Non-symbolic Representations Research Articles

Learning about time: Knowledge of formal timing symbols is related to individual differences in temporal precision.

Throughout the life span, we are capable of representing quantities in the absence of language, or nonsymbolically. Additionally, over the course of development, we learn many symbolic measurement systems for representing quantities such as time and number. Despite substantial evidence of a relation between the acquisition of symbolic and nonsymbolic numerical acuity (see Halberda, Mazzocco, & Feigenson, 2008), no work has explored whether a similar relation exists between understanding temporal units of measurement and timing precision. That is, does a child's understanding of words like "second," "minute," and "hour" have any relation to their ability to tell which of two events lasted longer? Six- and 7-year-old children (n = 102, Mage = 83.44 months, 52 females), who are in the process of learning temporal units of measurement, completed a temporal discrimination task (assessing nonsymbolic temporal acuity) and a symbolic timing assessment. Results revealed a positive correlation between children's nonsymbolic temporal acuity and their understanding of temporal units of measurement. Importantly, this correlation held when controlling for age and numerical acuity, suggesting a unique relation between children's temporal acuity and their understanding of temporal units of measurement. This study is the first to show a relation between symbolic and nonsymbolic representations of time. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

The Relationship Between Non-symbolic and Symbolic Numerosity Representations in Elementary School: The Role of Intelligence.

This study aimed to estimate the extent to which the development of symbolic numerosity representations relies on pre-existing non-symbolic numerosity representations that refer to the Approximate Number System. To achieve this aim, we estimated the longitudinal relationships between accuracy in the Number Line (NL) test and “blue–yellow dots” test across elementary school children. Data from a four-wave longitudinal study involving schoolchildren in grades 1–4 in Russia and Kyrgyzstan (N = 490, mean age 7.65 years in grade 1) were analyzed. We applied structural equation modeling and tested several competing models. The results revealed that at the start of schooling, the accuracy in the NL test predicted subsequent accuracy in the “blue–yellow dots” test, whereas subsequently, non-symbolic representation in grades 2 and 3 predicted subsequent symbolic representation. These results indicate that the effect of non-symbolic representation on symbolic representation emerges after a child masters the basics of symbolic number knowledge, such as counting in the range of twenty and simple arithmetic. We also examined the extent to which the relationships between non-symbolic and symbolic representations might be explained by fluid intelligence, which was measured by Raven’s Standard Progressive Matrices test. The results revealed that the effect of symbolic representation on non-symbolic representation was explained by fluid intelligence, whereas at the end of elementary school, non-symbolic representation predicted subsequent symbolic representation independently of fluid intelligence.

Symbol grounding of number words in the subitization range.

In three experiments, we explored whether number words are grounded in a nonsymbolic representation of numerosity. We used a sentence-picture verification task, where participants are required to check whether the concept given in a sentence corresponds to the subsequently presented object. We concurrently manipulated numerical congruency by orthogonally varying the number word attached to the concept and the quantity of objects. The number words and numerosities varied from one to four in Experiment 1 and from six to nine in Experiment 2. In Experiment 3, we employed number words six and eight with the constraint that, in the incongruent condition, a constant number-to-numerosity ratio of 2:1 was used. In Experiment 1, we found that participants were faster and more efficient when concept-object matches were accompanied by numerical congruency relative to incongruency. On the other hand, no such difference was observed in Experiments 2 and 3 for numbers falling outside of the subitization range. The results are consistent with the hypothesis that number words from one to four are grounded in a nonsymbolic representation of the size of small sets.

Non-symbolic numerosities do not automatically activate spatial–numerical associations: Evidence from the SNARC effect

The SNARC (spatial-numerical association of response codes) effect is the finding that people are generally faster to respond to smaller numbers with left-sided responses and larger numbers with right-sided responses. The SNARC effect has been widely reported for responses to symbolic representations of number such as digits. However, there is mixed evidence as to whether it occurs for non-symbolic representations of number, particularly when magnitude is irrelevant to the task. Mitchell et al. reported a SNARC effect when participants were asked to make orientation decisions to arrays of one-to-nine triangles (pointing upwards vs. pointing downwards) and concluded that SNARC effects occur for non-symbolic, non-canonical representations of number. They additionally reported that this effect was stronger in the subitising range. However, here we report four experiments that do not replicate either of these findings. Participants made upwards/inverted decisions to one-to-nine triangles where total surface area was either controlled across numerosities (Experiments 1, 2, and 4) or increased congruently with numerosity (Experiment 3). There was no evidence of a SNARC effect either across the full range or within the subset of the subitising range. The results of Experiment 4 (in which we presented the original stimuli of Mitchell et al.) suggested that visual properties of non-symbolic displays can prompt SNARC-like effects driven by visual cues rather than numerosity. Taken in the context of other recent findings, we argue that non-symbolic representations of number do not offer a direct and automatic route to numerical-spatial associations.

