Non-symbolic Representations Of Number Research Articles

From spontaneous focusing on numerosity to mathematics achievement: The mediating role of non-symbolic number processing and mapping between symbolic and non-symbolic representations of number

Young children who readily demonstrate a self-initiated orientation, or spontaneous focusing on numerosity (SFON), perform better in mathematics in later years. To further our understanding of the mechanisms behind this relation, the present longitudinal study with 150 Chinese preschoolers examined the potential mediating role of non-symbolic number processing and mapping between symbolic and non-symbolic representations of number. Mediation analysis indicates two independent pathways leading from SFON to math achievement—namely the non-symbolic number processing pathway and the number mapping pathway—providing a more comprehensive model to explain the predictability of SFON on children’s math achievement. Our findings indicate that children with a stronger tendency to focus on the cardinal information of the environment are better at processing set sizes as well as mapping non-symbolic quantity information onto numbers, leading to better math achievement.

Single-neuron representation of nonsymbolic and symbolic number zero in the human medial temporal lobe

The number zero holds a special status among numbers, indispensable for developing a comprehensive number theory.1,2,3,4 Despite its importance in mathematics, the neuronal foundation of zero in the human brain is unknown. We conducted single-neuron recordings in neurosurgical patients5,6,7 while they made judgments involving nonsymbolic number representations (dot numerosity), including the empty set, and symbolic numbers (Arabic numerals), including numeral zero. Neurons showed responsiveness to either the empty set or numeral zero, but not both. Neuronal activity to zero in both nonsymbolic and symbolic formats exhibited a numerical distance effect, indicating that zero representations are integrated together with countable numerosities and positive integers at the low end of the number line.8,9 A boundary in neuronal coding existed between the nonsymbolic empty set and small numerosities, correlating with the relative difficulty in discriminating numerosity zero behaviorally. Conversely, no such boundary was found for symbolic zero activity, suggesting that symbolic representations integrate zero with other numerals along the number line, reconciling its outlier role. The status of zero as a special nonsymbolic numerical quantity is reflected in the activity of neurons in the human brain, which seems to serve as a scaffold for more advanced representations of zero as a symbolic number.

Working memory and arithmetic impairments in children with FMR1 premutation and gray zone alleles.

ABSTRACT.Expansive mutations in familial mental retardation 1 (FMR1) gene have been associated with different phenotypes. Full mutations are associated with intellectual disability and autism spectrum disorder; premutations are associated with math learning difficulties and working memory impairments. In gray zone, neuropsychological development has not yet been described.Objectives:This study aimed to describe the frequency of FMR1 premutation and gray zone alleles in a school population sample representing a broad spectrum of variation in math achievement and detail school achievement and cognitive performance in the children identified with FMR1 premutation or gray zone alleles.Methods:We described a two-phase study. In the first phase, 2,195 school-age children were screened for math achievement. In the second phase, 378 children with normal intelligence were neuropsychologically assessed and genotyped for FMR1. Of these, 121 children (61 girls) performed below percentile 25 in mathematics (MD group) and 257 children (146 girls) performed above percentile 25 (control group).Results:Four pupils presented expanded alleles, one premutation and three gray zone alleles. The girl with the premutation and one boy with a gray zone allele presented impairments in working memory and arithmetic performance below percentile 6, compatible with the diagnosis of developmental dyscalculia. These children’s difficulties were not associated with inaccuracy of nonsymbolic number representations or literacy impairments. Dyscalculia in these children seems to be associated mainly with working memory impairments.Conclusions: FMR1 expansions in the gray zone may contribute to dyscalculia in otherwise healthy and normally intelligent children.

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.

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.

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.

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.

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.

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.

Continuous visual properties of number influence the formation of novel symbolic representations.

Numerical symbols are thought to be mapped onto preexisting nonsymbolic representations of number. A growing body of evidence suggests that nonsymbolic numerical processing is significantly influenced by the associated visual properties of continuous quantity (e.g., surface area, density), but their role in the acquisition of novel symbols is unknown. Forty undergraduate students were trained to associate novel abstract symbols with numerical magnitudes. Half of the symbols were associated with nonsymbolic arrays in which total surface area and numerosity were correlated ("congruent"), and the other symbols were associated with arrays in which total surface area was equated across numerosities ("incongruent"). As numbers are represented in multiple formats (words, digits, nonsymbolic arrays), we also tested whether providing auditory nonword labels facilitated symbol learning. Following training, participants engaged in speeded comparisons of the newly learnt symbols. Comparisons were affected by the ratio between the numerosities associated with each symbol, a characteristic marker of numerical processing. Furthermore, comparisons were hardest for large-ratio comparisons of symbols associated with incongruent area and numerosity pairing during learning. In turn, these findings call for the further investigation of visual parameters on the development of numerical cognition.

