models depicted on pictures and

that children had to re-create using blocks. Bonus points could be gained for some models as a

19


function of time, and the test ended after three consecutive failures. Raw scores varied from 0 to

42. The Block Design mainly required visual processing ability, including perceptions of spatial

relations and mental manipulations of visual patterns. This subscale was highly correlated with

the Full Scale WPPSI-R score (r = .62) in the norming sample. It had excellent internal

consistency and test–retest reliability (see Sattler, 2001). Cronbach alpha for this subtest was .89

in our sample.

Procedure

Interviewers from an independent research firm were hired for the data collections. They

were trained, filmed, provided feedback, and tested for administration reliability by our research

coordinator. Specifically, they were trained by the research coordinator to administer the

measures, and were filmed when practicing with children that were not part of the study. The

following week, they would watched the video with, and provided feedback by the research

coordinator. Feedback to the interviewers would also be provided regularly after administrations

on real participants. Measures were administered in French (most families) or in English. All

French speaking children were assessed using French norms and validation procedures. Table 1

summaries the data collection procedures: the variables (predictors and outcomes), child’s age-

at-assessment, informant, and instruments.

When children were 5 months old, a face-to-face interview with the person most

knowledgeable about the child provided data on the household income, and the mother’s and the

father’s diploma. The parenting expectancies’ subscale was filled out separately by both parents

at home when their child was 5, 17 and 29 months, each time after a home visit.

The person most knowledgeable about the child also provided data on the child’s early

cognitive development at 29 months, and a trained research assistant administered the VCR, the

20


PPVT, and the Block Design test following a standard procedure in a face-to-face interview

when children born in 1997-98 were 41 months.

The adapted-NKT was orally administered one-on-one by a trained research assistant at

school or at home. The baseline and first levels of the adapted-NKT were administered at ages 4,

5 (preschool period), and 6 (kindergarten); the second level was added to the previous ones at

age 7 (grade 1). The CAT was individually assessed at school or at home by a trained research

assistant at ages 8 (grade 2) and 10 (grade 4).

Data Analysis

Treatment of Missing Data

Missing NK, cognitive, and family scores ranged between less than 1% and 8.5%. Based

on the Little’s MCAR test, the missing data were not completely at random (χ2 = 1020.86, df =

516, p < .001). A series of t tests obtained with the MVA module in SPSS 20.0 for Windows

(SPSS Inc, Chicago, IL) showed that children with missing data tended to have lower NK, and

were from a significantly lower socioeconomic background. To control for this potential bias, we

used full information maximum likelihood (FIML) to treat missing data. All the statistics

reported were estimated using FIML.

Missing data on the math achievement measures ranged between 13.5% and 21.2%. We

used multiple imputations to produce estimates for missing data through PROC MI procedure in

SAS (SAS Institute, Inc, Cary, NC).

Developmental Trajectories of NK

Developmental trajectories of NK from ages 4 years to 7 years were estimated using

semi-parametric mixture models in the PROC TRAJ procedure in SAS (SAS Institute, Inc, Cary,

NC; Jones & Nagin, 2007). This procedure identifies clusters of individuals following similar

21


progressions of an outcome over time by fitting a group-based model. A quadratic relationship is

used to model the link between age and behavior (here NK). Model estimation results in three

key outputs: (a) the shape of each group’s trajectory, (b) the estimated proportion of the

population belonging to each trajectory group, and (c) for each individual in the estimation

sample, an estimate of the probability that he/she belongs to each of the trajectory groups

identified (based on posterior probability of group membership).

In our model, subjects were included when data were available for at least two time-

points out of four to minimize attrition due to missing data. Thus, a total of 1,597 children were

included in the analysis. Solutions yielding two to five trajectory-groups of various shapes

(intercept -0-, linear -1-, quadratic -2-, cubic -3-) were estimated, and the best fitting solution

was derived based on the values of the Bayesian Information Criterion (BIC), which reflects the

parsimony of the model, the Akaike information criterion (AIC), and the theoretical likelihood

(L). Once the best-fit model established, each child was assigned to a specific NK trajectory-

group based on the highest probability of belonging to a trajectory.

