Nonsymbolic, approximate arithmetic in children: abstract addition prior to instruction

To add or to remove? The role of working memory updating in preschool children's non-symbolic arithmetic abilities between addition and subtraction

Early computational capacity sets the foundation for mathematical learning. Preschool children have been shown to perform both non-symbolic addition and subtraction problems. However, it is still unknown how different operations affect the representational precision of the non-symbolic arithmetic solutions. The current study compared 83 4- and 5-year-olds' ability to solve non-symbolic addition and subtraction problems and examined the role of working memory underlying the two arithmetic processes. In the task, children were shown two sets of arrays that were sequentially occluded and were asked to either sum the arrays up (addition) or remove one array from the other (subtraction). The solution was then compared with a visible array. Children also completed two working memory tasks to measure their working memory storage and updating abilities. Results showed that children's representational precision in addition was higher than that in subtraction. Although children's performance in both arithmetic operations were associated with working memory updating, solving subtractive problems imposed additional cognitive resources in working memory updating. These findings reveal early developmental differences between addition and subtraction. Children's computational capacity in both addition and subtraction develops early in childhood, and the operation in subtraction demands more mental manipulation in working memory.

Predicting individual differences in preschoolers' numeracy and geometry knowledge: The role of understanding abstract relations between objects and quantities

Preschoolers' mathematics knowledge develops early and varies substantially. The current study focused on two ontogenetically early emerging cognitive skills that may be important predictors of later math skills (i.e., geometry and numeracy): children's understanding of abstract relations between objects and quantities as evidenced by their patterning skills and the approximate number system (ANS). Children's patterning skills, the ANS, numeracy, geometry, nonverbal intelligence (IQ), and executive functioning (EF) skills were assessed at age 4 years, and their numeracy and geometry knowledge was assessed again a year later at age 5 (N = 113). Above and beyond children's initial knowledge in numeracy and geometry, as well as IQ and EF, patterning skills and the ANS at age 4 uniquely predicted children's geometry knowledge at age 5, but only age 4 patterning uniquely predicted age 5 numeracy. Thus, although patterning and the ANS are related, they differentially explain variation in later geometry and numeracy knowledge. Results are discussed in terms of implications for early mathematics theory and research.

Number sense-arithmetic link in Grade 1 and Grade 2: A case of fluency

BACKGROUND

Recent research suggested fluent processing as an explanation on why number sense contributes to simple arithmetic tasks-'Fluency hypothesis'.

AIMS

The current study investigates whether number sense contributes to such arithmetic tasks when other cognitive factors are controlled for (including those that mediate the link); and whether this contribution varies as a function of participants' individual maths fluency levels.

SAMPLE

Four hundred and thirty-seven Chinese schoolchildren (186 females; Mage = 83.49 months) completed a range of cognitive measures in Grade 1 (no previous classroom training) and in Grade 2 (a year later).

METHODS

Number sense, arithmetic (addition and subtraction), spatial ability, visuo-spatial working memory, perception, reaction time, character reading and general intelligence were measured.

RESULTS

Our data showed that the link between number sense and arithmetic was weaker in Grade 1 (Beta = .15 for addition and .06 (ns) for subtraction) compared to Grade 2 (.23-.28), but still persisted in children with no previous maths training. Further, math's performance in Grade 1 did not affect the link between number sense and maths performance in Grade 2.

CONCLUSION

Our data extended previous findings by showing that number sense is linked with simple maths task performance even after controlling for multiple cognitive factors. Our results brought some evidence that number sense-arithmetic link is somewhat sensitive to previous formal maths education. Further research is needed, as the differences in effects between grades were quite small, and arithmetic in Grade 1 did not moderate the link at question in Grade 2.

Development of number line estsimation in Chinese preschoolers: a comparison between numerical and non-numerical symbols

To examine the level of number line estimation (NLE) in Chinese children with respect to representations of both numerical (Arabic numerals) and non-numerical symbols (dots), a total of 192 Chinese preschoolers aged between 4 and 5 years participated in four different NLE tasks. These tasks were paired to evaluate the accuracy and patterns of children's estimations in both numerical and non-numerical symbol contexts. Our findings indicate that, for Chinese preschoolers, relatively precise numerical symbol representations begin to emerge as early as 4 years of age. The accuracy of number line estimates for both 4- and 5-year-old children gradually increases in tasks involving both numerical and non-numerical symbols. Additionally, the development and patterns observed in the number line estimates of 4- and 5-year-old Chinese preschoolers are similar in both numerical symbol and non-numerical symbol tasks. These results indicate that the initiation of relatively precise numerical symbol representation and the turning point in the developmental trajectory, where the relatively precise representation for numerical symbols surpasses that of non-numerical ones, occur earlier in Chinese children than in their Western counterparts.