Annual Conference 2013 paper: Six-year-old children considered failing in maths

This study confirmed that children as young as 6-years-old can already be deemed as failing in mathematics and considered a range of factors which may be contributing towards that judgement. These included the core number sense, connections between symbolic and non-symbolic representations of number, mathematical language used in the home, memory, anxiety and relative young age of the pupil in their year group. Case studies of four children formed the basis of the study. Thematic analysis conducted on the data revealed lack of understanding, weak memory and behavioural aspects as the main themes to explain the pupils’ difficulties with maths attainment. Intervention led to different outcomes depending on the profile of the individual pupil. The apparent progress of pupils indicates that targeted intervention concentrating on reinforcing connections between finger gnosis, numerosity and symbolic number patterns appear to have positive effects on these children’s learning, leading to greater confidence and enhanced understanding.

Left parietal TMS disturbs priming between symbolic and non-symbolic number representations

An amodal number representation activated by all types of numerical input, irrespective of the input notation, has often been proposed to be located in the left or right intraparietal sulcus (IPS). Two cross-notational priming experiments were carried out to test the existence of a notation-independent magnitude representation in the left or right parietal lobes. In Experiment 1, stimuli were Arabic digits and number words. Results revealed no significant effect of repetitive transcranial magnetic stimulation (rTMS) over left or right IPS during prime presentation. In contrast, in Experiment 2, digits and dot patterns were intermixed and here the priming distance effect (PDE) was reduced in the right TMS condition and absent for stimulation over left IPS. These findings suggest: (1) that TMS over left but not right IPS disrupts processes that are crucial for priming when symbolic and non-symbolic stimuli are intermixed, and (2) that disruption of the left IPS on its own is not sufficient to disrupt cross-notational priming when purely symbolic number notations are used. Our results point towards a crucial role of the left hemisphere for the mapping between small symbolic and non-symbolic numerosities.

Non-symbolic halving in an Amazonian indigene group.

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS.

Predicting sights from sounds: 6-month-olds’ intermodal numerical abilities

Although the psychophysics of infants’ nonsymbolic number representations have been well studied, less is known about other characteristics of the approximate number system (ANS) in young children. Here three experiments explored the extent to which the ANS yields abstract representations by testing infants’ ability to transfer approximate number representations across sensory modalities. These experiments showed that 6-month-olds matched the approximate number of sounds they heard to the approximate number of sights they saw, looking longer at visual arrays that numerically mismatched a previously heard auditory sequence. This looking preference was observed when sights and sounds mismatched by 1:3 and 1:2 ratios but not by a 2:3 ratio. These findings suggest that infants can compare numerical information obtained in different modalities using representations stored in memory. Furthermore, the acuity of 6-month-olds’ comparisons of intermodal numerical sequences appears to parallel that of their comparisons of unimodal sequences.

Symbolic and non-symbolic distance effects in children and their connection with arithmetic skills

The ability to compare numerical magnitudes is assumingly related to children’s arithmetic skills. The role of symbolic and non-symbolic number representations in this relationship is, however, still a matter of debate. To address this issue we assessed addition and subtraction skills of 8–10-year-old children ( n = 35) and asked them to compare numerical magnitudes of dot patterns and Arabic digits in different numerical ranges. Results revealed that the relationship between numerical magnitude comparisons and arithmetic skills is not restricted to symbolic stimuli, but that it can also be detected for non-symbolic dot patterns. The range of numerosities for which this relationship was found and the manner in which the magnitude comparison was related to arithmetic skills differed regarding the dots and digits. These findings highlight the role of both symbolic and non-symbolic number representations in the development of arithmetic skills and strengthen the view of different developmental trajectories underlying these representations.

Visual enumeration: A bi-directional mapping process between symbolic and non-symbolic representations of number?