We also performed a three-way repeated-measure ANOVA 4 (time) X 4 (group) X 2

(sex) on the NK scores from ages 4 to 7 to evaluate within and between groups differences

across time. The results were adjusted with Bonferronni correction for multiple comparisons.

Prediction of Achievement in Mathematics from NK Trajectories

Using the PROC MIXED SAS procedure, we performed a 2 (time) X 4 (group) X 2 (sex)

repeated-measures ANOVA on math scores at ages 8 and 10 to test whether boys and girls of

various trajectories maintained their relative position from early NK to math achievement in

grades 2 and 4. The results were adjusted for multiple comparisons.

Prediction of Low NK Trajectory

22


The predictive associations between early-life family and child-cognitive factors and the

low trajectory of NK were first tested with SPSS 20.0 for Windows (SPSS Inc, Chicago, IL) by

running chi-square tests (for categorical variables) or ANOVAs (for continuous variables) in

which the low trajectory was compared to the other trajectories of NK (see Figure 1).

We next conducted a binary logistic regression analysis to test the unique prediction of

early-life family factors and child-cognitive risk factors to membership in the low trajectory of

NK. The models were tested with Mplus 7.11 (Muthén & Muthén, 2012). The predictors were

grouped in three blocks following a developmental-hierarchical time order. Early-life family

factors were considered in the first two blocks, first, household income, and both mother and

father diploma, then mother and father outcome expectancies. Child cognitive factors, including

the index of early cognitive development, the Block Design visual-spatial score, the VCR

memory-span score, and the PPVT receptive vocabulary score, were introduced in the last block.

Each block of predictors was entered sequentially. Non-significant variables were removed from

the regressive model, before adding a new block of variables. To derive the best fitting model,

the regressive models were compared to a baseline model that did not include predictors.

Goodness-of-fit indices were quantified using the Akaike’s information criterion (AIC), the

variance explained (R2), and the -2 Log Likelihood ratio (∆-2LL). The lowest AIC index, the

highest R2, and a significant deterioration of the model fit suggested by the ∆-2LL value

indicated a better fit of the regression model compared to the baseline model.

Results

Developmental Trajectories of NK

As presented in Figure 1, the solution yielding a four-group model (one linear, three

quadratic trajectories) was found to best fit the NK data (BIC = -12886.15; AIC = -12845.83; L =

23


-12830.83). Trajectory models of two (BIC = -13119.94; AIC = -13098.44; L = -13090.44), three

(BIC = -12922.03; AIC = -12889.77; L = -12877.77) and five groups (BIC = -12951.53; AIC = -

12897.77; L = -12877.77) were also examined, but not retained on the basis of the BIC, AIC and

Likelihood indices.

The four trajectory groups consisted of (a) a Low-Increasing group1 (10% of children, [n

= 153; boys = 71, girls = 82]), (b) a Moderate-Increasing group (39%, [n = 630; boys = 334, girls

= 296]), (c) a Moderate-Fast Increasing group (32%, [n = 506; boys = 241, girls = 265]), and (d)

a High-Increasing group (19%, [n = 308; boys = 132, girls = 176]). All trajectory groups

displayed significant increasing trends over time (mostly quadratic, with exception of the Low-

Increasing trajectory which showed linear trend). A chi-square analysis showed significant sex

differences in NK trajectory group membership, χ2 (3) = 9.46, p < .05, with a higher proportion

of boys in the Moderate-Increasing trajectory, and a higher proportion of girls in the High-

Increasing trajectory.

Results from the repeated-measure ANOVA on NK scores showed significant group by

time interaction, F(8.48, 1848.43) = 73.93, p < .001, η2 = .25, and sex by time interaction,

F(2.83, 1848.43) = 3.19, p < .05, η2 = .005, but no significant group by sex interaction.

