How do symbolic and non-symbolic numerical magnitude processing skills relate to individual differences in children's mathematical skills? A review of evidence from brain and behavior
TL;DR: A few neuroimaging studies revealed that brain activation during number comparison correlates with children's mathematics achievement level, but the consistency of such relationships for symbolic and non-symbolic processing is unclear.
About: This article is published in Trends in Neuroscience and Education.The article was published on 2013-06-01 and is currently open access. It has received 478 citations till now. The article focuses on the topics: Approximate number system & Developmental Dyscalculia.
Summary (2 min read)
Jump to: [INTRODUCTION] – [DEVELOPMENT OF NON-SYMBOLIC NUMBER PROCESSING] – [SYMBOLIC PROCESSING DEVELOPMENT] – [BRAIN IMAGING DATA] – [EDUCATIONAL INTERVENTIONS] – [Most of the existing interventions have been applied to kindergarteners or children] and [SUMMARY AND CONCLUSIONS]
INTRODUCTION
- One important way in which cognitive neuroscience has made successful connections to educational research is by drawing attention to the importance of numerical magnitude processing as a foundation for higher-level numerical and mathematical skills (e.g., Butterworth et al., 2011; De Smedt et al., 2010) .
- Such research can pinpoint more precisely the mathematical content that should be included in specific interventions.
- Beyond educational applications, establishing whether symbolic or non-symbolic numerical magnitude processing skills, or both, are predictive of children's mathematics achievement is of theoretical importance too.
- While non-symbolic representations of numerical magnitudes are thought be shared across species and can already be measured in early infancy (Cantlon, 2012) , symbolic representations are uniquely human and relatively recent cultural inventions to provide abstract representations of numerical magnitude.
- The authors provide an integrative review of the existing body of data that has dealt with this question.
DEVELOPMENT OF NON-SYMBOLIC NUMBER PROCESSING
- The nature and role of typically developing children's magnitude representations have been commonly explored with magnitude comparison tasks (Box 1).
- Individuals with more precise ANS representations perform more accurately and faster on magnitude comparison tasks and they show smaller effects of ratio or distance.
- Many studies have failed to find such a significant relationship (see Table 1 for a summary).
- One possible explanation for these contrasting findings is that there is no standardized version of the dot comparison task.
SYMBOLIC PROCESSING DEVELOPMENT
- The development of symbolic number processing has been typically investigated by means of magnitude comparison tasks that involve Arabic digits (Box 1).
- This relationship appears to be very consistent for overall RT on the symbolic comparison task.
- This contradictory pattern of results could partly be due to methodological differences.
- The other studies used the distance or the ratio effect as an indicator of ANS precision: According to Noël and Rousselle (2011) , this developmental profile suggests that the first deficit seen in DD children is specific to the magnitude processing of symbolic numbers and not to the ANS.
BRAIN IMAGING DATA
- There have been a growing number of efforts to uncover which brain regions might underlie the associations between numerical magnitude processing and mathematics achievement.
- In another set of recent studies (Cantlon & Li, 2013; Emerson & Cantlon, 2012) , children viewed educational videos (Sesame Street) that had mathematical content, while their brain activity was recorded using fMRI.
- These studies cannot specifically constrain their understanding of the brain regions that underlie the association between symbolic and nonsymbolic numerical magnitude processing and children's mathematics achievement, since they did not explicitly address such relationships.
- Taken together, while neuroimaging methods are being used to constrain their understanding of the association between numerical magnitude processing and mathematics skills in both children with and without DD, there are currently too few studies, often with relatively small sample sizes, to allow for clear-cut conclusions to be drawn.
EDUCATIONAL INTERVENTIONS
- Various attempts have been made to design educational interventions to foster the development of numerical magnitude processing.
- These types of interventions have been embedded in larger-scale kindergarten programs for children from low-income communities (Dyson et al., 2013; Griffin, 2004 ) and children at-risk for DD (Toll et al., 2013) .
- From these interventions, it is, however, not possible to determine the precise effects of stimulating numerical magnitude processing.
- More relevant are therefore intervention studies that only focused on very specific aspects of numerical magnitude processing, as reviewed in Table 4 and Box 3.
Most of the existing interventions have been applied to kindergarteners or children
- From low-income backgrounds, yet surprisingly few studies have focused on older children or children with DD.
- Wilson et al. (2006) and Kucian et al. (2011) showed that computerized interventions significantly improved children with DD's numerical magnitude processing skills.
