The precision of mapping between number words and the approximate number system predicts children’s formal math abilities

https://doi.org/10.1016/j.jecp.2016.06.003Get rights and content

Highlights

  • Verbal number estimation variability correlates with overall math abilities.

  • Estimation predicts formal math, even when ANS, age and vocabulary are controlled.

  • Estimation mediates the link between ANS precision and overall math abilities.

  • Estimation variability indexes ANS precision and the ANS-number word mapping.

Abstract

Children can represent number in at least two ways: by using their non-verbal, intuitive approximate number system (ANS) and by using words and symbols to count and represent numbers exactly. Furthermore, by the time they are 5 years old, children can map between the ANS and number words, as evidenced by their ability to verbally estimate numbers of items without counting. How does the quality of the mapping between approximate and exact numbers relate to children’s math abilities? The role of the ANS–number word mapping in math competence remains controversial for at least two reasons. First, previous work has not examined the relation between verbal estimation and distinct subtypes of math abilities. Second, previous work has not addressed how distinct components of verbal estimation—mapping accuracy and variability—might each relate to math performance. Here, we addressed these gaps by measuring individual differences in ANS precision, verbal number estimation, and formal and informal math abilities in 5- to 7-year-old children. We found that verbal estimation variability, but not estimation accuracy, predicted formal math abilities, even when controlling for age, expressive vocabulary, and ANS precision, and that it mediated the link between ANS precision and overall math ability. These findings suggest that variability in the ANS–number word mapping may be especially important for formal math abilities.

Introduction

Educated adults and children have at least two means of representing and processing numerical information; they can use the approximate number system (ANS), which forms quick, imprecise numerical estimates without relying on serial counting (Dehaene, Dehaene-Lambertz, & Cohen, 1998), or they can use an exact number system that represents numerical information precisely via counting and number words and that is essential for much of school mathematics (Dehaene, 1992).

The ANS and exact number representations can be distinguished in several ways. The ANS is present in humans from birth (Izard, Sann, Spelke, & Streri, 2009) and also has been demonstrated in non-human animals, including fish (Dadda, Piffer, Agrillo, & Bisazza, 2009), rodents (Meck, Church, & Gibbon, 1985), whales and dolphins (Abramson, Hernandez-Lloreda, Call, & Colmenares, 2013), and non-human primates (Cantlon & Brannon, 2006); its presence in these pre-verbal and non-verbal populations shows that the ANS does not rely on language or an understanding of external symbols. Non-verbal ANS comparison tasks—in which participants typically are presented with two arrays of stimuli that differ by varying numerical ratios (e.g., two arrays of dots, two sequences of tones) and must indicate which is more numerous—reveal that the ANS represents number in a noisy continuous fashion along a mental number line, with increasing overlap between neighboring numerical representations as the target quantity grows (Dehaene, 1992, Piazza et al., 2004, van Oeffelen and Vos, 1982). As a result, performance on ANS comparison tasks is ratio dependent; quantities that differ by larger ratios are more easily distinguished than quantities that differ by finer ratios. The precision of ANS representations (i.e., the degree of non-overlap between neighboring representations) is often measured as the Weber fraction (w), which improves over human development (Halberda and Feigenson, 2008, Halberda et al., 2012, Libertus and Brannon, 2010, Odic et al., 2013, Xu and Spelke, 2000) and also varies among individuals in a given age group (e.g., Halberda et al., 2008, Libertus et al., 2011, Starr et al., 2013).

In contrast, the exact number system is uniquely human and is slowly acquired over the course of several years as children learn to use number words to represent numerical concepts (Carey, 2009, Feigenson et al., 2004, Le Corre and Carey, 2007, Wynn, 1992). The exact number system appears to depend on language given that people living in communities that lack a verbal counting routine lack exact large number concepts (Frank et al., 2008, Gordon, 2004, Pica et al., 2004, Spaepen et al., 2011). Critically, whereas the ANS lacks the precision to precisely represent large numbers (e.g., to represent exactly 50 as distinct from 49 and 51), exact numbers support such precision.

Despite the differences between the ANS and exact number words, children eventually learn a correspondence between the two systems. The development of this mapping is protracted (Crollen et al., 2011, Le Corre and Carey, 2007, Odic et al., 2015, Pinheiro-Chagas et al., 2014) and might not be causally linked (Gunderson, Spaepen, & Levine, 2015), but once complete it allows children and adults both to label approximate number representations with exact number words and to produce approximate representations given an exact number word. For example, a 5-year-old child shown a display of 10 dots presented too quickly to count can translate the resulting imprecise ANS representation of approximately 10 dots to arrive at a verbal estimate of “ten,” and a child asked to produce a particular number of actions, such as quickly tapping “ten times without counting,” will produce an approximately correct number of taps (Odic et al., 2015).

