Elsevier

Cognition

Volume 187, June 2019, Pages 1-9
Cognition

Learning correspondences between magnitudes, symbols and words: Evidence for a triple code model of arithmetic development

https://doi.org/10.1016/j.cognition.2018.11.016Get rights and content

Abstract

According to the Triple Code Model, early arithmetic development depends on learning the mappings between non-verbal representations of magnitude (quantity) and symbolic verbal (number words) and visual (Arabic numerals) representations of number. We examined this hypothesis in a sample of 166 4- to 7-year old children. Children completed 4 paired-associate learning tasks and a broad range of measures assessing early numerical (symbolic and non-symbolic magnitude comparison, digit writing, arithmetic) and reading skills (letter-sound knowledge, phoneme awareness, rapid automatized naming, word reading). A path model showed that paired-associate learning tasks involving mapping magnitudes onto verbal or visual stimuli predicted arithmetic performance over and above other well-established predictors. This relationship was specific to arithmetic and, consistent with the Triple Code Model, highlights that mapping between non-symbolic magnitude representations and the corresponding symbolic forms (verbal and visual) is important to the development of arithmetic skills.

Introduction

The Triple Code Model (Dehaene & Cohen, 1995) posits that numerical processing involves three categories of mental representation: two symbolic forms in which numbers are represented as words (e.g., “three”) and Arabic numerals (e.g., “3”) and a non-symbolic analogue system which represents numbers as magnitudes (quantities). These three classes of representation are seen as separate but linked, and an important requirement in early arithmetic development is to learn the mappings between them. Thus, according to this model, deficits in developing any of the three classes of representation (number words, Arabic numerals, magnitudes), or problems learning the mappings between them, might lead to difficulties in arithmetic.

Learning the mappings between number words, Arabic numerals and nonverbal magnitude representations is a clear example of paired-associate learning. Paired-associate learning is the ability to form connections between arbitrary pairs of stimuli such as visual symbols and their verbal labels (Litt, de Jong, van Bergen, & Nation, 2013). The ability to form such mappings is known to predict reading ability (Clayton et al., 2018, Litt and Nation, 2014, Warmington and Hulme, 2012) but the extent to which paired-associate learning predicts arithmetic skills is as yet unclear. In this paper we explore whether differences in paired-associate learning (specifically learning mappings between magnitudes and verbal labels or visual symbols) are related to individual differences in children’s arithmetic skills.

Rudimentary non-symbolic representations of magnitude are present in early infancy (Xu & Spelke, 2000) and are associated with visual and phonological representations of number during language acquisition (von Aster & Shalev, 2007; but see Lyons, Bugden, Zheng, De Jesus, & Ansari, 2018). The Triple Code Model suggests that children’s understanding of magnitudes (coded in the approximate number system; ANS) is important as it provides the semantic underpinning for Arabic numeral and number word representations (Dehaene & Cohen, 1995); this idea has been supported by both computational modeling and neuroimaging studies (Lyons and Ansari, 2009, Verguts and Fias, 2004). In this view the ANS provides a critical foundation for the development of arithmetic skills. Recent meta-analyses report a modest relationship between performance on non-symbolic magnitude judgement tasks (a measure of the ANS) and arithmetic performance (rs = .20–.24; Chen and Li, 2014, Fazio et al., 2014, Schneider et al., 2017; but see De Smedt, Noël, Gilmore, & Ansari, 2013 who report only a weak relationship). It has also been argued that an impairment in the processing of magnitudes is a central deficit in mathematics disorder (i.e. deficit in representing magnitudes: Butterworth, 2010, Piazza et al., 2010; or accessing magnitude representations: Rousselle & Noël, 2007).

A variety of evidence shows that children’s number knowledge (knowledge of Arabic numerals and spoken number words) is predictive of arithmetic attainment. Numerosity judgements with Arabic numeral pairs (identifying the larger number in a pair of Arabic numerals) are more strongly related to arithmetic ability than analogous non-symbolic magnitude comparison tasks (e.g., Holloway and Ansari, 2009, Rousselle and Noël, 2007, Schneider et al., 2017). In addition, children who experience difficulties in arithmetic are slower to count dots (Landerl, Bevan, & Butterworth, 2004), and name digits, but not letters or geometric shapes, than typically developing children (Pauly et al., 2011, van der Sluis et al., 2004). This indicates that their difficulties are specific to words linked to magnitude representations. Furthermore, Gobel, Watson, Lervag, and Hulme (2014) showed that knowledge of Arabic numerals at 6-years uniquely predicted arithmetic skills at 7-years. These findings suggest that symbolic number knowledge is critical for the development of early arithmetic ability; a finding which is further highlighted by Martin, Cirino, Sharp, and Barnes (2014) who demonstrated that symbolic number knowledge predicted variations in mathematics performance, above and beyond counting ability.

