Symbolic Number Processing Research Articles

A color-digit Stroop task shows numerical influence on numerosity processing.

The numerical Stroop task involves presenting participants with two digits that differ in physical size and numerical value and asking them to report which digit had the larger size or value while ignoring the other dimension. Previous studies show that participants have difficulty ignoring the irrelevant dimension and thus have implications on the automaticity of numerical processing. The present study investigates the automatic influence of numerical value on numerosity processing in a novel Stroop-like task. In two experiments, participants were presented with digits made of colored stripes and asked to identify the number of different colors. In both experiments, interference and facilitation effects were found, supporting the automaticity of symbolic number processing and its influence on numerosity processing. These findings expand upon previous research on numerical as well as counting Stroop tasks, and have potential implications for studying interference and basic numerical processing in children and clinical populations.

Typical Mathematical Education Cannot Predict Non‐Verbal Number Sense

ABSTRACTFormal education would promote symbolic number processing ability, but the relationship between non‐symbolic number sense and mathematical education remains controversial. The current investigation hypothesized that non‐symbolic number sense is independent from the limited human experience of mathematical education, while symbolic number processing could be influenced by related closely to mathematical education. Experiment 1 compared both non‐symbolic number sense and arithmetic computation of undergraduates majoring in STEM fields and humanities. Experiment 2 compared homologous undergraduates from China and Kyrgyzstan in non‐symbolic number sense and mathematical cognitive processing. Both of two experiments found that mathematical education was significantly associated with symbolic mathematics, but not with non‐symbolic number sense. The results suggest that while mathematical education might improve symbolic mathematics, it does not alter non‐symbolic number sense.

(Dis)similarities between non-symbolic and symbolic number representations: Insights from vector space models

Empirical evidence in support of a shared system for non-symbolic and symbolic number processing has been inconclusive. The current study aims to address this question in a novel way, specifically by testing whether the efficient coding principle based on co-occurrence of number symbols in natural language holds for both non-symbolic and symbolic number processing. The efficient coding principle postulates that perception is optimized when stimuli frequently co-occur in a natural environment. We hypothesized that both numerical ratios and co-occurrence frequencies of symbolic numbers would significantly influence participants' performance on a non-symbolic and symbolic number comparison task. To test this hypothesis, we employed latent semantic analysis on a TASA corpus to quantify number co-occurrence in natural language and calculate language similarity estimates. We engaged 73 native English speakers (mean age = 19.36, standard deviation = 1.83) with normal or corrected vision and no learning disorders in a number comparison task involving non-symbolic (dot arrays) and symbolic stimuli (Arabic numerals and English number words). Results showed that numerical ratios significantly predicted participants' performances across all number formats (ps < 0.001). Language similarity estimates derived from everyday language also predicted performance on the non-symbolic task and the symbolic task involving number words (ps < 0.007). Our results highlight the complex nature of numerical processing, pointing to the co-occurrence of number symbols in natural language as an auxiliary factor in understanding the shared characteristics between non-symbolic and symbolic number representations. Given that our study focused on a limited number range (5 to 16) and a specific task type, future studies should explore a wider range of tasks and numbers to further test the role of the efficient coding principle in number processing.

The relationships among working memory, inhibitory control, and mathematical skills in primary school children: Analogical reasoning matters

The current study examined the mediating role of analogical reasoning in the relationships among working memory, inhibitory control, and children’s mathematical skills. Two hundred fifty-one students from first to third grades were tested on visual-spatial working memory, verbal working memory, inhibitory control, analogical reasoning, and different mathematical skills (i.e., symbolic number processing and mathematical reasoning). After controlling for age and gender, analogical reasoning significantly contributed to both symbolic number processing and mathematical reasoning. The relationship between verbal working memory and symbolic number processing, as well as the relationship between verbal working memory and mathematical reasoning, were significantly mediated by analogical reasoning. The exploratory analyses further revealed that there was no significant age difference in the roles of analogical reasoning. These results highlight the important role of analogical reasoning in explaining the relationships among working memory, inhibitory control, and children’s different mathematical abilities. The findings also indicate that analogical reasoning, the ability of identifying and processing critical relational information, may be a potential avenue to improve children’s mathematical skills.