Gibson and Gestalt: (re)presentation, processing, and construction

Seeking to avoid the typical binary choices between symbolic representations and no representations, or between functionally decomposable psychological processes and no psychological processes, or between direct perception of mind-independent physical properties and indirect perception of sense data, this article proposes that even a clear-thinking friend of Gibson can accept that perception of the environment is mediated by appearances and that such appearances are produced by functionally decomposable, rule-instantiating psychological processes. In so doing, it avoids both hyper-intellectualization of the perceptual process and the positing of sense data as immediate objects of perception. It considers notions of perceptual mediation from classical Gestalt psychology, while referencing recent phenomenological arguments showing that perception does not simply conform to mind-independent physical properties. Perception presents objects and scenes under a phenomenal aspect, in a manner suitable to be (and evaluable as) action-guiding. Finally, it examines neuroscientific decompositions of Gibsonian information pickup mechanisms, finding that such mechanisms are reasonably described as effecting processes of construction (on a not-necessarily-cognitivist conception of construction). As a separate point, such mechanisms are usefully described as including subpersonal, non-symbolic representations and processes of information integration.

Preschoolers and multi-digit numbers: A path to mathematics through the symbols themselves

Numerous studies from developmental psychology have suggested that human symbolic representation of numbers is built upon the evolutionally old capacity for representing quantities that is shared with other species. Substantial research from mathematics education also supports the idea that mathematical concepts are best learned through their corresponding physical representations. We argue for an independent pathway to learning “big” multi-digit symbolic numbers that focuses on the symbol system itself. Across five experiments using both between- and within-subject designs, we asked preschoolers to identify written multi-digit numbers with their spoken names in a two-alternative-choice-test or to indicate the larger quantity between two written numbers. Results showed that preschoolers could reliably map spoken number names to written forms and compare the magnitudes of two written multi-digit numbers. Importantly, these abilities were not related to their non-symbolic representation of quantities. These findings have important implications for numerical cognition, symbolic development, teaching, and education.

On the Difference Between Numerosity Processing and Number Processing.

The ANS theory on the processing of non-symbolic numerosities and the ANS mapping account on the processing of symbolic numbers have been the most popular theories on numerosity and number processing, respectively, in the last 20 years. Recently, both the ANS theory and the ANS mapping account have been questioned. In the current study, we examined two main assumptions of both the ANS theory and the ANS mapping account. ERPs were measured in 21 participants during four same-different match-to-sample tasks, involving non-symbolic stimuli, symbolic stimuli, or a combination of symbolic and non-symbolic stimuli (i.e., mapping tasks). We strictly controlled the visual features in the non-symbolic stimuli. Based on the ANS theory, one would expect an early distance effect for numerosity in the non-symbolic task. However, the results show no distance effect for numerosity. When analyzing the stimuli based on visual properties, an early distance effect for area subtended by the convex hull was found. This finding is in line with recent claims that the processing of non-symbolic stimuli may be dependent on the processing of visual properties instead of on numerosity (only). With regards to the processing of symbolic numbers, the ANS mapping account states that symbolic numbers are first mapped onto their non-symbolic representations before further processing, since the non-symbolic representation is at the basis of processing the symbolic number. If the non-symbolic format is the basic format of processing, one would expect that the processing of non-symbolic numerosities would not differ between purely non-symbolic tasks and mapping tasks, resulting in similar ERP waveforms for both tasks. Our results show that the processing of non-symbolic numerosities does differ between the tasks, indicating that processing of non-symbolic number is dependent on task format. This provides evidence against the ANS mapping account. Alternative theories for both the processing of non-symbolic numerosities and symbolic numbers are discussed.

Children's Non-symbolic and Symbolic Numerical Representations and Their Associations With Mathematical Ability.