Given the apparent importance of associations between different representations of number (Dehaene & Cohen, 1995), research has begun to explore how children's skills in mapping between different representations of number relate to their early arithmetic performance. Mundy and Gilmore, 2009, Brankaer et al., 2014 assessed the ability of 6- to 8-year-olds to map between symbolic representations of number (Arabic numerals and spoken words) and their corresponding magnitudes (e.g. map the digit 8 to a set of eight dots); Libertus, Odic, Feigenson, and Halberda (2016) examined younger children's (5- to 7-years-old) ability to map between magnitudes and exact number words. These studies found that children's mapping ability accounted for individual differences in arithmetic performance, over and above other known predictors (e.g. non-symbolic and/or symbolic number comparison). The critical role of cross-modal pairing has also been highlighted by Defever, De Smedt, and Reynvoet (2013): when asked to determine whether two stimuli depicted matching numerosities, children with mathematical learning difficulties were slower than typically developing children to respond to cross-modal pairs (matching a dot array to a corresponding Arabic numeral; 5 dots vs. “5”). In contrast, the two groups did not differ in the speed of responding on a comparable within-modal task (matching two dot arrays).

Extending this research, Hurst, Anderson, and Cordes (2017) examined 3- to 4-year-olds’ ability to map between the three aspects of the Triple Code Model: numeral-to-magnitude, word-to-magnitude, and numeral-to-word. Like Brankaer et al. (2014), they found that performance was lowest when mapping between quantities and their associated symbols. This was interpreted as suggesting that children may first map numerals to number words and only through this mapping do they subsequently learn the association between numerals and the quantities they represent.

Taken together, these studies show that children’s knowledge of the associations between different representations of number is related to arithmetic ability (Brankaer et al., 2014, Defever et al., 2013, Libertus et al., 2016, Mundy and Gilmore, 2009). Furthermore, they demonstrate that pairings which involve non-verbal numerosities are often difficult for children (Brankaer et al., 2014, Hurst et al., 2017). These studies, however, all examine previously learned associations between quantities and Arabic numerals/number words. In the current study we examine the acquisition of these mappings and the extent to which this ability relates to arithmetic.

Learning the mappings between alternative representations of number is a form of paired-associate learning. Only one study to date has directly investigated the relationship between paired-associate learning and arithmetic. Cui et al. (2017) found no significant relationship between visual-verbal paired-associate learning and addition and subtraction ability in 5- to 6-year-old Chinese children. Their visual-verbal paired-associate learning condition required children to learn pairings between spoken nonsense words and pictures of imaginary animals. However, the Triple Code Model (Dehaene & Cohen, 1995) suggests that it is learning the associations between nonverbal magnitudes and verbal labels or visual symbols that may be particularly crucial for arithmetic development. No prior studies have investigated this possibility; the current study will fill this gap.

This study examines the role of paired-associate learning as a predictor of arithmetic in 4.5- to 7-year-old children. We tested children on four paired-associate learning conditions using verbal (nonwords), visual (abstract symbols) and magnitude stimuli (dot arrays). There were four conditions: magnitude-verbal, magnitude-visual, visual-verbal, and verbal-visual. Children also completed measures of formal arithmetic (addition), symbolic and non-symbolic magnitude understanding (as assessed using digit and dot comparison tasks) and number knowledge (digit writing). In line with the theory that magnitude representations (coded in the ANS) are a critical foundation for symbolic number knowledge and arithmetic (Dehaene and Cohen, 1995, Piazza and Dehaene, 2004), we predicted that paired-associate learning will only be related to arithmetic ability in the conditions involving learning associations with magnitudes (i.e., magnitude-visual; magnitude-verbal).