Although all people who use exact number words can form a mapping between these words and the ANS, the ease with which people map between number formats and the quality of this mapping may differ from person to person (Lyons, Ansari, & Beilock, 2012). Much as we can index individual differences in ANS precision using a non-verbal ANS comparison task, we can index individual differences in the mapping between the ANS and exact number words using a verbal estimation task. 1In a typical version of this task, participants are shown some number of dots too quickly to count and must verbally report how many there were. This yields two measures of performance. First, performance reflects verbal estimation accuracy (e.g., a participant should produce a number word close to “twelve” for 12 dots, “twenty-three” for 23 dots, etc.). By plotting participants’ verbal responses against the target number, mapping accuracy can be measured through the intercept, slope, and error rate of their responses (i.e., error rate = response  target) (Crollen et al., 2011, Izard and Dehaene, 2008, Odic et al., 2015). If perfectly accurate, participants’ average intercept and error rate will be 0 and their slope will be 1 (i.e., for every additional item in the target array, participants should increment their estimate by one number word in their known list). Estimation accuracy depends on several factors, including children’s ability to calibrate the mapping between the ANS and number words (Sullivan & Barner, 2014), potential observer biases (Barth & Paladino, 2011), and the range of number words known (Lipton & Spelke, 2005).

Verbal estimation tasks also produce a second measure of performance: estimation variability. Estimation variability reflects the degree to which a participant’s verbal responses vary from trial to trial (Cordes et al., 2001, Odic et al., 2015). For example, a child who is shown 15 dots on three occasions and reports seeing “five,” “fifteen,” and “twenty-five” will be (on average) perfectly accurate, but his estimation variability will be high compared with a child who says “fourteen,” “fifteen,” and “sixteen.” Although often measured as a coefficient of variance (CV, the standard deviation of all responses for a given stimulus number divided by that number), we previously showed that estimation variability can be more robustly captured through a simple maximum-likelihood estimation method that returns an estimate that we call σ (Odic et al., 2015).2Children with a lower σ are less variable/more precise in their verbal estimates.

Do individual differences in the mapping between the ANS and exact number words affect math performance? One reason to think so is that the ANS itself appears to be related to math performance. Prior studies that measured ANS representations without requiring any mappings to words or symbols found evidence of a link between individual differences in ANS precision and school math abilities (for a review, see Feigenson, Libertus, & Halberda, 2013). For example, Halberda and colleagues (2008) showed that adolescents’ ANS precision in a number task using non-symbolic dot arrays related to their prior performance on standardized symbolic math assessments. Subsequent work revealed that this relation between the ANS and math is present prior to schooling (Bonny and Lourenco, 2013, Libertus et al., 2011) and is maintained throughout adulthood (DeWind and Brannon, 2012, Halberda et al., 2012, Libertus et al., 2012, Lourenco et al., 2012, Lyons and Beilock, 2011). Furthermore, ANS precision measured during infancy or during the early preschool years also predicts later math performance (Libertus et al., 2013a, Mazzocco et al., 2011b, Starr et al., 2013). Although some studies failed to find a link between ANS precision and math ability (e.g., Castronovo and Göbel, 2012, Gilmore et al., 2013, Price et al., 2012, Sasanguie et al., 2012), recent meta-analyses support the existence of this link (Chen and Li, 2014, Fazio et al., 2014, Schneider et al., 2016).