Motor activities to improve maths performance in pre-school children with typical development

Poor maths skills are associated with negative outcomes throughout life, such as lower academic qualifications, decreased professional success and socio-economic results. Mathematical skills emerge continuously throughout childhood and those that children acquire in pre-school are crucial for activities that support analytical thinking, problem-solving and reasoning and argumentation skills. Many of these activities are related to motor skills, since certain cognitive and motor areas of the brain are activated simultaneously when solving maths problems. Of all motor skills, visuomotor integration skills have been documented as those that are most consistently positively and significantly associated with maths performance in pre-school children. These skills are influenced by visual perception (spatial and attention skills), fine motor coordination and gross motor skills. Early intervention can improve visuomotor integration skills in pre-school children. Of all skills that make up visuomotor integration, spatial skills, in addition to being the first skills to influence numerical knowledge and the recognition of geometric shapes, are also those skills that form part of the majority of programs and activities to be worked on with pre-school children for the development of mathematical concepts. However, most intervention programs or activities to develop spatial skills are carried out in the classroom, usually through activities involving handling small objects. In this sense and given the significant association between visuomotor integration skills and gross motor skills, the main objective of this study was to list a set of activities to develop spatial skills, with a strong involvement of gross motor skills, in a classroom, playground or home context.

Development of precision of non-symbolic arithmetic operations in 4-6-year-old children

Children can represent the approximate quantity of sets of items using the Approximate Number System (ANS), and can perform arithmetic-like operations over ANS representations. Previous work has shown that the representational precision of the ANS develops substantially during childhood. However, less is known about the development of the operational precision of the ANS. We examined developmental change in the precision of the solutions to two non-symbolic arithmetic operations in 4-6-year-old U.S. children. We asked children to represent the quantity of an occluded set (Baseline condition), to compute the sum of two sequentially occluded arrays (Addition condition), or to infer the quantity of an addend after observing an initial array and then the array incremented by the unknown addend (Unknown-addend condition). We measured the precision of the solutions of these operations by asking children to compare their solutions to visible arrays, manipulating the ratio between the true quantity of the solution and the comparison array. We found that the precision of ANS representations that were not the result of operations (in the Baseline condition) was higher than the precision of solutions to ANS operations (in the Addition and Unknown-addend conditions). Further, we found that precision in the Baseline and Addition conditions improved significantly between 4 and 6 years, while precision in the Unknown-Addend condition did not. Our results suggest that ANS operations may inject "noise" into the representations they operate over, and that the development of the precision of different operations may follow different trajectories in childhood.

Diverse mathematical knowledge among indigenous Amazonians

We investigate number and arithmetic learning among a Bolivian indigenous people, the Tsimane', for whom formal schooling is comparatively recent in history and variable in both extent and consistency. We first present a large-scale meta-analysis on child number development involving over 800 Tsimane' children. The results emphasize the impact of formal schooling: Children are only found to be full counters when they have attended school, suggesting the importance of cultural support for early mathematics. We then test especially remote Tsimane' communities and document the development of specialized arithmetical knowledge in the absence of direct formal education. Specifically, we describe individuals who succeed on arithmetic problems involving the number five-which has a distinct role in the local economy-even though they do not succeed on some lower numbers. Some of these participants can perform multiplication with fives at greater accuracy than addition by one. These results highlight the importance of cultural factors in early mathematics and suggest that psychological theories of number where quantities are derived from lower numbers via repeated addition (e.g., a successor function) are unlikely to explain the diversity of human mathematical ability.

Association between motor and math skills in preschool children with typical development: Systematic review

Mathematics has been the subject in which many school-age children have revealed many difficulties. Research carried out in an attempt to understand the causes of failure in this area pointed to a positive association between mathematical performance and motor performance. Given the importance of math development in future school outcomes, knowing which specific motor components are most associated with math performance can help educators define better strategies for teaching mathematics. In this sense, the present systematic review study aimed to identify the components of motor skills most positively associated with mathematical performance in children with typical development who attend preschool. The PRISMA methodology (Preferred Reporting Items for Systematic Reviews and Meta-Analyses) was used in this study. The databases searched were ERIC, PubMED, SciELO, Scopus and Web of Science. A total of 2,909 articles were identified, of which 18 were included in this systematic review. The main results showed positive associations between fine motor skills, namely fine motor coordination and visuomotor integration, and mathematical performance. The math skill of numerical counting was the most associated with FMS. The main characteristics of the instruments used showed that the tasks of copying figures or drawings are the most used to assess visuomotor integration and the tasks of handling objects with pinch-like movements are the most used to assess fine motor coordination. Given the importance of mathematical performance in future school results, identifying early children with difficulties in fine motor skills will help educators to design better strategies for teaching mathematical skills. In this sense, the need to identify instruments to assess fine motor skills in preschool children with characteristics that facilitate their administration by the educator in the classroom context, i.e., requiring little administration time, not requiring much experience or training, the possibility of being administered to the group/class, few material resources, and the results can be easily interpreted, classified, and associated with mathematical performance.

Understanding the Effects of Transcranial Electrical Stimulation in Numerical Cognition: A Systematic Review for Clinical Translation

Atypical development of numerical cognition (dyscalculia) may increase the onset of neuropsychiatric symptoms, especially when untreated, and it may have long-term detrimental social consequences. However, evidence-based treatments are still lacking. Despite plenty of studies investigating the effects of transcranial electrical stimulation (tES) on numerical cognition, a systematized synthesis of results is still lacking. In the present systematic review (PROSPERO ID: CRD42021271139), we found that the majority of reports (20 out of 26) showed the effectiveness of tES in improving both number (80%) and arithmetic (76%) processing. In particular, anodal tDCS (regardless of lateralization) over parietal regions, bilateral tDCS (regardless of polarity/lateralization) over frontal regions, and tRNS (regardless of brain regions) strongly enhance number processing. While bilateral tDCS and tRNS over parietal and frontal regions and left anodal tDCS over frontal regions consistently improve arithmetic skills. In addition, tACS seems to be more effective than tDCS at ameliorating arithmetic learning. Despite the variability of methods and paucity of clinical studies, tES seems to be a promising brain-based treatment to enhance numerical cognition. Recommendations for clinical translation, future directions, and limitations are outlined.