Significant differences in NK across all trajectories were found (p < .001), except between the

Moderate-Fast Increasing trajectory (M = 15.45, SD = 1.55) and the High-Increasing trajectory

(M = 15.26, SD = 2.25) at age 6. The Low-Increasing trajectory improved significantly from ages

5 to 6, and ages 6 to 7 (p < .05), but not from ages 4 to 5 (p > .05). The Moderate-Increasing

trajectory improved between all ages (p < .05); the Moderate-Fast Increasing trajectory improved

on NK from ages 4 to 5, and ages 5 to 6 (p < .05); and the High-Increasing trajectory improved

1 By convention, two terms are used to qualify the trajectory; the first reflects the initial level (e.g., low, moderate, high) of the NK score, whereas the second indicates the developmental course (i.e., slope) of the phenotype (e.g., increase, decrease, no-change or persistent).

24


on NK from ages 4 to 5, and from 5 to 6 (p < .05), but not from ages 6 to 7 (p > .05).

Developmental patterns of NK also differed across sex at age 7 (p < .01). Boys significantly

progressed in their NK from ages 6 to 7 (p < .01), whereas girls did not (p > .05).

Children’s NK was interpreted in reference to the developmental level conversion chart

of the Number Knowledge Test (Okamoto & Case, 1996). This conversion chart provides a

rough index of the developmental levels normally reached as a function of age. Children

following the Low-Increasing trajectory, although progressing over time, systematically trailed

in terms of expected acquisition of NK. These children were characterized by a persistent lower

performance in NK from the end of preschool to grade 1. Over these three years (age 4 to 7

years), these children maintained the equivalent of a two-year delay in expected NK, thus

signaling a substantial risk of mathematics underachievement in later grades.

Prediction of Later Math Achievement from NK Trajectories

Results from the repeated-measures ANOVA on math scores at ages 8 and 10 showed

significant group by time interaction, confirming the expected difference in later mathematic

achievement according to NK trajectories (p < .01), but with the exception of the Moderate-Fast

Increasing and the High-Increasing trajectories at age 10 (p >.05). The significant sex by time

interactions, but no significant sex by group interaction, further painted an evolving picture of

sex differences in math: boys performed better in math than girls at age 8 (p < .05), but not at age

10 (p < .05). Most importantly, boys and girls of the Low-Increasing group both systematically

trailed in mastering mathematical skills in comparison to the other trajectory groups, which also

brings additional support to the developmental trajectories of NK.

Family and Child Cognitive Predictors of Low NK Trajectory

25


Descriptive statistics of the cognitive and family variables for each trajectory-group are

provided in Table 2. Table 3 shows the early family and child-cognitive profiles of the low

versus other trajectories of NK. Compared to children of other trajectories, children of the low

NK trajectory were from household of lower income, and were more likely to have parents

without high school diploma (30% of mothers, 38% of fathers); both mothers and fathers of

children in the Low-Increasing trajectory also perceived they had lower impact on their child

development than other parents. Children of the low NK trajectory also had lower memory-span,

visual-spatial skills, receptive vocabulary, as well as hampered cognitive development at 29

months.

Table 4 shows the results of the binary logistic regression models, which tested the

unique longitudinal contributions of these factors to membership in the Low-Increasing

trajectory of NK, again using the three other trajectories as the basis for comparison. The final,

best-fitting regression model accounted for 26% of the variance of membership in the Low-

Increasing trajectory. A combination of family and child cognitive factors accounted for this

prediction. Father diploma (but not mother diploma) and household income uniquely predicted

membership in the Low-Increasing trajectory, and their unique contribution was maintained even

after taking into account the other factors in the model, including cognitive variables. Mother

parenting expectancies (but not father’s) added to the prediction of low NK trajectory, but this

contribution was no longer significant in the final model, when child cognitive factors were

taken into account. Three of the four cognitive variables, early cognitive development, visual-

spatial skills, and memory span uniquely predicted membership in Low-Increasing trajectory of

NK. Thus, children from family with low income and father with poor education backgrounds,

who, at 29 months, trailed in reaching early learning milestones, and who had lower visual-

26


spatial and memory skills at 41 months were more likely to fall on the Low-Increasing trajectory

of NK. According to the Odds ratio statistics, low early cognitive development best predicted

low profiles of NK (OR = .55), followed by the father diploma (OR = .70), the memory span

(OR= .80), the visual-spatial skills (OR = .86), and household income (OR = .87).

