Numerical estrangement and integration between symbolic and non-symbolic numerical information: Task-dependence and its link to math abilities in adults
Introduction
Most adults have access to two different systems to represent numerical information: an exact number system that relies on different forms of number symbols (e.g., Arabic numerals, number words) to represent exact numerical information, and an approximate number system that represents imprecise numerical magnitudes from non-symbolic stimuli. The exact number system requires explicit exposure to and instruction in the use of a symbolic number system. For example, the more parents talk about numbers with their toddlers, the better their later understanding of the cardinality principle, i.e., that the last word in the count sequence refers to the total number of objects in the set that was counted (Levine, Suriyakham, Rowe, Huttenlocher, & Gunderson, 2010).
In contrast, the approximate number system (ANS) seems to be present from birth (Izard, Sann, Spelke, & Streri, 2009). Its precision is typically measured with non-symbolic number comparison tasks. In these tasks, participants choose which member of a dot array pair contains the larger number of dots while controlling for perceptual information that often correlates with numerical information (e.g., density, area, convex hull; Dietrich, Huber, & Nuerk, 2015). Behavioral performance on these non-symbolic number comparison tasks is typically dependent on the ratio between the numbers. Specifically, when the ratio is closer to 1 (e.g., 15 dots vs 16 dots, a 15:16 ratio), participants tend to respond slower and less accurately than when the numbers are at distant ratios (e.g., 15 dots vs 30 dots, a 1:2 ratio).
What might be the relation between symbolic and non-symbolic number systems? According to the ANS mapping account, one acquires the meaning of symbolic numbers by mapping them onto approximate non-symbolic magnitude (Dehaene, 2001; Piazza, 2010). This notion is supported by evidence that similar behavioral effects (Defever, Sasanguie, Gebuis, & Reynvoet, 2011; Holloway & Ansari, 2009) and brain activation patterns (Cantlon et al., 2009; Dehaene, Izard, & Piazza, 2005; Eger, Sterzer, Russ, Giraud, & Kleinschmidt, 2003) are observed when processing non-symbolic and symbolic numbers. An alternative hypothesis posits that small numerical symbols such as number words are first mapped onto an object tracking system (OTS), a precise representation with a limited capacity of up to four objects (Carey, 2011). Then, with increasing knowledge of the count list, one might use that knowledge to infer principles about number relations such that numbers coming later in the count list are larger (Reynvoet & Sasanguie, 2016). This process gradually results in a separate system for symbolic numbers where symbolic numbers are represented through order associations with other symbolic numbers.
While the exact and approximate number systems are thought to distinctly represent different forms of numerical information, previous research has yielded mixed results regarding the extent to which these systems are integrated in adulthood (Liu, Schunn, Fiez, & Libertus, 2015; Liu, Schunn, Fiez, & Libertus, 2018; Lyons, Ansari, & Beilock, 2012; Schneider et al., 2017). More importantly, even though previous studies have found that both symbolic and non-symbolic number processing are related to math abilities (e.g., Libertus, Odic, & Halberda, 2012; Sasanguie, Lyons, De Smedt, & Reynvoet, 2017; Schneider et al., 2017), the way in which numerical integration between non-symbolic and symbolic numerical information relates to adults' formal math skills remains largely uncharacterized. Therefore, in this study, we implemented a mixed-format number comparison task and a number-letter discrimination task to investigate the integration between symbolic and non-symbolic numerical information (henceforth referred to as “numerical integration”), as well as a standardized math assessment to test how numerical integration relates to adults' formal math abilities.