Symbolic and non-symbolic predictors of number line task in Italian kindergarteners.

The number line estimation task (NLE) is often used as a predictor for broader measures of mathematical achievement. In spite of its popularity, it is still not clear whether the task is based on symbolic or non-symbolic numerical competence. In particular, there is only a very limited amount of studies investigating the relationship between NLE performance and symbolic vs. non-symbolic math skills in children who have not yet begun formal schooling. This study investigates the strength of the association between NLE performance and symbolic and non-symbolic tasks in young kindergarteners. Ninety two 5-year-old children completed the NLE task (range 0-100) and a battery of early numerical competence tests including symbolic-lexical tasks, symbolic semantic tasks, and non-symbolic semantic tasks. The relationship between symbolic and non-symbolic early numerical competence and NLE performance was analyzed using a regression model based on the Bayesian Information Criterion (BIC). Results show that only symbolic semantic tasks are significant predictors of NLE performance. These results suggest that symbolic numerical knowledge is involved in number line processing among young children, whilst non-symbolic knowledge is not. This finding brings new data to the debate on the relationship between non-symbolic numeral knowledge and symbolic number processing and supports the evidence of a primary role of symbolic number processing already in young kindergarteners.

Electrophysiological Correlates of the Interaction of Physical and Numerical Size in Symbolic Number Processing: Insights from a Novel Go/Nogo Numerical Stroop Task.

The interaction of physical and numerical size has been investigated and repeatedly demonstrated in the numerical Stroop task, in which participants compare digits of different physical sizes. It is, however, not entirely clear yet what psychological processes contribute to this interaction. The aim of the present study is to investigate the role of inhibition in the interaction of physical and numerical size, by introducing a novel paradigm that is suitable to elicit inhibition-related event-related potential components. To this end, we combined the go/nogo paradigm with the numerical Stroop task while measuring EEG and reaction times. Participants were presented with Arabic number pairs and had to press a button if the number on one side was numerically larger and they had to refrain from responding if the number on the other side was numerically larger. The physical size of the number pairs was also manipulated, in order to create congruent, neutral, and incongruent trials. Behavioural results confirmed the well-established numerical distance and numerical Stroop effects. Analysis of electrophysiological data revealed the classical go/nogo electrophysiological effects with numerical stimuli, and showed that peak amplitudes were larger for nogo than for go trials on the N2, as well as on the P3 component, on frontal and midline electrodes. When analysing the congruency effects, the peak amplitude of N2 was larger in incongruent trials than in neutral and congruent trials, while there was no evidence of a congruency effect on the P3 component peaks. Further analysis of the electrophysiological data revealed an additional facilitatory effect in the go trials, as well as an additional interference effect in the nogo trials. Taken together, it seems that interference effects are probably resolved by inhibitory processes and that facilitatory effects are affected by different cognitive control processes required by go versus nogo trials.

The role of basic number processing in high mathematics achievement in primary school

While symbolic number processing is an important correlate for typical and low mathematics achievement, it remains to be determined whether children with high mathematics achievement also have excellent symbolic number processing abilities. We investigated this question in 64 children (aged 8 to 10), i.e., 32 children with persistent high achievement in mathematics (above the 90th percentile) and 32 average-achieving peers (between the 25th and 75th percentile). Children completed measures of symbolic number processing (comparison and order). We additionally investigated the roles of spatial visualization and working memory. High mathematics achievers were faster and more accurate in order processing compared to average achievers, but no differences were found in magnitude comparison. High mathematics achievers demonstrated better spatial visualization ability, while group differences in working memory were less clear. Spatial visualization ability was the only significant predictor of group membership. Our results therefore highlight the role of high spatial visualization ability in high mathematics achievement.