Discussion

This study was among the first to identify discrete patterns of developmental trajectories

of NK from late preschool to school entry, to document their related mathematic achievement in

elementary school, as well as their early family and cognitive predictors. Using a large

representative sample of children, four distinct developmental trajectories of NK were revealed

from ages 4 years to 7 years. Early development of NK was not linear, but rather varied in onset

level and rate of progression during the transition from late preschool to school entry.

Specifically, heterogeneity in children’s NK development was organized around a small Low-

Increasing performing group (10%), a Moderate-Increasing group (39%), a Moderate-Fast

Increasing group (32%) and a High-Increasing performing group (19%). Thus, as expected, two

groups were identified at the extreme (i.e., the Low-Increasing and the High-Increasing groups),

each accounting for a significant minority of children. Two other groups representing the

majority of children fell between these two groups and showed different change in NK over time.

As early as age 4, between-group differences in children’s level of NK were significant.

These differences were maintained throughout the elementary school years, suggesting long-term

prediction from early NK to later achievement. Of specific interest, children in the Low-

Increasing trajectory fell well behind other children – about a two-years behind –, and the gap

between these children and those in the other trajectories did not narrow during the course of the

school years. In grades 2 and 4, and compared to children in the other trajectories, children of the

27


Low-Increasing trajectory were trailing in mathematics, reflecting a stable pattern of low early

NK leading to low mathematic achievement during elementary school. This low and persistent

trajectory depicted the most atypical course of NK in this sample, and its prevalence (10%) was

consistent with those previously reported for mathematics learning disability (Barbaresi et al.,

2005; Shalev et al., 2000). In other words, these children were already at risk of mathematic

difficulties in preschool, and these difficulties were generally maintained in elementary school,

suggesting that deficits in NK likely hampers later mathematic and academic achievement.

Compared to children of the other trajectories of NK, children in the Low-Increasing

trajectory-group were characterized by lower early cognitive development. Already at 29

months, they had not acquired basic verbal and counting competencies (e.g., naming colors,

counting objects and making small sentences), and at 41 months visual-spatial skills and memory

span, to the same level as the other children. These hampered cognitive skills were predictive

even after controlling for household income, parents education and outcome expectancies.

These predictive associations were only partly consistent with previous findings showing

that child cognitive abilities such as visual-spatial skills and memory-span underlie the

emergence of early numeracy (Bull et al., 2008; Geary, 2011; Soto-Calvo et al., 2015). As

suggested by the triple code model (Dehaene & Cohen, 1995), the symbolic representation of

numbers develops sequentially. Numerical quantities are handled at first in a non-symbolic way

by the approximate magnitude mechanism (analog code), used for estimating quantities. Then,

increasing exposure to language and informal educational activities lead children to learn number

words (verbal code) and to identify number symbols (visual code), e.g., Arabic digits,

corresponding to the quantities (Dehaene et al., 1999). Accordingly, we found hindered visual-

28


spatial skills (which may be involved in the analog and visual codes) to predict low NK.

However, early language did not emerge as a significant predictor of NK.

Indeed, our results did not support poor receptive vocabulary skills (verbal code) at 41

months as a significant cognitive risk factor for low-NK profiles. One possible reason for this

failed prediction is that disruption in receptive vocabulary might characterize children with

severe mathematic learning disability, but not children at-risk of early NK difficulties. As for

NK, receptive vocabulary relies on both phonological and semantic components of language.

Deficiency was found in the language-based phonological loop for children diagnosed with a

severe mathematic learning disability, i.e., dyscalculia (Geary, 2011; Geary, Hoard, Byrd-

Craven, Nugent, & Numtee, 2007). However, contrary to children with dyscalculia, those

experiencing mathematic difficulties, but not reaching the clinical criteria for dyscalculia, were

found with a superior phonemic system to keep information in memory (Geary, 2011; Geary et

al., 2007).

