Association between basic numerical abilities and mathematics achievement

Numerical Processing and Executive Functioning in Early Versus Middle Childhood: A Japanese Sample

Many previous studies have investigated developmental differences in numerical processing by manipulating numerical distance and physical size in a number sequence. While it has been theorized that children's maturity level in executive functioning affects their numerical processing, the interaction between numerical processing and executive functioning through development remains unclear. We divided 60 Japanese school children, aged 8-12 years, into three age-related groups (second graders, fourth graders, and sixth graders) and had them perform physical and numerical comparison Stroop tasks. In the physical comparison task, the numerical Stroop effect (i.e., automatic numerical processing) was evident in each group, but, in the numerical comparison task, the numerical distance effect (i.e., intentional numerical processing) was evident in each group. Also, in the numerical comparison task, the size congruity effect (an index of the attentional and inhibitory control mechanisms of executive functioning) was more salient among second graders than among fourth or sixth graders. These results suggest that numerical processing matures and then plateaus just before primary school, while executive functioning continues to develop. Thus, these data provide evidence of a developmental dissociation between numerical processing and executive functioning.

Testing the Efficacy of Training Basic Numerical Cognition and Transfer Effects to Improvement in Children's Math Ability

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children's math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named "123 Bakery" which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic, and symbol-to-numerosity mapping. Fifty-six first graders were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post-training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, and symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; hereafter ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested.

The Use of Local and Global Ordering Strategies in Number Line Estimation in Early Childhood

A lot of research has been devoted to number line estimation in primary school. However, less is known about the early onset of number line estimation before children enter formal education. We propose that ordering strategies are building blocks of number line estimation in early childhood. In a longitudinal study, children completed a non-symbolic number line estimation task at age 3.5 and 5 years. Two ordering strategies were identified based on the children's estimation patterns: local and global ordering. Local ordering refers to the correct ordering of successive quantities, whereas global ordering refers to the correct ordering of all quantities across the number line. Results indicated a developmental trend for both strategies. The percentage of children applying local and global ordering strategies increased steeply from 3.5 to 5 years of age. Moreover, children used more advanced local and global ordering strategies at 5 years of age. Importantly, level of strategy use was related to more traditional number line estimation outcome measures, such as estimation accuracy and regression fit scores. These results provide evidence that children use dynamic ordering strategies when solving the number line estimation task in early stages of numerical development.

Keys to the Gate? Equal Sign Knowledge at Second Grade Predicts Fourth-Grade Algebra Competence

Algebraic competence is a major determinant of college access and career prospects, and equal sign knowledge is taken to be foundational to algebra knowledge. However, few studies have documented a causal effect of early equal sign knowledge on later algebra skill. This study assessed whether second-grade students' equal sign knowledge prospectively predicts their fourth-grade algebra knowledge, when controlling for demographic and individual difference factors. Children (N = 177; M_age  = 7.61) were assessed on a battery of tests in Grade 2 and on algebraic knowledge in Grade 4. Second-grade equal sign knowledge was a powerful predictor of these algebraic skills. Findings are discussed in terms of the importance of foregrounding equal sign knowledge to promote effective pedagogy and educational equity.

On the Difference Between Numerosity Processing and Number Processing

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

Number line estimation in highly math-anxious individuals

In this study, we aimed to investigate the difficulties highly math-anxious individuals (HMA) may face when having to estimate a number's position in a number line task. Twenty-four HMA and 24 low math-anxiety (LMA) individuals were presented with four lines with endpoints 0-100, 0-1,000, 0-100,000, and 267-367 on a computer monitor on which they had to mark the correct position of target numbers using the mouse. Although no differences were found between groups in the frequency of their best-fit model, which was linear for all lines, the analysis of slopes and intercepts for the linear model showed that the two groups differed in performance on the less familiar lines (267-367 and 0-100,000). Lower values for the slope and higher values for the intercept were found in the HMA group, suggesting that they tended to overestimate small numbers and underestimate large numbers on these non-familiar lines. Percentage absolute error analyses confirmed that HMA individuals were less accurate than their LMA counterparts on these lines, although no group differences were found in response time. These results indicate that math anxiety is related to worse performance only in the less familiar and more difficult number line tasks. Therefore, our data challenge the idea that HMA individuals might have less precise numerical representations and support the anxiety-complexity effect posited by Ashcraft and colleagues.

Individuality in the Early Number Skill Components Underlying Basic Arithmetic Skills

Early number skills underlie success in basic arithmetic. However, very little is known about the skill profiles among children in preprimary education and how the potential profiles are related to arithmetic development. This longitudinal study of 440 Finnish children in preprimary education (mean age: 75 months) modeled latent performance-level profile groups for the early number skill components that are proposed to be key predictors of arithmetic (symbolic number comparison, mapping, and verbal counting skills). Based on three assessment time points (September, January, and May), four profile groups were found: the poorest-performing (6%), low-performing (16%), near-average-performing (33%), and high-average-performing children (45%). Although the differences between the groups were statistically significant in all three number skill components and in basic arithmetic, the poorest-performing children seemed to have serious difficulties in accessing the semantic meaning of symbolic numbers that was required in the number comparison and mapping tasks in this study. Interestingly, the tasks demanding processing between quantities and symbols also most differentiated the poorest-performing children from the low-performing children. Due to remarkable and stable individual differences in early number skill components, the findings suggest systematic support and progress monitoring practices in preeducational settings to diminish and avoid potential difficulties in arithmetic and mathematics in general.