Nonsymbolic Numerosity Maps at the Occipitotemporal Cortex Respond to Symbolic Numbers.

Numerosity, the set size of a group of items, helps guide human and animals' behavior and decisions. Numerosity perception is thought to be a precursor of symbolic numerical cognition. Previously, we uncovered neural populations selectively tuned to numerosities organized in a network of topographic maps in human association cortex. Here we investigate whether these numerosity maps are also involved in the processing of symbolic numbers, using 7T fMRI and a number-detection task. We recruited 7 participants (3 females) and found that the numerosity map at the temporal-occipital cortex (NTO) also responds to symbolic numbers. Furthermore, we found that numerosity-tuned neuronal populations at the NTO map in the left hemisphere are tuned to symbolic numbers. These results reveal different functions of the numerosity maps and support a link between numerosity representation and symbolic number processing in the ventral temporal-occipital cortex.SIGNIFICANCE STATEMENT Humans and other animals share an intuitive "number sense" to approximately represent numerosity. However, humans possess a unique ability to process number symbols (e.g., Arabic numbers). It has been argued that the human understanding of symbolic numbers is rooted in our ability to numerosity perception. Here we investigate whether numerosity-tuned neuronal populations organized at a network of topographic maps also respond to symbolic numbers. We find one of the maps at the temporal-occipital cortex is involved in symbolic numerical cognition and the neuronal populations are tuned to numbers. These results provide evidence for a link between nonsymbolic numerosity and symbolic number processing.

Early neural markers for individual difference in mathematical achievement determined from rational number processing

The neural markers for individual differences in mathematical achievement have been studied extensively using magnetic resonance imaging; however, high temporal resolution electrophysiological evidence for individual differences in mathematical achievement require further elucidation. This study evaluated the event-related potential (ERP) when 48 college students with high or low mathematical achievement (HA vs. LA) matched non-symbolic and symbolic rational numbers. Behavioral results indicated that HA students had better performance in the discretized non-symbolic matching, although the two groups showed similar performances in the continuous matching. ERP data revealed that even before non-symbolic stimulus presentation, HA students had greater Bereitschaftspotential (BP) amplitudes over posterior central electrodes. After the presentation of non-symbolic numbers, HA students had larger N1 amplitudes at 160 ms post-stimulus, over left-lateralized parieto-occipital electrodes. After the presentation of symbolic numbers, HA students displayed more profound P1 amplitudes at 100 ms post-stimulus, over left parietal electrodes. Furthermore, larger BP and N1 amplitudes were associated with the shorter reaction times, and larger P1 amplitudes corresponded to lower error rates. The BP effect could indicate preparation processing, and early left-lateralized N1 and P1 effects could reflect the non-symbolic and symbolic number processing along the dorsal neural pathways. These results suggest that the left-lateralized P1 and N1 components elicited by matching non-symbolic and symbolic rational numbers can be considered as neurocognitive markers for individual differences in mathematical achievement.

Symbolic number comparison and number priming do not rely on the same mechanism

In elementary symbolic number processing, the comparison distance effect (in a comparison task, the task is more difficult with smaller numerical distance between the values) and the priming distance effect (in a number processing task, actual number is easier to process with a numerically close previous number) are two essential phenomena. While a dominant model, the approximate number system model, assumes that the two effects rely on the same mechanism, some other models, such as the discrete semantic system model, assume that the two effects are rooted in different generators. In a correlational study, here we investigate the relation of the two effects. Critically, the reliability of the effects is considered; therefore, a possible null result cannot be attributed to the attenuation of low reliability. The results showed no strong correlation between the two effects, even though appropriate reliabilities were provided. These results confirm the models of elementary number processing that assume distinct mechanisms behind number comparison and number priming.