It is also possible that specific language components may be differentially associated with

specific mathematic skills, such as NK. For example, a recent longitudinal study following

children from ages 6 to 9 years showed that vocabulary and listening comprehension predict to

geometry, but not arithmetic or algebra (Vukovic & Lesaux, 2013). LeFevre et al. (2010) have

also shown that linguistic skills such as elision and receptive vocabulary (PPVT) of preschool

and kindergarten children uniquely predicted number naming, but not nonlinguistic arithmetic

during the same year. Although strongly associated with numeration, linguistic skills were

weakly associated with magnitude comparison two years later. Accordingly, receptive

vocabulary may be associated to some components of NK such as counting skills, but not with

magnitude comparisons, or procedural understanding of whole numbers. The content of language

29


input also impact on NK development. Children who hear great amounts of spatial language

outperform on nonverbal spatial tasks associated to NK development, such as the Block Design

subtest of the WPPSI (Pruden, Levine, & Huttenlocher, 2011). This shows that the role of

language in NK development vary depending upon the language inputs and the NK outcomes.

One final reason for this lack of significance may be that poor receptive vocabulary may

have been accounted for by the early cognitive development index at 29 months, which also had

a strong language component. To evaluate this possibility, we simply removed this measure from

the regression, and found that receptive vocabulary at 41 months became significant (although

with an odd ratio barely different from one; results not shown). Thus, early cognitive

development at 29 months, which not only included language but also predicted receptive

language at 41 months, appeared as a key general factor in predicting low NK.

Interestingly, even when controlling for early-life family factors, father diploma, but not

mother diploma, still uniquely predicted low NK. Children of fathers with low educational

background were more at risk of low NK, suggesting that fathers with high education might

prevent the development of low numeracy and later mathematic difficulties. It does not mean

that mother diploma is not important for early NK, but it rather shows the increment of higher

paternal education, once the covariance with maternal education is taken into account. One

possible explanation for this result is that fathers with higher degrees might provide more support

for learning. Indeed, compared to parents from lower income and schooling, parents with higher

educational background were found to engage more frequently with their children in a broader

range of explicit mathematical-related activities (LeFevre et al., 2009; Levine et al., 2010;

Siegler, 2009; Siegler & Ramani, 2009). Accordingly, it is possible that fathers with low

educational background were less engaged with their children in mathematic-related activities,

30


resulting in slow and/or age inappropriate development of specific cognitive abilities involved in

NK and mathematics. Beyond parental involvement, fathers with low education and their low

NK child could share common vulnerability to poor cognitive abilities. The role of fathers in the

development of NK should be further explored in future research to address this question.

Fathers’ expectancies regarding their children development was, however, not

significantly associated to NK, whereas low mothers’ outcome expectancies were initially

predictive to low NK trajectory. However, it did not uniquely predict to low NK development

once child cognitive variables were entered in the model. Parenting style and perceptions had

previously been associated with children’s school-trajectory (LeFevre et al., 2009; Kouimtzi &

Stogiannidou, 2009). Here, mothers’ expectancies about her developing child may have been

associated to NK through one or many child-cognitive factors and thus, lost its significance when

entering the child cognitive abilities into the analyses. Parental beliefs and behaviors have been

associated to knowledge such as counting objects and learning vocabulary (Glascoe & Leew,

2010; Kouimtzi & Stogiannidou, 2009; LeFevre et al., 2009). Therefore, it is possible that

mother’s expectancies promote the development of NK through children’s cognitive abilities and

knowledge implied in mathematics. It may also be that our measure of parenting outcome

expectancies was too general, not specifically tapping into children’s learning of NK, but rather

providing a global perspective.

Implications for Research and Practices

These findings have implications for early identification of children at risk of persistent

mathematic difficulties and school underachievement, as well as for preventive intervention.