Most empirical evidence supports the view that non-symbolic and symbolic representations are foundations for advanced mathematical ability. However, the detailed development trajectories of these two types of representations in childhood are not very clear, nor are the different effects of non-symbolic and symbolic representations on the development of mathematical ability. We assessed 253 4- to 8-year-old children’s non-symbolic and symbolic numerical representations, mapping skills, and mathematical ability, aiming to investigate the developmental trajectories and associations between these skills. Our results showed non-symbolic numerical representation emerged earlier than the symbolic one. Four-year-olds were capable of non-symbolic comparisons but not symbolic comparisons; five-year-olds performed better at non-symbolic comparisons than symbolic comparisons. This performance difference disappeared at age 6. Children at age 6 or older were able to map between symbolic and non-symbolic quantities. However, as children learn more about the symbolic representation system, their advantage in non-symbolic representation disappeared. Path analyses revealed that a direct effect of children’s symbolic numerical skills on their math performance, and an indirect effect of non-symbolic numerical skills on math performance via symbolic skills. These results suggest that symbolic numerical skills are a predominant factor affecting math performance in early childhood. However, the influences of symbolic and non-symbolic numerical skills on mathematical performance both declines with age.

Altered Parietal Activation during Non-symbolic Number Comparison in Children with Prenatal Alcohol Exposure.

Number processing is a cognitive domain particularly sensitive to prenatal alcohol exposure, which relies on intact parietal functioning. Alcohol-related alterations in brain activation have been found in the parietal lobe during symbolic number processing. However, the effects of prenatal alcohol exposure on the neural correlates of non-symbolic number comparison and the numerical distance effect have not been investigated. Using functional magnetic resonance imaging (fMRI), we examined differences in brain activation associated with prenatal alcohol exposure in five parietal regions involved in number processing during a non-symbolic number comparison task with varying degrees of difficulty. fMRI results are presented for 27 Cape Colored children (6 fetal alcohol syndome (FAS)/partial FAS, 5 heavily exposed (HE) non-sydromal, 16 controls; mean age ± SD = 11.7 ± 1.1 years). Fetal alcohol exposure was assessed by interviewing mothers using a timeline follow-back approach. Separate subject analyses were performed in each of five regions of interest, bilateral horizontal intraparietal sulci (IPS), bilateral posterior superior parietal lobules (PSPL), and left angular gyrus (left AG), using the general linear model with predictors for number comparison and difficulty level. Mean percent signal change for each predictor was extracted for each subject for each region to examine group differences and associations with continuous measures of alcohol exposure. Although groups did not differ in performance, controls activated the right PSPL more during non-symbolic number comparison than exposed children, but this was not significant after controlling for maternal smoking, and the right IPS more than children with fetal alcohol syndrome (FAS) or partial FAS. More heavily exposed children recruited the left AG to a greater extent as task difficulty increased, possibly to compensate, in part, for impairments in function in the PSPL and IPS. Notably, in non-syndromal heavily exposed children activation was impaired in the right PSPL, but spared in the right IPS. These results extend previous findings of poor right IPS recruitment during symbolic number processing in FAS/PFAS, indicating that mental representation of relative quantity is affected by prenatal alcohol exposure for both symbolic and non-symbolic representations of quantity.

Children's Non-symbolic, Symbolic Addition and Their Mapping Capacity at 4-7 Years Old.

The study aimed to examine the developmental trajectories of non-symbolic and symbolic addition capacities in children and the mapping ability between these two. We assessed 106 4- to 7-year-old children and found that 4-year-olds were able to do non-symbolic addition but not symbolic addition. Five-year-olds and older were able to do symbolic addition and their performance in symbolic addition exceeded non-symbolic addition in grade 1 (approximate age 7). These results suggested non-symbolic addition ability emerges earlier and is less affected by formal mathematical education than symbolic addition. Meanwhile, we tested children’s bi-directional mapping ability using a novel task and found that children were able to map between symbolic and non-symbolic representations of number at age 5. Their ability in mapping non-symbolic to symbolic number became more proficient in grade 1 (approximate age 7). This suggests children at age 7 have developed a relatively mature symbolic representation system.