Predictors of middle school students’ growth in symbolic number comparison performance

The ability to efficiently compare number symbols, such as digits, is associated with mathematics competence across the lifespan. Performance on symbolic number comparison tasks differ across age groups; young students who are developing fluency with digits improve on symbolic number comparison, and performance is better in adults than children. However, whether this improvement continues for older students who are fluent with number symbols, and what cognitive factors relate to this improvement, is unknown. This study used a longitudinal sample of U.S. middle school students (n = 394) to examine whether symbolic number comparison performance changes over middle school (i.e., students aged 11-14), whether there are individual differences in students’ rate of change, and potential predictors of that change. Students completed measures of single-digit symbolic number comparison, nonsymbolic number comparison, executive function (EF), and mathematics competence in Grade 5 (M = 11.02 years; SD = 0.32), and double-digit symbolic number comparison in Grades 6-8. Results showed that, on average, students’ symbolic number comparison performance improved from Grades 6-8. Grade 5 Symbolic number comparison performance predicted Grade 8 symbolic number comparison and rate of change over Grades 6-8. Grade 5 nonsymbolic number comparison, EF, and mathematics competence predicted Grade 8 symbolic number comparison performance. Results suggest that numerical magnitude processing, executive functions, and mathematics competence are related to symbolic number processing well into middle school, and that students continue to refine their ability to process number symbols into adolescence.

A common neural substrate for number comparison, hand reaching and grasping: A SDM-PSI meta-analysis of neuroimaging studies

The reliance of number processing on sensorimotor mechanisms involved in hand action has been extensively documented by behavioural studies. Nonetheless, where and how the computations of number and hand action interact in the brain has received limited attention. In this study we investigated the brain networks underlying symbolic number comparison and the hand action of reaching and grasping, capitalizing on functional imaging studies meta-analyzed with the seed-based d mapping with permutation of subject images meta-analytic method (SDM-PSI). The main objective was to test whether and to what extent symbolic number processing recruits the same sensorimotor network involved in the hand action of reaching and grasping. We included 42 studies (756 participants) adopting symbolic number comparison tasks and 58 studies (867 participants) investigating hand reaching and hand grasping. The conjunction analysis of brain networks common to number processing, reaching, and grasping revealed spatial convergence over frontoparietal areas. Specifically, four clusters were identified, in and around the left and right intraparietal sulci, in the left precentral gyrus, and in the supplementary motor area. The degree of overlap was extensive, since the reach/grasp network mostly included the number areas. A qualitative analysis of functional characterization capitalizing on the Neurosynth database depicted a strong multifunctionality of the regions of overlap between numbers and hand action: these brain areas were also associated to a variety of functions within the domains of memory and imagery, visuospatial attention, and language. Overall, these results characterize the neuroanatomical substrate of the interaction between reaching, grasping, and symbolic number comparison.

Toward a unified account of nonsymbolic and symbolic representations of number: Insights from a combined psychophysical-computational approach.

There is an ongoing, vibrant debate about whether numerical information in both nonsymbolic and symbolic notations would be supported by different neurocognitive systems or rather by a common preverbal approximate number system, which is ratio dependent and follows Weber's law. Here, we propose that the similarities between nonsymbolic and symbolic number processing can be explained based on the principle of efficient coding. To probe this hypothesis we employed a new empirical approach, by predicting the behavioural performance in number comparison tasks with symbolic (i.e., number words) and nonsymbolic (i.e., arrays of dots) information not only from numerical ratio, but for the first time also from natural language data. That is, we used data extracted from vector-space models that are informative about the distributional pattern of number-words usage in natural language. Results showed that linguistic estimates predicted the behavioural performance in both symbolic and nonsymbolic tasks. However, and critically, our results also showed a task-dependent dissociation: linguistic data better predicted the performance in the symbolic task, whereas real numerical ratio better predicted the performance in the nonsymbolic task. These findings indicate that efficient coding of environmental regularities is an explanatory principle of human behavior in tasks involving numerical information. They also suggest that the ability to discriminate a stimulus from similar ones varies as a function of the specific statistical structure of the considered learning environment.