Over the last 30 years, numerical estimation has been largely studied. Recently, Castronovo and Seron (2007) proposed the bi-directional mapping hypothesis in order to account for the finding that dependent on the type of estimation task (perception vs. production of numerosities), reverse patterns of performance are found (i.e., under- and over-estimation respectively). Here, we further investigated this hypothesis by submitting adult participants to three types of numerical estimation task: 1) a perception task, in which participants had to estimate the numerosity of a non-symbolic collection; 2) a production task, in which participants had to approximately produce the numerosity of a symbolic numerical input; 3) a reproduction task, in which participants had to reproduce the numerosity of a non-symbolic numerical input. Our results gave further support to the finding that different patterns of performance are found according to the type of estimation task: 1) under-estimation in the perception task; 2) over-estimation in the production task; 3) accurate estimation in the reproduction task. Moreover, correlation analyses revealed that the more a participant under-estimated in the perception task, the more he/she over-estimated in the production task. We discussed these empirical data by showing how they can be accounted by the bi-directional mapping hypothesis (Castronovo & Seron, 2007).

Behavioral and Neural Basis of Number Sense in Infancy

Approximate number discrimination in adult human and nonhuman animals is governed by Weber's Law: The ratio between the values determines discriminability. Here, we review recent evidence from behavioral and neuroimaging studies that suggests that number sense in human infancy shares the same hallmark feature of Weber's Law and may rely on the same neural substrates as previously found in adults, children, and nonhuman animals. These findings support the notion of ontogenetic and phylogenetic continuity in number sense. New methods described here may help uncover how infants' early number sense supports the development of a mature number sense. Moreover, they may aid in understanding how children learn to map nonsymbolic number representations onto symbols for number by providing dependent measures that capture individual variability.

Children’s mapping between symbolic and nonsymbolic representations of number

When children learn to count and acquire a symbolic system for representing numbers, they map these symbols onto a preexisting system involving approximate nonsymbolic representations of quantity. Little is known about this mapping process, how it develops, and its role in the performance of formal mathematics. Using a novel task to assess children’s mapping ability, we show that children can map in both directions between symbolic and nonsymbolic numerical representations and that this ability develops between 6 and 8 years of age. Moreover, we reveal that children’s mapping ability is related to their achievement on tests of school mathematics over and above the variance accounted for by standard symbolic and nonsymbolic numerical tasks. These findings support the proposal that underlying nonsymbolic representations play a role in children’s mathematical development.

Nonsymbolic, approximate arithmetic in children: Abstract addition prior to instruction.

Do children draw upon abstract representations of number when they perform approximate arithmetic operations? In this study, kindergarten children viewed animations suggesting addition of a sequence of sounds to an array of dots, and they compared the sum to a second dot array that differed from the sum by 1 of 3 ratios. Children performed this task successfully with all the signatures of adults' nonsymbolic number representations: accuracy modulated by the ratio of the sum and the comparison quantity, equal performance for within- and cross-modality tasks and for addition and comparison tasks, and performance superior to that of a matched subtraction task. The findings provide clear evidence for nonsymbolic numerical operations on abstract numerical quantities in children who have not yet been taught formal arithmetic.

A developmental fMRI study of nonsymbolic numerical and spatial processing

This functional magnetic resonance imaging (fMRI) study systematically investigates whether there is a neurofunctional overlap of nonsymbolic numerical and spatial cognition in (intra)parietal regions in children and adults. The study also explores the association between finger use and (nonsymbolic) number processing across development. Twenty-four healthy individuals (12 children, 12 adults) were asked to make nonsymbolic numerical and spatial (experimental tasks) as well as color discriminations (control task). Using identical stimulus material across the three tasks disentangled nonsymbolic number representations from general attentional mechanisms, visual–spatial processing and response selection requirements. In both age groups, behavioral distance effects were obtained upon processing numerical (but not spatial and/or color) stimuli. Baseline imaging effects revealed age-dependent, partly overlapping activations of nonsymbolic numerical and spatial processing in the right posterior superior parietal lobe (PSPL) in adults only. Interestingly, regions more activated in children relative to adults were centred on bilateral supramarginal gyrus (SMG) and lateral portions of the anterior intraparietal sulcus (IPS), further extending to adjacent right post- and precentral gyrus, the latter of which has been reported to support grasping previously (Simon et al., 2002). Overall, our results are first evidence for an age-dependent neurofunctional link between areas supporting finger use and nonsymbolic number processing and furthermore, might be suggestive of a special role of fingers for the development of number magnitude representations and early arithmetic.