Boosting Numerical Cognition in Children and Adolescents with Mathematical Learning Disabilities by a Brain-Based Intervention: A Study Protocol for a Randomized, Sham-Controlled Clinical Trial

Numbers are everywhere, and supporting difficulties in numerical cognition (e.g., mathematical learning disability (MLD)) in a timely, effective manner is critical for their daily use. To date, only low-efficacy cognitive-based interventions are available. The extensive data on the neurobiology of MLD have increased interest in brain-directed approaches. The overarching goal of this study protocol is to provide the scientific foundation for devising brain-based and evidence-based treatments in children and adolescents with MLD. In this double-blind, between-subject, sham-controlled, randomized clinical trial, transcranial random noise stimulation (tRNS) plus cognitive training will be delivered to participants. Arithmetic, neuropsychological, psychological, and electrophysiological measures will be collected at baseline (T0), at the end of the interventions (T1), one week (T2) and three months later (T3). We expect that tRNS plus cognitive training will significantly improve arithmetic measures at T1 and at each follow-up (T2, T3) compared with placebo and that such improvements will correlate robustly and positively with changes in the neuropsychological, psychological, and electrophysiological measures. We firmly believe that this clinical trial will produce reliable and positive results to accelerate the validation of brain-based treatments for MLD that have the potential to impact quality of life.

Extracting Summary Statistics of Rapid Numerical Sequences

We examine the ability of observers to extract summary statistics (such as the mean and the relative-variance) from rapid numerical sequences of two digit numbers presented at a rate of 4/s. In four experiments (total N = 100), we find that the participants show a remarkable ability to extract such summary statistics and that their precision in the estimation of the sequence-mean improves with the sequence-length (subject to individual differences). Using model selection for individual participants we find that, when only the sequence-average is estimated, most participants rely on a holistic process of frequency based estimation with a minority who rely on a (rule-based and capacity limited) mid-range strategy. When both the sequence-average and the relative variance are estimated, about half of the participants rely on these two strategies. Importantly, the holistic strategy appears more efficient in terms of its precision. We discuss implications for the domains of two pathways numerical processing and decision-making.

Relations between Subdomains of Home Math Activities and Corresponding Math Skills in 4-Year-Old Children

Most studies on the subject have investigated relations between home math activities and child math skills, without paying much attention to the specific skills that such activities foster and their alignment with children’s math assessments. The present study examined specific relations between subdomains of home math activities and children’s corresponding math skills (e.g., home counting/cardinality activities related to children’s counting/cardinality skills). Participants were 78 mostly middle-income, White parents and their four-year-old children (M age = 53.19 months; 45% girls). Parents completed a 24-item survey about the frequency of home activities supporting five subdomains of math: counting/cardinality, set comparison, number identification, adding/subtracting, and patterning. Children’s skills in these same five subdomains were assessed using the Preschool Early Numeracy Scale (PENS) and the Early Patterning Assessment. Specific relations were observed in set comparison, adding/subtracting, and patterning, such that higher frequency of home activities in these subdomains related to advanced child math skills in the corresponding subdomains. No specific relations were found in counting/cardinality and number identification. Overall home math activities averaged across the five math subdomains positively related to children’s overall math skills. Findings highlight the importance of engagement in specific math activities in the home environment and their significance for corresponding child math development.

Canonical representations of fingers and dots trigger an automatic activation of number semantics: an EEG study on 10-year-old children

Over the course of development, children must learn to map a non-symbolic representation of magnitude to a more precise symbolic system. There is solid evidence that finger and dot representations can facilitate or even predict the acquisition of this mapping skill. While several behavioral studies demonstrated that canonical representations of fingers and dots automatically activate number semantics, no study so far has investigated their cerebral basis. To examine these questions, 10-year-old children were presented a behavioral naming task and a Fast Periodic Visual Stimulation EEG paradigm. In the behavioral task, children had to name as fast and as accurately as possible the numbers of dots and fingers presented in canonical and non-canonical configurations. In the EEG experiment, one category of stimuli (e.g., canonical representation of fingers or dots) was periodically inserted (1/5) in streams of another category (e.g., non-canonical representation of fingers or dots) presented at a fast rate (4 Hz). Results demonstrated an automatic access to number semantics and bilateral categorical responses at 4 Hz/5 for canonical representations of fingers and dots. Some differences between finger and dot configuration's processing were nevertheless observed and are discussed in light of an effortful-automatic continuum hypothesis.