First, it gives a clear portrait of children from preschool age that, without a formal diagnosis,

showed persistently lower math skills compared to their typically achieving peers. The specific

31


cognitive skills – memory span, and visual-spatial skills predicted NK over and above the family

characteristics, such as household income and father education, and the early cognitive

development and knowledge (of colors, counting and speaking) at 29 months. Thus, children

already performing poorly on these characteristics at early age should deserve special attention to

alleviate later NK and math difficulties. The attention given to these children “at risk” should

also persist in early grade school. One potential approach to promote achievement among

children with initial risk of underachievement is to capitalize on teacher’s instructions. For

example, higher order instruction, such as teaching strategies that highlight deduction and critical

thinking in conjunction with skill acquisition, tends to improve achievement more than a rote

basic skills approach, especially among students academically at risk (Clements, Agodini, &

Harris, 2013; Connor, Morrison, Fishman, Schatschneider, & Underwood, 2007; Hamre &

Pianta, 2005; Xue & Meisels, 2004).

Second, this study also revealed continuity from early NK to mathematical skills during

elementary school, and points to early NK as a potential target for early prevention. This result

reinforces the predictive value of NK and highlights the transition from late preschool to school

entry as a crucial period to assess and promote early numeracy. Given this, children’s NK skills

should be assessed before school starts in order to provide additional support as soon as

difficulties emerge in this area.

Third, the present study identified the specific cognitive and family predictors of low NK

skills from preschool to school entry. However, to improve our understanding of the mechanisms

underlying early mathematical development, future studies should further investigate the

association between the approximate magnitude system and the level of NK, over and above

those cognitive and family predictors. Previous research supported the approximate magnitude

32


system at preschool age to uniquely predict later performance in mathematics (Fuchs & McNeil,

2013; Libertus et al., 2011; Mazzocco, Feigneson, & Halberda, 2011). However, the extend to

which children with low ability to approximate magnitude difference also perform poorly in their

early NK is still unknown.

Limitations

The present study should be interpreted in the context of its limitations. First, the

administration of the adapted-NKT departed from Okamoto and Case (1996), which may have

resulted in a ceiling effect at specific ages. At ages 4, 5, and 7 this procedure did not result in a

significant divergence from Okamoto and Case (1996), as most children did not reach the 60%

mark allowing them to move to the next level (see Method). Post-hoc examination of the score

distributions did not signal a meaningful ceiling effect, as only 0.3%, 0.6%, and 1.5% of children

reached the scale, respectively. However, at age 6 years, a majority of children successfully

reached 60% of success at the top level (i.e., Level 1), and 5.6% of children scored the scale

maximum. This suggests a ceiling effect, that is, a compression of scores at the top of the scale.

This ceiling effect at age 6, may have partly accounted for the curvilinear and asymptotic form of

the two highest trajectories (i.e., the High-Increasing and the Moderate-Fast Increasing

trajectories; see Figure 1), as well as of their decreasing gap at ages 6 and 7. Although this may

have affected the shape of these higher-performing trajectory groups, it was unlikely to play an

important role in children’s trajectory membership, especially for the Low-Increasing group, as it

was based on four longitudinal data points. What may have played against the reliability of all

trajectory groups was the lower internal consistency of the adapted-NKT at age 5. But again, the

impact of this limited reliability on the proper NK characterization of children is likely to have

been attenuated by its longitudinal aggregation to three other, more reliable data points.

33


Interestingly, despite the low reliability at age 5, individual differences in the adapted-NKT were

highly stable across the four ages (r = .74 - .92).

It is also possible that the NK measure did not reflect what is learn in math at school, and

that children in the low trajectory of NK would have looked different if assessed through other

measures. However, with respect to the assessment of performance in mathematics (NK and

standardized test of mathematics achievement), our findings suggest that the information

provided by multiple informants was roughly equivalent.