Symbols Are Special: An fMRI Adaptation Study of Symbolic, Nonsymbolic, and Non-Numerical Magnitude Processing in the Human Brain.

How are different formats of magnitudes represented in the human brain? We used functional magnetic resonance imaging adaptation to isolate representations of symbols, quantities, and physical size in 45 adults. Results indicate that the neural correlates supporting the passive processing of number symbols are largely dissociable from those supporting quantities and physical size, anatomically and representationally. Anatomically, passive processing of quantities and size correlate with activation in the right intraparietal sulcus, whereas symbolic number processing, compared with quantity processing, correlates with activation in the left inferior parietal lobule. Representationally, neural patterns of activation supporting symbols are dissimilar from neural activation patterns supporting quantity and size in the bilateral parietal lobes. These findings challenge the longstanding notion that the culturally acquired ability to conceptualize symbolic numbers is represented using entirely the same brain systems that support the evolutionarily ancient system used to process quantities. Moreover, these data reveal that regions that support numerical magnitude processing are also important for the processing of non-numerical magnitudes. This discovery compels future investigations of the neural consequences of acquiring knowledge of symbolic numbers.

Non-Symbolic Numerosity and Symbolic Numbers are not Processed Intuitively in Children: Evidence From an Event-Related Potential Study

The approximate number system (ANS) theory and the ANS mapping account have been the most prominent theories on non-symbolic numerosity processing and symbolic number processing respectively, over the last 20 years. Recently, there is a growing debate about these theories, mainly based on research in adults. However, whether the ANS theory and ANS mapping account explain the processing of non-symbolic numerosity and symbolic number in childhood has received little attention. In the current ERP study, we first examined whether non-symbolic numerosity processing in 9-to-12-year-old children (N = 34) is intuitive, as proposed by the ANS theory. Second, we examined whether symbolic number processing is rooted in non-symbolic numerosity processing, as proposed the ANS mapping account. ERPs were measured during four same-different match-to-sample tasks with non-symbolic numerosities, symbolic numbers, and combinations of both. We found no evidence for intuitive processing of non-symbolic numerosity. Instead, children processed the visual features of non-symbolic stimuli more automatically than the numerosity itself. Moreover, children do not seem to automatically activate non-symbolic numerosity when processing symbolic numbers. These results challenge the ANS theory and ANS mapping account in 9-to-12-year-old children.

The relationship between math anxiety and math achievement in young children is mediated through working memory, not by number sense, and it is not direct

Math anxiety is considered a predictor of math achievement, although the cognitive mechanism whereby math anxiety impairs math achievement is unclear. The paper presents the results of cross-sectional (N = 241) and longitudinal (N = 369) studies conducted among early school-aged children on the cognitive mechanism whereby math anxiety impairs math achievement. The following hypotheses were tested: (1) math anxiety directly affects math achievement; (2) in accordance with processing efficiency and attentional cognitive theories, math anxiety indirectly affects math achievement through working memory; (3) in accordance with the cognitive deficit model, math anxiety indirectly affects math achievement through number sense. The results mostly confirm the mediating role of working memory and undermine the mediating role of number sense and the direct path in the relationship between math anxiety and math achievement. Because previous studies undertaken in adults show the direct path from math anxiety to math achievement and the role of symbolic number processing in explaining the relationship between the two, the methodological and developmental aspects of the obtained results are discussed in the paper.

Impaired large numerosity estimation and intact subitizing in developmental dyscalculia.

It is believed that the approximate estimation of large sets and the exact quantification of small sets (subitizing) are supported by two different systems, the Approximate Number System (ANS) and Object Tracking System (OTS), respectively. It is a current matter of debate whether they are both impaired in developmental dyscalculia (DD), a specific learning disability in symbolic number processing and calculation. Here we tackled this question by asking 32 DD children and 32 controls to perform a series of tasks on visually presented sets, including exact enumeration of small sets as well as comparison of large, uncountable sets. In children with DD, we found poor sensitivity in processing large numerosities, but we failed to find impairments in the exact enumeration of sets within the subitizing range. We also observed deficits in visual short-term memory skills in children with dyscalculia that, however, did not account for their low ANS acuity. Taken together, these results point to a dissociation between quantification skills in dyscalculia, they highlight a link between DD and low ANS acuity and provide support for the notion that DD is a multifaceted disability that covers multiple cognitive skills.