- Both studies did not include a control group who did not receive the intervention, which makes it difficult to evaluate whether these improvements were related to the intervention or to other factors, such as maturation or repeated testing.
- A next step will be to investigate how brain activity changes in response to the educational interventions reviewed above, an approach that has been successfully applied in the field of reading (McCandliss, 2010) .
- Only one study has examined the effect of a computerized numerical training program "Rescue Calcularis" on brain activity in children with and without DD (Kucian et al., 2011) and revealed significant neuroplastic changes of the intervention in both groups.
SUMMARY AND CONCLUSIONS
- One of the most robust findings in the literature that sought to uncover the association between numerical magnitude processing and mathematics achievement is that children who are better in determining which of two symbolic numbers is the largest have higher achievement in mathematics.
- Relatedly, children with DD show significant deficits in their ability to compare symbolic numbers.
- In view of this, it can be argued that such relationships are more robust and that the difficulty in finding relationships between non-symbolic numerical magnitude processing and mathematics achievement may indicate that the kinds of representations and processes measured by these tasks are not particularly critical for children's development of school-relevant mathematical competencies.
- Such research is, however, necessary to unravel the developmental trajectory of these associations.
- From a practical point of view, the existence of computer games to foster children's understanding of numerical magnitudes is extremely relevant for the early intervention of atrisk children.
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Figures (4)
Table 2 The nature of the relationship between symbolic (digit) comparison task performance and mathematics achievement in typically developing participants. The digit comparison measure(s) used and number range of the task are given in brackets. 
Table 4 Cognitive interventions that focused on numerical magnitude processing 
Table 1 The nature of the relationship between nonsymbolic (dot) comparison task performance and mathematics achievement in typically developing participants. The dot comparison measure(s) used and number range of the task are given in brackets. 
Table 3 Comparison of children with developmental dyscalculia (DD) and matched controls (C) on symbolic or non-symbolic number comparison tasks.
Citations
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TL;DR: The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains.
Abstract: Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non-symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two-level random-effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non-symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non-symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school-aged children and for adults.
511 citations
Cites background or methods or result from "How do symbolic and non-symbolic nu..."
...…review of the literature suggests that the association between numerical magnitude processing and broader mathematical competence might be more robust and consistent for studies with the symbolic magnitude processing tasks than for studies with the non-symbolic task (De Smedt et al., 2013)....
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...There are some intervention studies in which numerical magnitude comparison skills have been successfully trained (see De Smedt et al., 2013, for a review)....
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...…magnitude processing relate to broader mathematical competence might provide helpful background information for educational interventions aiming at improving learners’ numerical processing skills as preparation for more advanced mathematical learning (De Smedt et al., 2013; Feigenson et al., 2013)....
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...This is in line with suggestions made by De Smedt et al. (2013) in their narrative review of the literature, who raised the possibility that the association between magnitude processing and broader mathematical competence might be more robust for studies with the symbolic magnitude processing tasks…...
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...This hampers the integration of empirical findings across studies in narrative reviews of the literature (De Smedt et al., 2013; Feigenson, Libertus & Halberda, 2013)....
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TL;DR: It is empirically demonstrated that the symbolic number system is modulated more by development and education than the nonsymbolic system, and in contrast to the nonsedical system, the symbolic system ismodulated by language.
Abstract: Symbolic (ie, with Arabic numerals) approximate arithmetic with large numerosities is an important predictor of mathematics It was previously evidenced to onset before formal schooling at the kindergarten age (Gilmore et al, 2007) and was assumed to map onto pre-existing nonsymbolic (ie, abstract magnitudes) representations With a longitudinal study (Experiment 1), we show, for the first time, that nonsymbolic and symbolic arithmetic demonstrate different developmental trajectories In contrast to Gilmore et al’s (2007) findings, Experiment 1 showed that symbolic arithmetic onsets in grade 1, with the start of formal schooling, not earlier Gilmore et al (2007) had examined English-speaking children, whereas we assessed a large Dutch-speaking sample The Dutch language for numbers can be cognitively more demanding, for example, due to the inversion property in numbers above twenty Thus, for instance, the number 48 is named in Dutch “achtenveertig” (eight and forty) instead of “forty eight” To examine the effect of the language of numbers, we conducted a cross-cultural study with English- and Dutch-speaking children that had similar SES and math achievement skills (Experiment 2) Results demonstrated that Dutch-speaking kindergarteners lagged behind English-speaking children in symbolic arithmetic, not nonsymbolic and demonstrated a WM overload in symbolic arithmetic, not nonsymbolic Also, we show for the first time that the ability to name two-digit numbers highly correlates with symbolic approximate arithmetic not nonsymbolic Our experiments empirically demonstrate that the symbolic number system is modulated more by development and education than the nonsymbolic system Also, in contrast to the nonsymbolic system, the symbolic system is modulated by language
363 citations
Cites background from "How do symbolic and non-symbolic nu..."