Second, some family predictors did not specifically tap into the mathematic domain and

were only indirectly linked to NK (e.g., parenting outcome expectancies). Our parenting outcome

expectancies also had less than optimal internal consistency. Similarly, our child cognitive

predictors (e.g., visual-spatial measure) were not numerical in nature and only one item in the

early cognitive development measure was mathematically focused. The early cognitive

development measure was confined to three items and based on the assessment of only one rater,

the person “most knowledgeable about the child”, limiting its quantitative and conceptual

validity. Finally, our conclusions may only apply to children from the province of Quebec,

Canada. It should also be noted that the present results may be specific to this developmental

window, and predictors of low-NK trajectories could differ at different ages (Haworth, Kovas,

Petrill, & Plomin, 2007).

To conclude, this study convincingly showed that children with low cognitive abilities

and impoverished living conditions early in life are at greater risk of low NK from late preschool

and school entry, and of persistent lower math achievement in elementary school.

Acknowledgements

This research was supported by grants from the Québec Ministry of Health, the Fonds

34


Québécois de la Recherche sur la Société et la Culture (FQRSC), the Social Science and

Humanities Research Council (SSHRC), the Canadian Institutes for Health Research (CIHR),

and grant 11.G34.31.003 from the Russian Federation. We are grateful to the parents of the

children participants to the Québec Longitudinal Study of Child Development (QLSCD).

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4 5 6 70

2

4

6

8

10

12

14

16

18High/increasingModerate/fast increasingModerate/increasingLow/increasing median scores

Age

Num

ber K

now

ledg

e Sc

ores

Figure 1. Developmental trajectories of number knowledge from 4 to 7 years of age (N = 1597):

Low-Increasing (n =153; 10%), Moderate-Increasing (n =630; 39%), Moderate-Fast Increasing

(n =506; 32%), High-Increasing (n =308; 19%), and the median number knowledge score. Data

courtesy of the Quebec Institute of Statistics.

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Running Head: PREDICTORS OF LOW NUMBER KNOWLEDGE

Table 1Longitudinal measures by multi-informants and child’s age-at-assessment

Note. PMK = Person most knowledgeable about the child

Variables Informant Child age at data collection

Method of Assessment

Family income PMK 5 months Questionnaire completed during aface-to-face interview

Mother diploma PMK 5 months Questionnaire completed during aface-to-face interview

Father diploma PMK 5 months Questionnaire completed during aface-to-face interview

Outcome expectancies (mother)

Mother 5, 17, 29 monthsAverage score across time

Self-administered questionnaire

Outcome expectancies (father)

Father 5, 17, 29 monthsAverage score across time

Self-administered questionnaire

Early cognitive development

PMK 29 months Questionnaire completed during aface-to-face interview

Memory-span Child 41 months Visually Cued Recall TaskReceptive vocabulary Child 41 months Peabody Picture Vocabulary TestVisual-spatial skills Child 41 months Block Design subtest of the WPPSI-RNumber knowledge Child 4, 5, 6, 7 years Number Knowledge TestMath achievement Child 8, 10 years Mathematics subtest of the Canadian

Achievement Test

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Table 2Descriptive statistics of family and children variables for each trajectory-group

Note. Low-I refers to the Low-Increasing trajectory of NK, Moderate-I refers to the Moderate-Increasing trajectory of NK, Moderate-Fast I refers to the Moderate-Fast Increasing trajectory of NK, High-I refers to the High-Increasing trajectory of NK.Percentage is reported for categorical variables (%); Mean and Standard deviation are reported for continuous variables† [means (SD)].Data are courtesy of the Quebec Institute of Statistics

Family and children variables Low-I(10%, n = 153)

Moderate-I(39%, n = 630)

Moderate-Fast I(32%, n = 506)

High-I(19%, n = 308)