The integration of symbolic and non-symbolic representations of exact quantity in preschool children

Preschoolers (n=62) completed tasks that tapped their knowledge of symbolic and non-symbolic exact quantities, their ability to translate among different representations of exact quantity (i.e., digits, number words, and non-symbolic quantities), and their non-symbolic, digit, and spoken number comparison skills (e.g., which is larger, 2 or 4?). As hypothesized, children's knowledge about non-symbolic exact quantities, spoken number words, and digits predicted their ability to map between symbolic and non-symbolic exact quantities. Further, their knowledge of the mappings between digits and non-symbolic quantities predicted symbolic number comparison (i.e., of spoken number words or written digits). Mappings between written digits and non-symbolic exact quantities developed later than the other mappings. These results support a model of early number knowledge in which integration across symbolic and non-symbolic representations of exact quantity underlies the development of children's number comparison skills.

Mathematics at Your Fingertips: Testing a Finger Training Intervention to Improve Quantitative Skills

Previous research indicates that the use of fingers as representations of ordinal and cardinal number is an important part of young children's mathematics learning. Further to this, some studies have shown that a finger training intervention can improve young children's quantitative skills. In this article, we argue that fingers represent a means for children to bridge between different external representations of number (including verbal, symbolic and non-symbolic representations). Therefore we predicted that an intervention that combined finger training with experience playing games involving multiple representations would lead to greater increases in quantitative skills than either aspect of the intervention alone. One hundred and thirty-seven children aged between six and seven years old took part in an intervention study over the course of four weeks. The study tested the impact of five different conditions on participants' quantitative skills, their finger gnosis, and their ability to compare magnitudes of two non-symbolic representations of number. Relative to a control group, those children receiving a finger training intervention saw gains in finger gnosis skills (the ability to differentiate fingers when touched, without visual cues). Those children who played number games saw an increase in their non-symbolic magnitude comparison skills. However, only those children who experienced both aspects of the intervention saw increases in quantitative skills, which were assessed using an instrument informed by Gelman and Gallistel's (1978) five principles of counting. The findings show that a finger training intervention, when combined with intensive exposure to multiple representations of number, can support young children's development of quantitative skills. This adds to evidence in the literature regarding the role of fingers in children's mathematics learning, and may have implications for pedagogical approaches.

Arithmetic Training Does Not Improve Approximate Number System Acuity.

The approximate number system (ANS) is thought to support non-symbolic representations of numerical magnitudes in humans. Recently much debate has focused on the causal direction for an observed relation between ANS acuity and arithmetic fluency. Here we investigate if arithmetic training can improve ANS acuity. We show with an experimental training study consisting of six 45-min training sessions that although feedback during arithmetic training improves arithmetic performance substantially, it does not influence ANS acuity. Hence, we find no support for a causal link where symbolic arithmetic training influences ANS acuity. Further, although short-term number memory is likely involved in arithmetic tasks we did not find that short-term memory capacity for numbers, measured by a digit-span test, was effected by arithmetic training. This suggests that the improvement in arithmetic fluency may have occurred independent of short-term memory efficiency, but rather due to long-term memory processes and/or mental calculation strategy development. The theoretical implications of these findings are discussed.

Representations of numerical and non-numerical magnitude both contribute to mathematical competence in children.

A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non-symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5- and 6-year-old children using comparison tasks of non-symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non-numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non-symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non-numerical magnitude.

Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers.

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.

Examining pitch and numerical magnitude processing in congenital amusia: A quasi-experimental pilot study

ABSTRACTIntroduction: Congenital amusia is a developmental disorder associated with deficits in pitch height discrimination or in integrating pitch sequences into melodies. This quasi-experimental pilot study investigated whether there is an association between pitch and numerical processing deficits in congenital amusia. Since pitch height discrimination is considered a form of magnitude processing, we investigated whether individuals with amusia present an impairment in numerical magnitude processing, which would reflect damage to a generalized magnitude system. Alternatively, we investigated whether the numerical processing deficit would reflect a disconnection between nonsymbolic and symbolic number representations. Method: This study was conducted with 11 adult individuals with congenital amusia and a control comparison group of 6 typically developing individuals. Participants performed nonsymbolic and symbolic magnitude comparisons and number line tasks. Results were available from previous testing using the Montreal Battery of Evaluation of Amusia (MBEA) and a pitch change detection task (PCD). Results: Compared to the controls, individuals with amusia exhibited no significant differences in their performance on both the number line and the nonsymbolic magnitude tasks. Nevertheless, they showed significantly worse performance on the symbolic magnitude task. Moreover, individuals with congenital amusia, who presented worse performance in the Meter subtest, also presented less precise nonsymbolic numerical representation. Conclusions: The relationship between meter and nonsymbolic numerical discrimination could indicate a general ratio processing deficit. The finding of preserved nonsymbolic numerical magnitude discrimination and mental number line representations, with impaired symbolic number processing, in individuals with congenital amusia indicates that (a) pitch height and numerical magnitude processing may not share common neural representations, and (b) in addition to pitch processing, individuals with amusia may present a deficit in accessing nonsymbolic numerical representations from symbolic representations. The symbolic access deficit could reflect a widespread impairment in the establishment of cortico-cortical connections between association areas.