Can you trust your number sense: Distinct processing of numbers and quantities in elementary school children

Theories of number development have traditionally argued that the acquisition and discrimination of symbolic numbers (i.e., number words and digits) are grounded in and are continuously supported by the Approximate Number System (ANS)—an evolutionarily ancient system for number. In the current study, we challenge this claim by investigating whether the ANS continues to support the symbolic number processing throughout development. To this end, we tested 87 first- (Age M = 6.54 years, SD = 0.58), third- (Age M = 8.55 years, SD = 0.60) and fifth-graders (Age M = 10.63 years, SD = 0.67) on four audio-visual comparison tasks (1) Number words–Digits, (2) Tones–Dots, (3) Number words–Dots, (4) Tones–Digits, while varying the Number Range (Small and Large), and the Numerical Ratio (Easy, Medium, and Hard). Results showed that larger and faster developmental growth in the performance was observed in the Number Words–Digits task, while the tasks containing at least one non-symbolic quantity showed smaller and slower developmental change. In addition, the Ratio effect (i.e., the signature of ANS being addressed) was present in the Tones–Dots, Tones–Digits, and Number Words–Dots tasks, but was absent in the Number Words–Digits task. These findings suggest that it is unlikely that the ANS continuously underlines the acquisition and the discrimination of the symbolic numbers. Rather, our results indicate that non-symbolic quantities and symbolic numbers follow qualitatively distinct developmental paths, and argue that the latter ones are processed in a semantic network which starts to emerge from an early age.

Professional mathematicians do not differ from others in the symbolic numerical distance and size effects

The numerical distance effect (it is easier to compare numbers that are further apart) and size effect (for a constant distance, it is easier to compare smaller numbers) characterize symbolic number processing. However, evidence for a relationship between these two basic phenomena and more complex mathematical skills is mixed. Previously this relationship has only been studied in participants with normal or poor mathematical skills, not in mathematicians. Furthermore, the prevalence of these effects at the individual level is not known. Here we compared professional mathematicians, engineers, social scientists, and a reference group using the symbolic magnitude classification task with single-digit Arabic numbers. The groups did not differ with respect to symbolic numerical distance and size effects in either frequentist or Bayesian analyses. Moreover, we looked at their prevalence at the individual level using the bootstrapping method: while a reliable numerical distance effect was present in almost all participants, the prevalence of a reliable numerical size effect was much lower. Again, prevalence did not differ between groups. In summary, the phenomena were neither more pronounced nor more prevalent in mathematicians, suggesting that extremely high mathematical skills neither rely on nor have special consequences for analogue processing of symbolic numerical magnitudes.

Mapping letters to numbers: Potential mechanisms of literal symbol processing

This study examined impermanent symbol-referent connections (e.g., x = 5) with literal symbols – important symbols for higher-level mathematics that may be difficult to process due to interference from pre-existing associations from literacy. We examined literal symbol processing at the group and individual levels, and executive functioning and symbol-referent mapping ability as potential cognitive mechanisms. Participants completed same-different judgments using numbers, literal symbols, and novel symbols; symbol-referent mapping; and executive function tasks. On average, participants' judgments with literal symbols took longer than with other symbols (i.e., a literal symbol processing cost). Individual differences in this cost were associated with variability in literal symbol mapping performance, but not with executive functioning. Results suggest the processing cost results from learning symbol-referent connections with symbols that have prior associations, and identifies initial symbol-referent mapping ability as a potential mechanism that contributes to this difficulty. Findings provide novel characterization of symbolic number processing relevant for higher-level mathematics.