Socio-demographic (5 months)Household income, 30 000$ or less per year 44.44 34.00 19.40 13.40Maternal education, no high school diploma 30.06 22.70 10.50 7.50Paternal education, no high school diploma 37.87 23.00 16.80 8.60Parenting expectancies (5 to 29 months)Outcome expectancies (mother)† 7.81 (1.79) 8.28 (1.59) 8.59 (1.38) 8.69 (1.25)Outcome expectancies (father)† 7.90 (1.87) 8.34 (1.51) 8.56 (1.33) 8.80 (1.17)Child cognitive abilities (41 months)Visual-spatial skills† 4.03 (3.16) 5.44 (3.21) 6.88 (3.75) 8.54 (4.23)Receptive vocabulary† 21.20 (10.58) 26.69 (12.67) 32.90 (14.56) 38.22 (15.27)Memory-span† 1.98 (1.37) 2.65 (1.80) 3.72 (2.36) 4.25 (2.38)Early cognitive development† 1.58 (0.99) 2.07 (0.92) 2.40 (0.80) 2.61 (0.67)

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Table 3 Family and children characteristics associated with trajectories of low number knowledge from 4 to 7 years of age

Family and children variablesMissing Low-I Other trajectories

P(n = 1597) (10%, n = 153) (90%, n = 1444)

Socio-demographic (5 months) Household income, 30 000$ or less per year 1.25 44.44 24.56 <.001 Maternal education, no high school diploma 0.06 30.06 15.18 <.001 Paternal education, no high school diploma 7.70 37.87 17.66 <.001Parenting expectancies (5 to 29 months) Outcome expectancies (mother)† 0.25 7.81 (1.79) 8.47 (1.46) <.001 Outcome expectancies (father)† 7.14 7.90 (1.87) 8.52 (1.39) <.001Child cognitive abilities (41 months) Visual-spatial skills† 5.57 4.03 (3.16) 6.60 (3.82) <.001 Receptive vocabulary† 6.51 21.20 (10.58) 31.38 (14.66) <.001 Memory-span† 8.45 1.98 (1.37) 3.37 (2.24) <.001 Early cognitive development† 0.56 1.58 (0.99) 2.30 (0.86) <.001Note. Significant at p < .05; Low-I refers to the Low-Increasing trajectory of NK. Other trajectories refers to the Moderate-Increasing, Moderate-Fast increasing, and the High-Increasing trajectories of NK. Missing indicates the percent of missing data (%). Chi-square tests between the Low trajectory and the other trajectories were used for categorical variables (%).ANOVAs between the Low trajectory and the other trajectories were used for continuous variables† [mean, (SD)].Data are courtesy of the Quebec Institute of Statistics

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Table 4Associations between significant covariates (from Table 2) and the low trajectory of number knowledge (n=153)

Variables β p OR -2LL Δ -2LL p AIC R2

Block 1: Socio-demographic (5 months) -29681.14 71.67 0.00 59406.28 0.09Household income -0.20 0.00 0.85Mother diploma -0.08 0.16 0.87Father diploma -0.22 0.00 0.68Block 2: Parenting expectancies (5-29 months) -29675.01 84.33 0.00 59396.02 0.11Household income -0.19 0.00 0.85Father diploma -0.23 0.00 0.66Outcome expectancies (mother)† -0.11 0.00 0.87Outcome expectancies (father)† -0.08 0.08 0.90Block 3: Child cognitive abilities (41 months) -29621.23 180.81 0.00 59294.47 0.24Household income -0.13 0.01 0.89Father Diploma -0.17 0.00 0.72Outcome expectancies (mother)† -0.07 0.11 0.91Visual-spatial skills† -0.28 0.00 0.86Receptive vocabulary† -0.12 0.06 0.98Memory-span† -0.20 0.01 0.83Early cognitive development† -0.23 0.00 0.58Final model -29624.90 173.61 0.00 59297.79 0.26Household income -0.15 0.00 0.87Father Diploma -0.18 0.00 0.70Visual-spatial skills† -0.29 0.00 0.86Memory-span† -0.23 0.00 0.80Early cognitive development† -0.26 0.00 0.55

Note. Significant at p < .05; β: Standardized parameter estimates, OR: odds ratio, and †: continuous variables The -2 Log Likelihood ratio (∆-2LL), the Akaike’s information criterion (AIC), and the variance explained (R2) indicates the adequacy of the model. Data are courtesy of the Quebec Institute of Statistics

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