Approximate multiplication in young children prior to multiplication instruction

Prior work indicates that children have an untrained ability to approximately calculate using their approximate number system (ANS). For example, children can mentally double or halve a large array of discrete objects. Here, we asked whether children can perform a true multiplication operation, flexibly attending to both the multiplier and multiplicand, prior to formal multiplication instruction. We presented 5- to 8-year-olds with nonsymbolic multiplicands (dot arrays) or symbolic multiplicands (Arabic numerals) ranging from 2 to 12 and with nonsymbolic multipliers ranging from 2 to 8. Children compared each imagined product with a visible comparison quantity. Children performed with above-chance accuracy on both nonsymbolic and symbolic approximate multiplication, and their performance was dependent on the ratio between the imagined product and the comparison target. Children who could not solve any single-digit symbolic multiplication equations (e.g., 2 × 3) on a basic math test were nevertheless successful on both our approximate multiplication tasks, indicating that children have an intuitive sense of multiplication that emerges independent of formal instruction about symbolic multiplication. Nonsymbolic multiplication performance mediated the relation between children's Weber fraction and symbolic math abilities, suggesting a pathway by which the ANS contributes to children's emerging symbolic math competence. These findings may inform future educational interventions that allow children to use their basic arithmetic intuition as a scaffold to facilitate symbolic math learning.

Approximate arithmetic training does not improve symbolic math in third and fourth grade children

BACKGROUND

Prior studies reported that practice playing an approximate arithmetic game improved symbolic math performance relative to active control groups in adults and preschool children (e.g. Park & Brannon, 2013, 2014; Park et al., 2016; Szkudlarek & Brannon, 2018). However, Szkudlarek, Park and Brannon (2021) recently failed to replicate those findings in adults. Here we test whether approximate arithmetic training yields benefits in elementary school children who have intermediate knowledge of arithmetic.

METHOD

We conducted a randomized controlled trial with a pre and post-test design to compare the effects of approximate arithmetic training and visuo-spatial working memory training on standardized math performance in third and fourth grade children.

RESULTS

We found that approximate arithmetic training did not yield any significant gains on standardized measures of symbolic math performance.

CONCLUSION

A Bayesian analysis supports the conclusion that approximate arithmetic provides no benefits for symbolic math performance.

Effects of Visual Training of Approximate Number Sense on Auditory Number Sense and School Math Ability

Research with children and adults suggests that people's math performance is predicted by individual differences in an evolutionarily ancient ability to estimate and compare numerical quantities without counting (the approximate number system or ANS). However, previous work has almost exclusively used visual stimuli to measure ANS precision, leaving open the possibility that the observed link might be driven by aspects of visuospatial competence, rather than the amodal ANS. We addressed this possibility in an ANS training study. Sixty-eight 6-year-old children participated in a 5-week study that either trained their visual ANS ability or their phonological awareness (an active control group). Immediately before and after training, we assessed children's visual and auditory ANS precision, as well as their symbolic math ability and phonological awareness. We found that, prior to training, children's precision in a visual ANS task related to their math performance - replicating recent studies. Importantly, precision in an auditory ANS task also related to math performance. Furthermore, we found that children who completed visual ANS training showed greater improvements in auditory ANS precision than children who completed phonological awareness training. Finally, children in the ANS training group showed significant improvements in math ability but not phonological awareness. These results suggest that the link between ANS precision and school math ability goes beyond visuospatial abilities and that the modality-independent ANS is causally linked to math ability in early childhood.

Alzheimer's disease disrupts domain-specific and domain-general processes in numerosity estimation

INTRODUCTION

This study investigated how Alzheimer's Disease (AD) affects numerosity estimation abilities (e.g., finding the approximate number of items in a collection).

METHOD

Across two experiments, performance from HOA (i.e., Healthy Older Adults; N = 48) and AD patients (N = 50) was compared on dot comparison tasks. Participants were presented with two dot arrays and had to select the more numerous dot array in comparison tasks. They also took a Simon task and a number-line tasks (i.e., number-line tasks in which they had to indicate the position of a number on a line 0 to 100 or on a line 0 to 1,000 in the number-line task).

RESULTS

In Experiment 1, (a) AD patients obtained significantly poorer performance while comparing collections of dots, especially harder (small-ratio) collections, (b) these deficits correlated with poorer performance on the number-line task for larger numerosities (i.e., 0 to 1,000), and (c) AD patients showed poorer performance on incongruent (where numerosity and area occupied by dots mismatched) than on congruent items (where both features matched), while HOA showed no congruency effects. Experiment 2 showed (a) congruency effects in both groups when convex hull was tested as an incongruent feature, and (b) comparable sequential modulations of congruency effects in both groups.

CONCLUSIONS

Our findings showed that numerosity abilities decline in AD patients, and that this decline results from impaired domain-specific processes (i.e., numerosity processing) and domain-general processes (i.e., inhibition). These findings have important implications to further our understanding of how specific and general cognitive processes contribute to numerosity estimation/comparison performance, and how such contributions change during Alzheimer's disease.

Hemispheric asymmetries in processing numerical meaning in arithmetic

Hemispheric asymmetries in arithmetic have been hypothesized based on neuropsychological, developmental, and neuroimaging work. However, it has been challenging to separate asymmetries related to arithmetic specifically, from those associated general cognitive or linguistic processes. Here we attempt to experimentally isolate the processing of numerical meaning in arithmetic problems from language and memory retrieval by employing novel non-symbolic addition problems, where participants estimated the sum of two dot arrays and judged whether a probe dot array was the correct sum of the first two arrays. Furthermore, we experimentally manipulated which hemisphere receive the probe array first using a visual half-field paradigm while recording event-related potentials (ERP). We find that neural sensitivity to numerical meaning in arithmetic arises under left but not right visual field presentation during early and middle portions of the late positive complex (LPC, 400-800 ms). Furthermore, we find that subsequent accuracy for judgements of whether the probe is the correct sum is better under right visual field presentation than left, suggesting a left hemisphere advantage for integrating information for categorization or decision making related to arithmetic. Finally, neural signatures of operational momentum, or differential sensitivity to whether the probe was greater or less than the sum, occurred at a later portion of the LPC (800-1000 ms) and regardless of visual field of presentation, suggesting a temporal and functional dissociation between magnitude and ordinal processing in arithmetic. Together these results provide novel evidence for differences in timing and hemispheric lateralization for several cognitive processes involved in arithmetic thinking.