If having more precise ANS representations is linked to better symbolic math performance, this raises the question of whether the mapping between the ANS and exact number words predicts additional variance in math abilities. Better mappings could plausibly provide children with a more accurate or reliable sense of the approximate magnitude corresponding to a number word or symbol, thereby improving mathematical intuitions. Indeed, children’s exact number abilities—especially those related to their emerging understanding of exact number symbols—have also been shown to be important for math achievement. For example, Holloway and Ansari, 2009, Brankaer et al., 2014, and Sasanguie and colleagues (Sasanguie et al., 2014, Sasanguie et al., 2012) found that children’s performance on a symbolic number comparison task (e.g., showing participants two Arabic numerals and asking them to quickly judge which is greater) correlated with math abilities. Similarly, Kolkman, Kroesbergen, and Leseman (2013) found that kindergarteners’ performance on a symbolic number comparison task and a symbolic number line estimation task (but not on a non-symbolic number comparison task or a non-symbolic number line estimation task) predicted math performance in first grade. Other studies have found that children with math learning disabilities show significantly worse performance on symbolic number tasks than children without math learning disabilities (De Smedt and Gilmore, 2011, Iuculano et al., 2008, Landerl and Kölle, 2009, Rousselle and Noel, 2007). These findings have sometimes been taken to indicate that the mapping of number words and symbols to their meanings is critical for math achievement—perhaps even more important than the ANS itself for the math skills tested (for a review, see De Smedt, Noel, Gilmore, & Ansari, 2013). However, some have recently questioned whether symbolic number comparison tasks actually reflect any use of approximate number representations. Lyons, Nuerk, and Ansari (2015) suggested that ratio effects in symbolic tasks (e.g., taking longer to respond that 19 is more than 17 than to respond that 19 is more than 11) do not reflect a mapping to approximate number representations but instead stem from more general issues of symbolic processing (e.g., frequency effects of particular number symbols). If so, then previously observed correlations between ratio effects in symbolic number tasks and math performance might not reveal anything about the role of the mapping between the ANS and number words in mathematical thinking.

One way to address this issue is to use tasks that more directly probe the mapping between the ANS and exact number words. The verbal estimation task described earlier requires explicit translation between non-symbolic quantities and symbolic quantities and, as such, may be better suited to asking whether individual differences in the ANS–number word mapping relate to math abilities. However, previous studies using verbal estimation tasks have yielded mixed results with regard to math performance. Some verbal estimation studies have found a significant relation between ANS–number word mapping and math ability in children. For example, Mazzocco, Feigenson, and Halberda (2011a) found that in a sample of ninth graders individual differences in ANS precision and verbal estimation independently explained variability in children’s standardized math test performance. Similarly, Pinheiro-Chagas and colleagues (2014) found that 10-year-old children’s ANS and verbal estimation performance each correlated with calculation skills, with verbal estimation partly mediating the relation between ANS and calculation. Mundy and Gilmore (2009) found that the accuracy of 6- to 8-year-old children’s mappings between non-symbolic quantities and number words predicted a small but significant amount of variance in math performance. In addition, Bartelet, Vaessen, Blomert, and Ansari (2014) found that kindergarteners’ verbal number estimation skills predicted their arithmetic performance in first grade, whereas their efficiency on a non-symbolic ANS comparison task did not. However, other studies found no evidence of this association; Lyons, Price, Vaessen, Blomert, and Ansari (2014) observed no reliable link between success at estimating the numerosity of a dot array and math ability in first through sixth graders. In addition, other studies found mixed evidence within a single sample; in a study by Castronovo and Göbel (2012), adults’ verbal estimation error rates correlated with math abilities, but their estimation variability did not. Together, these findings do not offer a coherent picture of the relation among the ANS, verbal estimation, and mathematics.

At least two possibilities may help to explain these divergent findings. First, verbal estimation abilities may be more important for some math skills than for others. In preschool and early elementary school instruction, mathematics is often described as composed of informal and formal components (Baroody, 1987, Baroody and Wilkins, 1999). Informal math knowledge is acquired from children’s everyday experiences with collections or events in their environment. For example, preschool-aged children’s informal math abilities include counting, comparing number words to determine which is greater, and determining the answers to simple arithmetic problems using tokens or fingers. In contrast, formal math knowledge requires an understanding of symbols and mathematical conventions and, hence, is learned through explicit instruction. Children’s formal math abilities include reading and writing Arabic numerals, understanding the place value system, and recalling memorized addition, subtraction, and multiplication facts.

The distinction between formal and informal aspects of mathematics plays an important role in interpreting the correlations between ANS precision and math ability. In a recent study, Libertus, Feigenson, and Halberda (2013b) found that 3- to 7-year-old children’s ANS precision predicted their informal math abilities, but not their formal math abilities, up to 2 years later. In adults, Lourenco and colleagues (2012) showed that ANS precision correlated with performance on the Woodcock–Johnson (WJ) calculation subtest and KeyMath geometry subtest but not with performance on the WJ math fluency, applied math problems, and quantitative concepts subtests (but see Inglis, Attridge, Batchelor, & Gilmore, 2011, for a failure to find a link between any WJ math subtest and ANS acuity in adults). In contrast, Holloway and Ansari (2009) found that children’s performance on a symbolic number comparison task, but not on a non-symbolic ANS comparison task, significantly correlated with performance on the WJ math fluency subtest and marginally correlated with the calculation subtest. Taken together, these findings suggest that the link between ANS precision and exact number representations does not play a uniform role across all aspects of mathematics. Verbal estimation abilities may likely be more closely related to those aspects of math that heavily rely on understanding the meaning of number words and symbols (i.e., formal math abilities).