Trajectories of Symbolic and Nonsymbolic Magnitude Processing in the First Year of Formal Schooling.

Sensitivity to numerical magnitudes is thought to provide a foundation for higher-level mathematical skills such as calculation. It is still unclear how symbolic (e.g. Arabic digits) and nonsymbolic (e.g. Dots) magnitude systems develop and how the two formats relate to one another. Some theories propose that children learn the meaning of symbolic numbers by scaffolding them onto a pre-existing nonsymbolic system (Approximate Number System). Others suggest that symbolic and nonsymbolic magnitudes have distinct and non-overlapping representations. In the present study, we examine the developmental trajectories of symbolic and nonsymbolic magnitude processing skills and how they relate to each other in the first year of formal schooling when children are becoming more fluent with symbolic numbers. Thirty Grade 1 children completed symbolic and nonsymbolic magnitude processing tasks at three time points in Grade 1. We found that symbolic and nonsymbolic magnitude processing skills had distinct developmental trajectories, where symbolic magnitude processing was characterized by greater gains than nonsymbolic skills over the one-year period in Grade 1. We further found that the development of the two formats only related to one another in the first half of the school year where symbolic magnitude processing skills influenced later nonsymbolic skills. These findings indicate that symbolic and nonsymbolic abilities have different developmental trajectories and that the development of symbolic abilities is not strongly linked to nonsymbolic representations by Grade 1. These findings also suggest that the relationship between symbolic and nonsymbolic processing is not as unidirectional as previously thought.

The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions.

How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the "symbol-grounding problem," is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture of the symbol-grounding problem.

Electrophysiological markers of newly acquired symbolic numerical representations: the role of magnitude and ordinal information

It is not yet understood how children acquire the meaning of numerical symbols and most existing research has focused on the role of approximate non-symbolic representations of number in this process (see Piazza, (Trends in Cognitive 14(12):542–551, 2010). However, numerical symbols necessitate an understanding of both order and magnitude, therefore order likely also plays a role in learning about number. Here, we used an artificial symbol-learning paradigm to contrast learning approximate numerical magnitude with learning numerical order. Thirty-two adult participants were randomly assigned to either the magnitude training group, in which they were trained to associate novel abstract symbols with non-symbolic numerical magnitudes, or the order training group, in which they were taught the ordinal sequence of the symbols, in analogy to the count sequence. Subsequently, electroencephalographic (EEG) data were recorded while participants completed a magnitude comparison task with the newly learned symbols. Comparing these newly acquired symbols affected event related potentials (ERPs) in a way that resembled comparisons of real numerical symbols [e.g. Temple & Posner, (Proc Natl Acad Sci USA 95(13):7836–7841, 1998)]. Furthermore, these ERP effects did not differ across learning groups, suggesting that adults formed similar representations regardless of method of instruction. In turn, the current findings highlight the potential role of ordinal information in symbolic acquisition.

Flowcharting the Meaning of Logic Formulas

In logic, representation of a domain (e.g., physical reality) comprises the things its expressions (formulas) refer to and their relationships. Recent research has examined the realm of nonsymbolic representations, especially diagrams. It is claimed that, in general, diagrams have advantages over linguistic descriptions. Current diagrammatic representations of logic formulas do not completely depict their underlying semantics, and they lack a basic static structure that incorporates elementary dynamic features, creating a conceptual gap that can lead to misinterpretation. This paper demonstrates a methodology for mapping the sense of a logic formula and producing diagrams that integrate linguistic conception, truth-values, and meaning and can be used in teaching, communication, and understanding, especially with students specializing in logic representation and reasoning.