Frequency-based Dissociation of Symbolic and Nonsymbolic Numerical Processing during Numerical Comparison

Recent evidence suggests that during numerical calculation, symbolic and nonsymbolic processing are functionally distinct operations. Nevertheless, both roughly recruit the same brain areas (spatially overlapping networks in the parietal cortex) and happen at the same time (roughly 250 msec poststimulus onset). We tested the hypothesis that symbolic and nonsymbolic processing are segregated by means of functionally relevant networks in different frequency ranges: high gamma (above 50 Hz) for symbolic processing and lower beta (12–17 Hz) for nonsymbolic processing. EEG signals were quantified as participants compared either symbolic numbers or nonsymbolic quantities. Larger EEG gamma-band power was observed for more difficult symbolic comparisons (ratio of 0.8 between the two numbers) than for easier comparisons (ratio of 0.2) over frontocentral regions. Similarly, beta-band power was larger for more difficult nonsymbolic comparisons than for easier ones over parietal areas. These results confirm the existence of a functional dissociation in EEG oscillatory dynamics during numerical processing that is compatible with the notion of distinct linguistic processing of symbolic numbers and approximation of nonsymbolic numerical information.

Discrimination of ordinal relationships in temporal sequences by 4-month-old infants

The ability to discriminate the ordinal information embedded in magnitude-based sequences has been shown in 4-month-old infants, both for numerical and size-based sequences. At this early age, however, this ability is confined to increasing sequences, with infants failing to extract and represent decreasing order. Here we investigate whether the ability to represent order extends to duration-based sequences in 4-month-old infants, and whether it also shows the asymmetry signature previously observed for number and size. Infants were tested in an order discrimination task in which they were habituated to either increasing or decreasing variations in temporal duration, and were then tested with novel sequences composed of new temporal items whose durations varied following the familiar and the novel orders in alternation. Across three experiments, we manipulated the duration of the single temporal items and therefore of the whole sequences, which resulted in imposing more or less constraints on infants' working memory, or general processing capacities. Results showed that infants failed at discriminating the ordinal direction in temporal sequences when the sequences had an overall long duration (Experiment 1), but succeeded when the duration of the sequences was shortened (Experiments 2 and 3). Moreover, there was no sign of the asymmetry signature previously reported for number and size, as successful discrimination was present for infants habituated to both increasing and decreasing sequences. These results suggest that sensitivity to temporal order is present very early in development, and that its functional properties are not shared with other magnitude dimensions, such as size and number.

The effect of multimodal information on children's numerical judgments

Although much research suggests that adults, infants, and nonhuman primates process number (among other properties) across distinct modalities, limited studies have explored children's abilities to integrate multisensory information when making judgments about number. In the current study, 3- to 6-year-old children performed numerical matching or numerical discrimination tasks in which numerical information was presented either unimodally (visual only), cross-modally (comparing audio with visual), or bimodally (simultaneously presenting audio and visual input). In three experiments, we investigated children's multimodal numerical processing across distinct task demands and difficulty levels. In contrast to previous work, results indicate that even the youngest children (3 and 4 years) performed above chance across all three modality presentations. In addition, the current study contributes two other novel findings, namely that (a) children exhibit a cross-modal disadvantage when numerical comparisons are easy and that (b) accuracy on bimodal trial types led to even more accurate numerical judgments under more difficult circumstances, particularly for the youngest participants and when precise numerical matching was required. Importantly, findings from this study extend the literature on children's numerical cross-modal abilities to reveal that, like their adult counterparts, children readily track and compare visual and auditory numerical information, although their abilities to do so are not perfect and are affected by task demands and trial difficulty.

The Developing Mental Number Line: Does Its Directionality Relate to 5- to 7-Year-Old Children's Mathematical Abilities?

Spatial representations of number, such as a left-to-right oriented mental number line, are well documented in Western culture. Yet, the functional significance of such a representation remains unclear. To test the prominent hypothesis that a mental number line may support mathematical development, we examined the relation between spatial-numerical associations (SNAs) and math proficiency in 5- to 7-year-old children. We found evidence of SNAs with two tasks: a non-symbolic magnitude comparison task, and a symbolic "Where was the number?" (WTN) task. Further, we found a significant correlation between these two tasks, demonstrating convergent validity of the directional mental number line across numerical format. Although there were no significant correlations between children's SNAs on the WTN task and math ability, children's SNAs on the magnitude comparison task were negatively correlated with their performance on a measure of cross-modal arithmetic, suggesting that children with a stronger left-to-right oriented mental number line were less competent at cross-modal arithmetic, an effect that held when controlling for age and a set of general cognitive abilities. Despite some evidence for a negative relation between SNAs and math ability in adulthood, we argue that the effect here may reflect task demands specific to the magnitude comparison task, not necessarily an impediment of the mental number line to math performance. We conclude with a discussion of the different properties that characterize a mental number line and how these different properties may relate to mathematical ability.