Similar to the non-symbolic number comparison tasks described above, symbolic number comparison tasks have participants identify the larger of two Arabic numerals (or other symbolic number formats). In general, participants are slower at comparing two symbolic numbers with small numerical distance (e.g., judging 6 is smaller than 7) than with large numerical distance (e.g., judging 6 is smaller than 9). This is known as the distance effect (Defever et al., 2011; Sasanguie, De Smedt, Defever, & Reynvoet, 2012). Both distance and ratio effects are typically explained by the overlapping magnitude representations on a mental number line. On this mental number line, magnitudes are represented in a Gaussian distribution such that the closer the numbers are on this mental number line, the harder they are to discriminate. Thus, the similarity of the distance and ratio effects observed in symbolic and non-symbolic number processing tasks has been taken as evidence for an integration between exact and approximate numerical representations such that non-symbolic number representations are employed during numerical comparisons, even when the numerical information is symbolic (Dehaene, 2001; Dehaene & Akhavein, 1995).
Other evidence suggests that the two number systems might not be so tightly integrated (e.g., Lyons et al., 2012; Sasanguie, De Smedt, & Reynvoet, 2017). For instance, Lyons et al. (2012) asked adults to compare numbers presented either in symbolic formats (Arabic numerals), non-symbolic format (dot arrays), or mixed formats (dots vs numerals). The authors argued that if the symbolic and non-symbolic numerical information were indeed integrated, mixed-format comparisons should result in comparable accuracy and response time to same-format comparisons. However, they found significantly higher response time and lower accuracy in mixed-format comparisons relative to same-format comparisons. The authors reasoned that additional cognitive effort was needed in the mixed-format comparison likely due to a lack of integration between non-symbolic and symbolic number representations, i.e., an estrangement between non-symbolic and symbolic number representations. A switch cost for mixed-format trials has also been found in an audio-visual comparison paradigm where participants indicated the numerically larger of two stimuli when presented with spoken number words, tone sequences, Arabic numerals or dot arrays (Marinova, Sasanguie, & Reynvoet, 2018). In a follow-up study, Marinova, Sasanguie, and Reynvoet (2021) manipulated three experimental factors (the number range, the ratio difficulty, and the presentation modality) in this audio-visual comparison task and found ratio effects in all tasks containing non-symbolic number stimuli, but not in the task containing symbolic numbers only, and a switch cost was also observed for mixed-format conditions. These findings thus further support two distinct number processing systems that are not tightly integrated.
The comparison tasks used by Lyons et al. (2012) explicitly asked participants to process the two formats of number to make the comparison. However, making explicit magnitude-based judgements in this comparison task may force the translation of symbolic number representations into non-symbolic ones or vice versa, leading to increased response time and reduced accuracy in the mixed-format comparisons. To explore whether numerical integration is evident without explicit magnitude comparisons, Liu et al. (2015) implemented a number-letter discrimination task. In this task, adult participants were asked to decide whether two-item symbol strings were composed of Arabic numerals or letters, with the stimuli superimposed on dot arrays designed to match or mismatch quantities denoted by the numeral strings. Importantly, the dot array (including its quantity) was irrelevant for completing the task. Nevertheless, participants responded more accurately and faster when the Arabic numerals matched (versus mismatched) the dot quantities, suggesting that numerical integration occurs between non-symbolic and symbolic numbers even when the task does not require decisions about number magnitude or the non-symbolic number to be processed. In a follow-up experiment using event-related potentials (ERPs), adult participants passively viewed the same images as in the above-mentioned study (Liu et al., 2018). The amplitude of the N1, an ERP component linked to number processing, was greater for matching than mismatching dot quantities and Arabic numerals. This suggests that the human brain readily integrates non-symbolic and symbolic number representation even in the absence of a task that requires magnitude judgements.