...When symbolic approximation is being proven to be an important, consistent predictor of children’s math achievement (De Smedt et al., 2013; Xenidou-Dervou et al., 2013), we demonstrate that the ability to name large numbers plays an important role in its developmental onset....
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...To our knowledge, this is the first evidence for the effect of the inversion property on the onset of symbolic approximation; a core system for the development of mathematical achievement (De Smedt et al., 2013; Xenidou-Dervou et al., 2013)....
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...Early symbolic processing skills have been consistently proven to be significant predictors of math achievement (for a review see De Smedt et al., 2013; see also Göbel et al., 2014b; Lyons et al., 2014), even beyond general processing skills, such as working memory (WM) abilities (Xenidou-Dervou et…...
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...Given the extensive research that indicates the importance of symbolic processing skills in the development of children’s math achievement (De Smedt et al., 2013; XenidouDervou et al., 2013; Lyons et al., 2014), future studies should place more focus on the role that language plays in developing…...
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...The ANS is often assumed to be linked with the development of our symbolic mathematical abilities (for a review see Feigenson et al., 2013; but see also the review by De Smedt et al., 2013)....
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TL;DR: A meta-analysis of cross-sectional and longitudinal studies revealed a moderate but statistically significant association between number acuity and math performance, and suggested that many previous studies were underpowered due to small sample sizes.
340 citations
TL;DR: Analysis of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender.
Abstract: Recent findings that earlier fraction knowledge predicts later mathematics achievement raise the question of what predicts later fraction knowledge. Analyses of longitudinal data indicated that whole number magnitude knowledge in first grade predicted knowledge of fraction magnitudes in middle school, controlling for whole number arithmetic proficiency, domain general cognitive abilities, parental income and education, race, and gender. Similarly, knowledge of whole number arithmetic in first grade predicted knowledge of fraction arithmetic in middle school, controlling for whole number magnitude knowledge in first grade and the other control variables. In contrast, neither type of early whole number knowledge uniquely predicted middle school reading achievement. We discuss the implications of these findings for theories of numerical development and for improving mathematics learning.
163 citations
TL;DR: The integrative theory of numerical development as discussed by the authors posits that a coherent theme does exist, and that this theme unifies numerical development from infancy to adulthood, and thus the mental number line expands rightward to encompass larger whole numbers, leftward to include negatives, and interstitially to include fractions and decimals.
Abstract: Understanding of numerical development is growing rapidly, but the volume and diversity of findings can make it difficult to perceive any coherence in the process. The integrative theory of numerical development posits that a coherent theme does exist—progressive broadening of the set of numbers whose magnitudes can be accurately represented—and that this theme unifies numerical development from infancy to adulthood. From this perspective, development of numerical representations involves four major acquisitions: (a) representing magnitudes of nonsymbolic numbers increasingly precisely, (b) linking nonsymbolic to symbolic numerical representations, (c) extending understanding to increasingly large whole numbers, and (d) extending understanding to all rational numbers. Thus, the mental number line expands rightward to encompass larger whole numbers, leftward to encompass negatives, and interstitially to include fractions and decimals.
163 citations
Cites background from "How do symbolic and non-symbolic nu..."
...…two including meta-analyses, indicate that relations between ANS acuity and math achievement are weaker and less consistent than relations between representations of symbolic numerical magnitude and math achievement (Chen & Li, 2014; De Smedt et al., 2013; Fazio, Bailey, Thompson, & Siegler, 2014)....
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References
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TL;DR: The horizontal segment of the intraparietal sulcus appears as a plausible candidate for domain specificity: It is systematically activated whenever numbers are manipulated, independently of number notation, and with increasing activation as the task puts greater emphasis on quantity processing.