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.

The Central Executive Mediates the Relationship Between Children's Approximate Number System Acuity and Arithmetic Strategy Utilization in Computational Estimation

Studies investigating the relationship between working memory (WM) and approximate number system (ANS) acuity in the area of arithmetic strategy utilization are scarce. The choice/no choice method paradigm was used in the present study to determine whether and how ANS acuity and WM components affected strategy utilization. The results showed that the central executive (CE) mediated the relationship between ANS acuity and strategy utilization. Furthermore, quantile regression analyses revealed that the association between CE and strategy choice was robust from the first to highest quantile. Notably, the relationship between ANS acuity and strategy choice was significant at the median and higher quantiles (i.e., 0.5, 0.75, and 0.85 quantiles), but not significant at lower quantiles (i.e., 0.15 and 0.25 quantiles). These results suggest that domain-general skills play a crucial role in the relationship between children's ANS acuity and mathematical ability. The impact of ANS acuity and CE on strategy choice was dependent on the distribution of the strategy utilization level. These results provide a further understanding of the utilization of cognitive strategies.

Cognitive predictors of children's development in mathematics achievement: A latent growth modeling approach

Research has identified various domain-general and domain-specific cognitive abilities as predictors of children's individual differences in mathematics achievement. However, research into the predictors of children's individual growth rates, namely between-person differences in within-person change in mathematics achievement is scarce. We assessed 334 children's domain-general and mathematics-specific early cognitive abilities and their general mathematics achievement longitudinally across four time-points within the first and second grades of primary school. As expected, a constellation of multiple cognitive abilities contributed to the children's starting level of mathematical success. Specifically, latent growth modeling revealed that WM abilities, IQ, counting skills, nonsymbolic and symbolic approximate arithmetic and comparison skills explained individual differences in the children's initial status on a curriculum-based general mathematics achievement test. Surprisingly, however, only one out of all the assessed cognitive abilities was a unique predictor of the children's individual growth rates in mathematics achievement: their performance in the symbolic approximate addition task. In this task, children were asked to estimate the sum of two large numbers and decide if this estimated sum was smaller or larger compared to a third number. Our findings demonstrate the importance of multiple domain-general and mathematics-specific cognitive skills for identifying children at risk of struggling with mathematics and highlight the significance of early approximate arithmetic skills for the development of one's mathematical success. We argue the need for more research focus on explaining children's individual growth rates in mathematics achievement.

Visual Form Perception Can Be a Cognitive Correlate of Lower Level Math Categories for Teenagers

Numerous studies have assessed the cognitive correlates of performance in mathematics, but little research has been conducted to systematically examine the relations between visual perception as the starting point of visuospatial processing and typical mathematical performance. In the current study, we recruited 223 seventh graders to perform a visual form perception task (figure matching), numerosity comparison, digit comparison, exact computation, approximate computation, and curriculum-based mathematical achievement tests. Results showed that, after controlling for gender, age, and five general cognitive processes (choice reaction time, visual tracing, mental rotation, spatial working memory, and non-verbal matrices reasoning), visual form perception had unique contributions to numerosity comparison, digit comparison, and exact computation, but had no significant relation with approximate computation or curriculum-based mathematical achievement. These results suggest that visual form perception is an important independent cognitive correlate of lower level math categories, including the approximate number system, digit comparison, and exact computation.

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.

Numbers and functional lateralization: A visual half-field and dichotic listening study in proficient bilinguals

Potential links between language and numbers and the laterality of symbolic number representations in the brain are still debated. Furthermore, reports on bilingual individuals indicate that the language-number interrelationships might be quite complex. Therefore, we carried out a visual half-field (VHF) and dichotic listening (DL) study with action words and different forms of symbolic numbers used as stimuli to test the laterality of word and number processing in single-, dual-language and mixed -task and language- contexts. Experiment 1 (VHF) showed a significant right visual field/left hemispheric advantage in response accuracy for action word, as compared to any form of symbolic number processing. Experiment 2 (DL) revealed a substantially reversed effect - a significant right ear/left hemisphere advantage for arithmetic operations as compared to action word processing, and in response times in single- and dual-language contexts for number vs. action words. All these effects were language independent. Notably, for within-task response accuracy compared across modalities significant differences were found in all studied contexts. Thus, our results go counter to findings showing that action-relevant concepts and words, as well as number words are represented/processed primarily in the left hemisphere. Instead, we found that in the auditory context, following substantial engagement of working memory (here: by arithmetic operations), there is a subsequent functional reorganization of processing single stimuli, whether verbs or numbers. This reorganization - their weakened laterality - at least for response accuracy is not exclusive to processing of numbers, but the number of items to be processed. For response times, except for unpredictable tasks in mixed contexts, the "number problem" is more apparent. These outcomes are highly relevant to difficulties that simultaneous translators encounter when dealing with lengthy auditory material in which single items such as number words (and possibly other types of key words) need to be emphasized. Our results may also shed a new light on the "mathematical savant problem".