The relationship between paired-associate learning and reading is used as a control condition to investigate whether paired-associate learning involving magnitudes is specifically related to arithmetic. Paired-associate learning tasks that involve a verbal output are known to be correlates of reading ability (Clayton et al., 2018, Litt and Nation, 2014) so we predict that paired-associate learning conditions involving a verbal output (i.e., visual-verbal; magnitude-verbal) will be correlates of reading performance.

Section snippets

Participants

Participants were 166 children from a primary school in Brisbane, Australia (81 boys; mean age = 70.62 months, range 54–87 months, SD = 7.34). Children were in their first (Preparatory Year; n = 94, mean age = 65.30, SD = 3.99) or second year of school (Grade 1; n = 72, mean age = 77.57, SD = 4.23). The Australian Catholic University Human Research Ethics Committee (2015-141H) provided ethical approval. Parental consent was obtained using opt-out consent and children provided verbal assent

Results

The means, standard deviations, reliabilities and measures of skewness, and kurtosis for all variables are shown in Table 1 (all data are available on the OSF website; Malone, Heron-Delaney, Burgoyne, & Hulme, 2016). Correlations between variables are shown in Table 2. Most of the known predictors of reading and arithmetic correlate in the expected direction, with the exception of magnitude comparison. For these, the correlations were slightly stronger between magnitude comparison conditions

Discussion

This study examined the associations between different forms of paired-associate learning and variations in arithmetic skills in young children. As hypothesised, paired-associate learning involving mappings between magnitudes and verbal labels or visual symbols predicted individual differences in arithmetic skills in children, even after controlling for the effects of other established predictors of arithmetic. In contrast, paired-associate learning tasks involving mappings with verbal output

Conclusion

This study is the first to investigate whether differences in paired-associate learning ability (specifically, learning associations between magnitudes and verbal labels or visual symbols) are related to individual differences in arithmetic in children. Children who were better at learning mappings between magnitudes and verbal labels or visual symbols also demonstrated better arithmetic skills, even after controlling for a range of better established correlates of arithmetic. Our findings have

Author note

This research was supported by a grant from the Australian Catholic University (14HS4006CH).

References (55)

  • C. Hulme et al.

    Paired-associate learning, phoneme awareness, and learning to read

    Journal of Experimental Child Psychology

    (2007)
  • K. Landerl et al.

    Developmental dyscalculia and basic numerical capacities: A study of 8- to 9-year-old students

    Cognition

    (2004)
  • M.E. Libertus et al.

    The precision of mapping between number words and the Approximate Number System predicts children’s formal math abilities

    Journal of Experimental Psychology

    (2016)
  • R.A. Litt et al.

    Dissociating crossmodal and verbal demands in paired associate learning (PAL): What drives the PAL-reading relationship?

    Journal of Experimental Child Psychology

    (2013)
  • R.A. Litt et al.

    The nature and specificity of paired associate learning deficits in children with dyslexia

    Journal of Memory and Language

    (2014)
  • R.B. Martin et al.

    Number and counting skills in kindergarten as predictors of grade 1 mathematics skills

    Learning and Individual Differences

    (2014)
  • E. Mundy et al.

    Children’s mapping between symbolic and nonsymbolic representations of number

    Journal of Experimental Child Psychology

    (2009)
  • R.E. Núñez

    Is there really an evolved capacity for number?

    Trends in Cognitive Sciences

    (2017)
  • H. Pauly et al.

    Domain-specific rapid automatized naming deficits in children at risk for learning disabilities

    Journal of Neurolinguistics

    (2011)
  • M. Piazza et al.

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

    Cognition

    (2010)
  • L. Rousselle et al.

    Magnitude comparison is preschoolers: What counts? Influence of perceptual variables

    Journal of Experimental Child Psychology

    (2004)
  • 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)
  • S. van der Sluis et al.

    Inhibition and shifting in children with learning deficits in arithmetic and reading

    Journal of Experimental Child Psychology

    (2004)
  • I. Xenidou-Dervou et al.

    Nonsymbolic and symbolic magnitude comparison skills as longitudinal predictors of mathematical achivement

    Learning and Instruction

    (2017)
  • F. Xu et al.

    Large number discrimination in 6-month-old infants

    Cognition

    (2000)
  • 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)
  • S. Carey

    Where our number concepts come from

    Journal of Philosophy

    (2009)
  • Cited by (0)

    View full text