Another important question in math cognition that has yet to be fully answered is whether number processing in symbolic, non-symbolic, or both formats is crucial for formal math achievement, especially in adults, and earlier findings are mixed. On the one hand, symbolic number knowledge has been consistently found to be correlated with formal math performance (see De Smedt, Noël, Gilmore, & Ansari, 2013 for a review). For example, Castronovo and Göbel (2012) found that adults with greater math achievement showed faster and more accurate performance in symbolic number comparisons. Moreover, Lyons and Beilock (2011) found that symbolic number-ordering ability significantly predicted adults' complex mental arithmetic performance. In their study, adults identified whether triads of Arabic numerals (ranging from 1 to 9) were all in increasing order from left to right regardless of the numerical distance between numbers (e.g., “1, 2, 3” increases as does “1, 2, 5”). Participants' mental-arithmetic performance was evaluated using four different mental arithmetic tasks (i.e., addition, subtraction, multiplication, and division). The symbolic number-ordering and mental arithmetic task performance was correlated, even when controlling for numeral identification, performance on symbolic and non-symbolic number comparison tasks, and performance on letter ordering and working memory tasks. Finally, in a meta-analysis, Schneider et al. (2017) also found consistent associations between symbolic number processing and mathematical competence in children and adults.
On the other hand, support for a link between non-symbolic number processing and math abilities, especially in adults, is more mixed (Braham & Libertus, 2018; Libertus et al., 2012; Park & Brannon, 2013; Price, Palmer, Battista, & Ansari, 2012). For example, Libertus et al. (2012) found a positive association between precision of the approximate number system and performance on the quantitative portion of the Scholastic Aptitude Test (SAT), a standardized college entrance exam. To measure precision of non-symbolic number representations, college students completed a non-symbolic number comparison task, in which they decided whether there were more blue or yellow dots in a visual display. A robust correlation between precision of the approximate number system and quantitative scores on the SAT was found even when controlling for performance on the verbal portion of the SAT. In their meta-analysis, Schneider et al. (2017) also found a significant association between non-symbolic number processing and mathematical competence in children and adults, albeit a weaker one than observed between symbolic number processing and mathematical competence.
However, some other studies have not found a correlation between non-symbolic number processing and math achievement in adults (Castronovo & Göbel, 2012; Inglis, Attridge, Batchelor, & Gilmore, 2011; Price et al., 2012). For instance, Price et al. (2012) used three different methods to present stimuli in a non-symbolic number comparison task (i.e., simultaneously presented dot arrays that were either intermixed or spatially separated, and sequentially presented dot arrays) to assess the precision of participants' approximate number system. They did not find correlations between the performance on any of the three versions of the non-symbolic number comparison task and a measure of math fluency (Woodcock Johnson Math Fluency subtest, Woodcock, McGrew, Mather, et al., 2001).
One possible explanation for the inconsistencies in the literature regarding the link between the approximate number system and math abilities may be that it is mediated by the integration between symbolic and non-symbolic numerical information. Earlier studies have found that greater precision in mapping of non-symbolic to symbolic numerical representations positively correlates with children's and adolescents' math achievement (De Smedt, Verschaffel, & Ghesquière, 2009; Holloway & Ansari, 2009; Mazzocco, Feigenson, & Halberda, 2011). In other words, relatively weak integration between non-symbolic and symbolic numerical information may result in noisier mappings between them, which may lead to greater difficulty and less efficiency in accessing the exact representations of numerical information when solving math problems (Holloway & Ansari, 2009). Consistent findings have also been reported in adults, whose symbolic number-ordering ability mediates the link between approximate number system precision and complex mathematical skills (Lyons & Beilock, 2011). Moreover, prior work has suggested that the integration between symbolic and non-symbolic numerical information in adults may be supported by greater engagement of the parietal lobe compared to children who rely more heavily on frontal areas. This age-related shift to greater parietal engagement may reflect the maturation of underlying number representation and increasing flexibility in mapping between numerical symbols and the magnitudes they represent (Ansari, Garcia, Lucas, Hamon, & Dhital, 2005).