The relationship between non-symbolic multiplication and division in childhood

Children without formal education in addition and subtraction are able to perform multi-step operations over an approximate number of objects. Further, their performance improves when solving approximate (but not exact) addition and subtraction problems that allow for inversion as a shortcut (e.g., a + b - b = a). The current study examines children's ability to perform multi-step operations, and the potential for an inversion benefit, for the operations of approximate, non-symbolic multiplication and division. Children were trained to compute a multiplication and division scaling factor (*2 or /2, *4 or /4), and were then tested on problems that combined two of these factors in a way that either allowed for an inversion shortcut (e.g., 8*4/4) or did not (e.g., 8*4/2). Children's performance was significantly better than chance for all scaling factors during training, and they successfully computed the outcomes of the multi-step testing problems. They did not exhibit a performance benefit for problems with the a*b/b structure, suggesting that they did not draw upon inversion reasoning as a logical shortcut to help them solve the multi-step test problems.

The Development of Symbolic and Non-Symbolic Number Line Estimations: Three Developmental Accounts Contrasted Within Cross-Sectional and Longitudinal Data

Three theoretical accounts have been put forward for the development of children's response patterns on number line estimation tasks: the log-to-linear representational shift, the two-linear-to-linear transformation and the proportion judgment account. These three accounts have not been contrasted, however, within one study, using one single criterion to determine which model provides the best fit. The present study contrasted these three accounts by examining first, second and sixth graders with a symbolic and non-symbolic number line estimation task (Experiment 1). In addition, first and second graders were tested again one year later (Experiment 2). In case of symbolic estimations, the proportion judgment account described the data best. Most young children's non-symbolic estimation patterns were best described by a logarithmic model (within the log-to-lin account), whereas those of most older children were best described by the simple power model (within the proportion judgment account).

The developmental onset of symbolic approximation: beyond nonsymbolic representations, the language of numbers matters

Symbolic (i.e., with Arabic numerals) approximate arithmetic with large numerosities is an important predictor of mathematics. It was previously evidenced to onset before formal schooling at the kindergarten age (Gilmore et al., 2007) and was assumed to map onto pre-existing nonsymbolic (i.e., abstract magnitudes) representations. With a longitudinal study (Experiment 1), we show, for the first time, that nonsymbolic and symbolic arithmetic demonstrate different developmental trajectories. In contrast to Gilmore et al.’s (2007) findings, Experiment 1 showed that symbolic arithmetic onsets in grade 1, with the start of formal schooling, not earlier. Gilmore et al. (2007) had examined English-speaking children, whereas we assessed a large Dutch-speaking sample. The Dutch language for numbers can be cognitively more demanding, for example, due to the inversion property in numbers above 20. Thus, for instance, the number 48 is named in Dutch “achtenveertig” (eight and forty) instead of “forty eight.” To examine the effect of the language of numbers, we conducted a cross-cultural study with English- and Dutch-speaking children that had similar SES and math achievement skills (Experiment 2). Results demonstrated that Dutch-speaking kindergarteners lagged behind English-speaking children in symbolic arithmetic, not nonsymbolic and demonstrated a working memory overload in symbolic arithmetic, not nonsymbolic. Also, we show for the first time that the ability to name two-digit numbers highly correlates with symbolic approximate arithmetic not nonsymbolic. Our experiments empirically demonstrate that the symbolic number system is modulated more by development and education than the nonsymbolic system. Also, in contrast to the nonsymbolic system, the symbolic system is modulated by language.

Numerical magnitude processing deficits in children with mathematical difficulties are independent of intelligence

Developmental dyscalculia (DD) is thought to arise from difficulties in the ability to process numerical magnitudes. Most research relied on IQ-discrepancy based definitions of DD and only included individuals with normal IQ, yet little is known about the role of intelligence in the association between numerical magnitude processing and mathematical difficulties (MD). The present study examined numerical magnitude processing in matched groups of 7- to 8-year-olds (n=42) who had either discrepant MD (poor math scores, average IQ), nondiscrepant MD (poor math scores, below-average IQ) or no MD. Both groups of children with MD showed similar impairments in numerical magnitudes processing compared to controls, suggesting that the association between numerical magnitude processing deficits and MD is independent of intelligence.

Human infants' preference for left-to-right oriented increasing numerical sequences

While associations between number and space, in the form of a spatially oriented numerical representation, have been extensively reported in human adults, the origins of this phenomenon are still poorly understood. The commonly accepted view is that this number-space association is a product of human invention, with accounts proposing that culture, symbolic knowledge, and mathematics education are at the roots of this phenomenon. Here we show that preverbal infants aged 7 months, who lack symbolic knowledge and mathematics education, show a preference for increasing magnitude displayed in a left-to-right spatial orientation. Infants habituated to left-to-right oriented increasing or decreasing numerical sequences showed an overall higher looking time to new left-to-right oriented increasing numerical sequences at test (Experiment 1). This pattern did not hold when infants were presented with the same ordinal numerical information displayed from right to left (Experiment 2). The different pattern of results was congruent with the presence of a malleable, context-dependent baseline preference for increasing, left-to-right oriented, numerosities (Experiment 3). These findings are suggestive of an early predisposition in humans to link numerical order with a left-to-right spatial orientation, which precedes the acquisition of symbolic abilities, mathematics education, and the acquisition of reading and writing skills.