Abstract: Did evolution endow the human brain with a predisposition to represent and acquire knowledge about numbers? Although the parietal lobe has been suggested as a potential substrate for a domain-specific representation of quantities, it is also engaged in verbal, spatial, and attentional functions that may contribute to calculation. To clarify the organisation of number-related processes in the parietal lobe, we examine the three-dimensional intersection of fMRI activations during various numerical tasks, and also review the corresponding neuropsychological evidence. On this basis, we propose a tentative tripartite organisation. The horizontal segment of the intraparietal sulcus (HIPS) appears as a plausible candidate for domain specificity: It is systematically activated whenever numbers are manipulated, independently of number notation, and with increasing activation as the task puts greater emphasis on quantity processing. Depending on task demands, we speculate that this core quantity system, analogous t...
2,281 citations
Book•
01 Jan 2009TL;DR: Carey argues that the key to understanding cognitive development lies in recognizing conceptual discontinuities in which new representational systems emerge that have more expressive power than core cognition and are also incommensurate with core cognition as mentioned in this paper.
Abstract: Only human beings have a rich conceptual repertoire with concepts like tort, entropy, Abelian group, mannerism, icon and deconstruction. How have humans constructed these concepts? And once they have been constructed by adults, how do children acquire them? While primarily focusing on the second question, in The Origin of Concepts, Susan Carey shows that the answers to both overlap substantially. Carey begins by characterizing the innate starting point for conceptual development, namely systems of core cognition. Representations of core cognition are the output of dedicated input analyzers, as with perceptual representations, but these core representations differ from perceptual representations in having more abstract contents and richer functional roles. Carey argues that the key to understanding cognitive development lies in recognizing conceptual discontinuities in which new representational systems emerge that have more expressive power than core cognition and are also incommensurate with core cognition and other earlier representational systems. Finally, Carey fleshes out Quinian bootstrapping, a learning mechanism that has been repeatedly sketched in the literature on the history and philosophy of science. She demonstrates that Quinian bootstrapping is a major mechanism in the construction of new representational resources over the course of childrens cognitive development. Carey shows how developmental cognitive science resolves aspects of long-standing philosophical debates about the existence, nature, content, and format of innate knowledge. She also shows that understanding the processes of conceptual development in children illuminates the historical process by which concepts are constructed, and transforms the way we think about philosophical problems about the nature of concepts and the relations between language and thought.
1,688 citations
TL;DR: There are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual Differences in the present correlate with children’s past scores on standardized maths achievement tests, extending all the way back to kindergarten.
Abstract: Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.
1,220 citations
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...RT, NDE(RT), Acc: [5–28] NDE(RT) RT, Acc: [1–9] RT, Acc: [10–58] NDE(RT) RT, NDE(RT): [20–72] NDE(RT) RT: [20–72] W: [12–40] RTs Acc, RT NDE(Acc): [1–9] Acc, RT...
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...[39] 8–10 RT: [1–9] RT, Acc: [21–98] Landerl et al....
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TL;DR: It is shown that the resolution of the Approximate Number System continues to increase throughout childhood, with adultlike levels of acuity attained surprisingly late in development.
Abstract: Behavioral, neuropsychological, and brain imaging research points to a dedicated system for processing number that is shared across development and across species. This foundational Approximate Number System (ANS) operates over multiple modalities, forming representations of the number of objects, sounds, or events in a scene. This system is imprecise and hence differs from exact counting. Evidence suggests that the resolution of the ANS, as specified by a Weber fraction, increases with age such that adults can discriminate numerosities that infants cannot. However, the Weber fraction has yet to be determined for participants of any age between 9 months and adulthood, leaving its developmental trajectory unclear. Here we identify the Weber fraction of the ANS in 3-, 4-, 5-, and 6-year-old children and in adults. We show that the resolution of this system continues to increase throughout childhood, with adultlike levels of acuity attained surprisingly late in development.
770 citations
Additional excerpts
...RT, NDE(RT), Acc: [5–28] NDE(RT) RT, Acc: [1–9] RT, Acc: [10–58] NDE(RT) RT, NDE(RT): [20–72] NDE(RT) RT: [20–72] W: [12–40] RTs Acc, RT NDE(Acc): [1–9] Acc, RT...
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...[39] 8–10 RT: [1–9] RT, Acc: [21–98] Landerl et al....
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...[41] 8–10 RT: [1–9], RT: [21–98] Piazza et al....
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TL;DR: The authors found that dyscalculia is the result of specific disabilities in basic numerical processing, rather than the consequence of deficits in other cognitive abilities, with no special features consequent on their reading or language deficits.
727 citations