As reviewed above, there are uncertainties in the literature regarding the integration between symbolic and non-symbolic numerical information, and to what extent this numerical integration relates to adults' formal math abilities. Therefore, the goals of the current study were twofold: 1) to determine whether evidence for numerical integration is task dependent, and 2) to explore how numerical integration relates to adults' math abilities. To this end, we administered a number comparison task previously used by Lyons et al. (2012) and a number-letter discrimination task previously employed by Liu et al. (2015) to the same group of adult participants. There are two main differences between our comparison task and the comparison task used by Lyons et al. (2012). First, they used a blocked design in which participants were aware of the format of the upcoming stimulus set, whereas our trial types were randomly intermixed such that participants did not know whether a trial was a same- or mixed-format trial ahead of time. Second, in their analyses, Lyons and colleagues did not separate performance based on presentation order of the mixed trials (i.e., dot first or numeral first). Instead, we examined participants' performances on mixed trials separately based on the order in which different formats are presented (i.e., dot first vs Arabic numeral first). The number-letter discrimination task was identical to the one used by Liu et al. (2015). Even though these two conceptually different tasks measure numerical integration differently, we hypothesized that their indices of numerical integration would correlate. Additionally, we administered a standardized assessment of math ability (Woodcock Johnson III Tests of Achievement, Woodcock et al., 2001), and predicted that the two indices of numerical integration would correlate with adults' math abilities.
Section snippets
Participants
One hundred twenty-two adults participated in this study (77 female, age range: 18–35 years, mean age (Mage) = 23.39 years, standard deviation (SD) = 4.72). Participants were recruited from the Pittsburgh community, and written informed consent was obtained from all participants prior to completing any research activities as approved by the local Institutional Review Board. All participants were native English speakers with normal or corrected-to-normal vision. Participants received monetary
Number comparison task
For the number comparison task, we were interested in how different combinations of number formats (dots and numerals) affect participants' RT and accuracy. To this end, RT and accuracy were submitted to two separate repeated-measures ANOVAs with condition (four levels: DD, NN, DN and ND) as the within-subject variable. For RT, there was a significant main effect of condition, F (3, 336) = 125.5, p < .001, ηp2= 0.53. Participants responded faster for same-format conditions (DD, NN) than
Discussion
In the current study, we had two aims: to test whether numerical integration or estrangement is task-dependent and associated with adults' math abilities. To answer those questions, we administered two tasks to measure numerical integration/estrangement, and a standardized math assessment to the same group of adult participants. In the number comparison task with non-symbolic dot arrays and Arabic numerals, participants indicated which of the sequentially presented stimuli was numerically
Conclusions
The goals of this current study were twofold: 1) to determine whether evidence for numerical integration/ estrangement is task dependent, and 2) to explore the relation between numerical integration/estrangement and adults' math abilities using different measures. To answer those questions, we administered both a number comparison task similar to the one used by Lyons et al. (2012) and a number-letter discrimination task adapted from Liu et al. (2015) to the same group of adult participants. In
Declarations of interest
None.
Acknowledgement and funding
This work was supported by the National Science Foundation (grant number: 1734735). We would like to thank Taylor Casteel for collecting the data, and Corrine Durisko and Griffin Koch for their helpful discussion.
References (46)
- et al.
Format-dependent representations of symbolic and non-symbolic numbers in the human cortex as revealed by multi-voxel pattern analyses
NeuroImage
(2014) - et al.
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
Trends in Neuroscience and Education
(2013) - et al.
The predictive value of numerical magnitude comparison for individual differences in mathematics achievement
Journal of Experimental Child Psychology
(2009) - et al.
Children’s representation of symbolic and nonsymbolic magnitude examined with the priming paradigm
Journal of Experimental Child Psychology
(2011) - et al.
A Supramodal number representation in human intraparietal cortex
Neuron
(2003) - et al.
Differences in the acuity of the approximate number system in adults: The effect of mathematical ability
Acta Psychologica
(2013) - 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) - et al.
Mental representation: What can pitch tell us about the distance effect?
Cortex
(2008) - et al.
Intuitive sense of number correlates with math scores on college-entrance examination
Acta Psychologica
(2012) - et al.
Numerical ordering ability mediates the relation between number-sense and arithmetic competence
Cognition
(2011)