Children's mapping between non-symbolic and symbolic numerical magnitudes and its association with timed and untimed tests of mathematics achievement

The ability to map between non-symbolic numerical magnitudes and Arabic numerals has been put forward as a key factor in children’s mathematical development. This mapping ability has been mainly examined indirectly by looking at children’s performance on a symbolic magnitude comparison task. The present study investigated mapping in a more direct way by using a task in which children had to choose which of two choice quantities (Arabic digits or dot arrays) matched the target quantity (dot array or Arabic digit), thereby focusing on small quantities ranging from 1 to 9. We aimed to determine the development of mapping over time and its relation to mathematics achievement. Participants were 36 first graders (M = 6 years 8 months) and 46 third graders (M = 8 years 8 months) who all completed mapping tasks, symbolic and non-symbolic magnitude comparison tasks and standardized timed and untimed tests of mathematics achievement. Findings revealed that children are able to map between non-symbolic and symbolic representations and that this mapping ability develops over time. Moreover, we found that children’s mapping ability is related to timed and untimed measures of mathematics achievement, over and above the variance accounted for by their numerical magnitude comparison skills.

Brief non-symbolic, approximate number practice enhances subsequent exact symbolic arithmetic in children

Recent research reveals a link between individual differences in mathematics achievement and performance on tasks that activate the approximate number system (ANS): a primitive cognitive system shared by diverse animal species and by humans of all ages. Here we used a brief experimental paradigm to test one causal hypothesis suggested by this relationship: activation of the ANS may enhance children's performance of symbolic arithmetic. Over 2 experiments, children who briefly practiced tasks that engaged primitive approximate numerical quantities performed better on subsequent exact, symbolic arithmetic problems than did children given other tasks involving comparison and manipulation of non-numerical magnitudes (brightness and length). The practice effect appeared specific to mathematics, as no differences between groups were observed on a comparable sentence completion task. These results move beyond correlational research and provide evidence that the exercise of non-symbolic numerical processes can enhance children's performance of symbolic mathematics.

Novel Inversions in Auditory Sequences Provide Evidence for Spontaneous Subtraction of Time and Number

Animals, including fish, birds, rodents, non-human primates, and pre-verbal infants are able to discriminate the duration and number of events without the use of language. In this paper, we present the results of six experiments exploring the capability of adult rats to count 2–6 sequentially presented white-noise stimuli. The investigation focuses on the animal’s ability to exhibit spontaneous subtraction following the presentation of novel stimulus inversions in the auditory signals being counted. Results suggest that a subtraction operation between two opposite sensory representations may be a general processing strategy used for the comparison of stimulus magnitudes. These findings are discussed within the context of a mode-control model of timing and counting that relies on an analog temporal-integration process for the addition and subtraction of sequential events.

The development of numerical magnitude processing and its association with working memory in children with mild intellectual disabilities

The present research examined numerical magnitude processing and its association with working memory in children with mild intellectual disabilities (MID). We investigated the performance of 8-year-old children with MID on a symbolic (Arabic digits) and non-symbolic (dot patterns) magnitude comparison task by means of a chronological-age/ability-level-match design. We also examined whether the predicted problems with numerical magnitude comparison could be explained by working memory by using three working memory tasks. Findings revealed that children with MID performed more poorly than their chronological age-matched peers on both the symbolic and non-symbolic magnitude comparison tasks, suggesting impairments in these children's ability to represent numerical magnitudes. They also performed more poorly on working memory compared to their typically developing age- and ability-matched peers, but when these differences in working memory performance were additionally controlled for, the group differences on the numerical magnitude comparison tasks remained. Both symbolic numerical magnitude processing and central executive functioning predicted addition performance in children with MID.

Understanding less than nothing: children's neural response to negative numbers shifts across age and accuracy

We examined the brain activity underlying the development of our understanding of negative numbers, which are amounts lacking direct physical counterparts. Children performed a paired comparison task with positive and negative numbers during an fMRI session. As previously shown in adults, both pre-instruction fifth-graders and post-instruction seventh-graders demonstrated typical behavioral and neural distance effects to negative numbers, where response times and parietal and frontal activity increased as comparison distance decreased. We then determined the factors impacting the distance effect in each age group. Behaviorally, the fifth-grader distance effect for negatives was significantly predicted only by positive comparison accuracy, indicating that children who were generally better at working with numbers were better at comparing negatives. In seventh-graders, negative number comparison accuracy significantly predicted their negative number distance effect, indicating that children who were better at working with negative numbers demonstrated a more typical distance effect. Across children, as age increased, the negative number distance effect increased in the bilateral IPS and decreased frontally, indicating a frontoparietal shift consistent with previous numerical development literature. In contrast, as negative comparison task accuracy increased, the parietal distance effect increased in the left IPS and decreased in the right, possibly indicating a change from an approximate understanding of negatives' values to a more exact, precise representation (particularly supported by the left IPS) with increasing expertise. These shifts separately indicate the effects of increasing maturity generally in numeric processing and specifically in negative number understanding.

Working memory in nonsymbolic approximate arithmetic processing: a dual-task study with preschoolers

Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted a dual-task study with preschoolers with active phonological, visual, spatial, and central executive interference during the completion of a nonsymbolic approximate addition dot task. With regard to the role of WM, we found a clear performance breakdown in the central executive interference condition. Our findings provide insight into the underlying cognitive processes involved in storing and manipulating nonsymbolic approximate numerosities during early arithmetic.