A second potential explanation for the discrepancies in the literature is that only some indexes of verbal number estimation may be linked to math abilities. In particular, the accuracy of the mapping between the ANS and exact number words may play a distinct role from the variability in this mapping. If, for example, children use the mapping between number words and the ANS to perform quick mental arithmetic (accessing approximate representations to estimate the answers to math problems that use exact number symbols), a highly variable mapping would lead to many more errors over time even if the mapping is (on average) accurate. Similarly, a highly variable mapping may lead to a low-confidence internal signal, leading children to blindly guess or to use other non-optimal strategies (e.g., see Odic, Hock, & Halberda, 2014). As a result, individual differences in the variability of children’s mappings between ANS representations and exact number words may affect math abilities differently than estimation accuracy.

The goals of the current study were twofold. First, we aimed to identify whether children’s symbolic math abilities depend on both verbal estimation accuracy (intercept, slope, and error rate) and verbal estimation variability (σ) even when controlling for ANS precision. Second, we aimed to investigate the degree to which children’s ANS precision and verbal estimation accuracy and variability are differentially linked to informal versus formal math abilities. To this end, we tested 5- to 7-year-old children on a verbal number estimation task (to measure their accuracy through intercepts, slopes, and error rates and to measure their variability through σ), a non-symbolic ANS comparison task (to measure their ANS precision or w), and a standardized assessment of informal and formal math abilities. We focused on this age range because previous work shows that 5-year-olds have established a mapping between the ANS and number words (Le Corre and Carey, 2007, Odic et al., 2015) and because 5- to 7-year-olds are at the point of transitioning from mostly informal math skills to more formal ones.

Section snippets

Participants

We tested 51 children (23 girls; average age = 6 years 8 months, range = 5 years 1 month to 8 years 0 months). Data from 2 additional children were excluded because the children were unable to complete all three tasks. Most children came from families of middle to high socioeconomic status, all from the Baltimore, Maryland, area in the eastern United States. Parents provided informed written consent prior to their children’s participation, and all children received a small gift (e.g., toy, book) to thank

Results

We first report the data from each task, comparing performance in our sample with performance previously reported in the literature. Subsequently, we examine the relations among the three tasks. Where the underlying distribution of scores was found to be left or right skewed, we used Spearman rank-order coefficients instead of Pearson coefficients (Inglis & Gilmore, 2014). We report the correlations that were identified a priori as our primary interest in the main text, but a full overview of

Discussion

By their fifth birthday, children have two ways of representing number: intuitive non-verbal representations produced by the ANS and exact number words and symbols whose meanings are acquired gradually, usually over the preschool years. Although each of these systems has been shown to independently relate to early mathematics ability, the role of the mapping between the two systems (e.g., as measured by the accuracy and variability of children’s verbal number estimations) has remained

Acknowledgments

We thank Rebecca Zhu for help with testing participants, and all families and children for their participation. This research was supported by National Institute of Child Health and Human Development (NICHD) Grant R01 HD057258 to L.F. and J.H. Creation of the Panamath software was supported by National Science Foundation (NSF) Grant DRL0937675 to J.H. The authors declare no conflicts of interest.

References (100)

  • S. Dehaene

    Varieties of numerical abilities

    Cognition

    (1992)
  • S. Dehaene et al.

    Abstract representations of numbers in the animal and human brain

    Trends in Neuroscience

    (1998)
  • L.K. Fazio et al.

    Relations of different types of numerical magnitude representations to each other and to mathematics achievement

    Journal of Experimental Child Psychology

    (2014)
  • L. Feigenson et al.

    Core systems of number

    Trends in Cognitive Sciences

    (2004)
  • M.C. Frank et al.

    Number as a cognitive technology: Evidence from Piraha language and cognition

    Cognition

    (2008)
  • E.A. Gunderson et al.

    Approximate number word knowledge before the cardinal principle

    Journal of Experimental Child Psychology

    (2015)
  • I.D. Holloway et al.

    Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s mathematics achievement

    Journal of Experimental Child Psychology

    (2009)
  • G. Huntley-Fenner

    Children’s understanding of number is similar to adults’ and rats’: Numerical estimation by 5–7-year-olds

    Cognition

    (2001)
  • M. Inglis et al.

    Indexing the approximate number system

    Acta Psychology

    (2014)
  • V. Izard et al.

    Calibrating the mental number line

    Cognition

    (2008)
  • M.E. Kolkman et al.

    Early numerical development and the role of non-symbolic and symbolic skills

    Learning and Instruction

    (2013)
  • K. Landerl et al.

    Typical and atypical development of basic numerical skills in elementary school

    Journal of Experimental Child Psychology

    (2009)
  • M. Le Corre et al.

    One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles

    Cognition

    (2007)
  • M.E. Libertus et al.

    Is approximate number precision a stable predictor of math ability?

    Learning and Individual Differences

    (2013)
  • M.E. Libertus et al.

    Numerical approximation abilities correlate with and predict informal but not formal school mathematics abilities

    Journal of Experimental Child Psychology

    (2013)
  • M.E. Libertus et al.

    Intuitive sense of number correlates with math scores on college-entrance examination

    Acta Psychologica

    (2012)
  • I.M. Lyons et al.

    Numerical ordering ability mediates the relation between number-sense and arithmetic competence

    Cognition

    (2011)
  • E. Mundy et al.

    Children’s mapping between symbolic and nonsymbolic representations of number

    Journal of Experimental Child Psychology

    (2009)
  • D. Odic et al.

    Children’s mappings between number words and the approximate number system

    Cognition

    (2015)
  • M. Piazza et al.

    Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia

    Cognition

    (2010)
  • M. Piazza et al.

    Tuning curves for approximate numerosity in the human intraparietal sulcus

    Neuron

    (2004)
  • G.R. Price et al.

    Nonsymbolic numerical magnitude comparison: Reliability and validity of different task variants and outcome measures, and their relationship to arithmetic achievement in adults

    Acta Psychologica

    (2012)
  • L. Rousselle et al.

    Basic numerical skills in children with mathematics learning disabilities: A comparison of symbolic vs. non-symbolic number magnitude processing

    Cognition

    (2007)
  • J.H. Ryoo et al.

    Examining factor structures on the Test of Early Mathematics Ability–3: A longitudinal approach

    Learning and Individual Differences

    (2015)
  • K. Wynn

    Children’s acquisition of the number words and the counting system

    Cognitive Psychology

    (1992)
  • F. Xu et al.

    Large number discrimination in 6-month-old infants

    Cognition

    (2000)
  • A.J. Baroody

    Children’s mathematical thinking: A developmental framework for preschool, primary, and special education teachers

    (1987)
  • A.J. Baroody et al.

    The development of informal counting, number, and arithmetic skills and concepts

  • H. Barth et al.

    The development of numerical estimation: Evidence against a representational shift

    Developmental Science

    (2011)
  • Y. Benjamini et al.

    Controlling the false discovery rate: A practical and powerful approach to multiple testing

    Journal of the Royal Statistical Society B: Methodological

    (1995)
  • C. Blair et al.

    Relating effortful control, executive function, and false belief understanding to emerging math and literacy ability in kindergarten

    Child Development

    (2007)
  • J.L. Booth et al.

    Numerical magnitude representations influence arithmetic learning

    Child Development

    (2008)
  • C. Brankaer et al.

    Children’s mapping between non-symbolic and symbolic numerical magnitudes and its association with timed and untimed tests of mathematics achievement

    PLoS ONE

    (2014)
  • J.F. Cantlon et al.

    Shared system for ordering small and large numbers in monkeys and humans

    Psychological Science

    (2006)
  • S. Carey

    The origin of concepts

    (2009)
  • J. Castronovo et al.

    Impact of high mathematics education on the number sense

    PLoS ONE

    (2012)
  • C.A. Clark et al.

    Longitudinal associations between executive control and developing mathematical competence in preschool boys and girls

    Child Development

    (2013)
  • S. Cordes et al.

    Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

    Psychonomic Bulletin & Review

    (2001)
  • V. Crollen et al.

    Under- and over-estimation: A bi-directional mapping process between symbolic and non-symbolic representations of number?

    Experimental Psychology

    (2011)
  • N.K. DeWind et al.

    Malleability of the approximate number system: Effects of feedback and training

    Frontiers in Human Neuroscience

    (2012)
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