Symbolic estrangement or symbolic integration of numerals with quantities: Methodological pitfalls and a possible solution

Previous studies, which examined whether symbolic and non-symbolic quantity representations are processed by two independent systems or by one common system, reached contradicting findings, possibly due to methodological differences. Indeed, some researchers advocate the two systems approach, based on the presence of notation-specific switch cost in conditions where adults have to compare pairs of symbolic and non-symbolic quantities, in combination with the absence of such a cost in conditions containing quantities of the same notation. However, other researchers used matching instructions, and reported a facilitation in the mixed notation conditions, suggesting that the two systems are automatically integrated. In the current study, we conducted three experiments, in which we examined the existence of two separate quantity systems, but we used various experimental manipulations (e.g., task instructions, presentation order) to unravel the previous inconsistent findings. In Experiment 1, we investigated the role of task instructions by presenting participants with pure and mixed notation trials with both comparison and matching tasks. In Experiment 2, we tested the role of blocked and randomized presentation order for the pure and mixed trials. Our data showed that cost for switching between the symbolic and non-symbolic quantities is present, but is prone to a certain methodological drawback: when the differences between the processing times for two sequentially presented stimuli of different notations are not taken into account, this masks the cost for switching between the two systems. To overcome this problem, in Experiment 3 we used an audio-visual paradigm. Overall, our results provide further evidence for the existence of distinct quantity representations, independently of task instructions or presentation order. Additionally, considering this methodological pitfall we argue that the audio-visual paradigm is better suited when investigating the integration between symbolic and non- symbolic quantities.

How the Abstract Becomes Concrete: Irrational Numbers Are Understood Relative to Natural Numbers and Perfect Squares

Mathematical cognition research has largely emphasized concepts that can be directly perceived or grounded in visuospatial referents. These include concrete number systems like natural numbers, integers, and rational numbers. Here, we investigate how a more abstract number system, the irrationals denoted by radical expressions like 2, is understood across three tasks. Performance on a magnitude comparison task suggests that people interpret irrational numbers (specifically, the radicands of radical expressions) as natural numbers. Strategy self-reports during a number line estimation task reveal that the spatial locations of irrationals are determined by referencing neighboring perfect squares. Finally, perfect squares facilitate the evaluation of arithmetic expressions. These converging results align with a constellation of related phenomena spanning tasks and number systems of varying complexity. Accordingly, we propose that the task-specific recruitment of more concrete representations to make sense of more abstract concepts (referential processing) is an important mechanism for teaching and learning mathematics.

Acoustic features of auditory medical alarms-An experimental study of alarm volume

Audible alarms are a ubiquitous feature of all high-paced, high-risk domains such as aviation and nuclear power where operators control complex systems. In such settings, a missed alarm can have disastrous consequences. It is conventional wisdom that for alarms to be heard, "louder is better," so that alarm levels in operational environments routinely exceed ambient noise levels. Through a robust experimental paradigm in an anechoic environment to study human response to audible alerting stimuli in a cognitively demanding setting, akin to high-tempo and high-risk domains, clinician participants responded to patient crises while concurrently completing an auditory speech intelligibility and visual vigilance distracting task as the level of alarms were varied as a signal-to-noise ratio above and below hospital background noise. There was little difference in performance on the primary task when the alarm sound was -11 dB below background noise as compared with +4 dB above background noise-a typical real-world situation. Concurrent presentation of the secondary auditory speech intelligibility task significantly degraded performance. Operator performance can be maintained with alarms that are softer than background noise. These findings have widespread implications for the design and implementation of alarms across all high-consequence settings.

The analysis of EEG coherence reflects middle childhood differences in mathematical achievement

Symbolic numerical magnitude processing is crucial to arithmetic development, and it is thought to be supported by the functional activation of several brain-interconnected structures. In this context, EEG beta oscillations have been recently associated with attention and working memory processing that underlie math achievement. Due to that EEG coherence represents a useful measure of brain functional connectivity, we aimed to contrast the EEG coherence in forty 8-to-9-year-old children with different math skill levels (High: HA, and Low achievement: LA) according to their arithmetic scores in the Fourth Edition of the Wide Range Achievement Test (WRAT-4) while performing a symbolic magnitude comparison task (i.e. determining which of two numbers is numerically larger). The analysis showed significantly greater coherence over the right hemisphere in the two groups, but with a distinctive connectivity pattern. Whereas functional connectivity in the HA group was predominant in parietal areas, especially involving beta frequencies, the LA group showed more extensive frontoparietal relationships, with higher participation of delta, theta and alpha band frequencies, along with a distinct time-frequency domain expression. The results seem to reflect that lower math achievements in children mainly associate with cognitive processing steps beyond stimulus encoding, along with the need of further attentional resources and cognitive control than their peers, suggesting a lower degree of numerical processing automation.

Frequency of Home Numeracy Activities Is Differentially Related to Basic Number Processing and Calculation Skills in Kindergartners

Home numeracy has been shown to play an important role in children's mathematical performance. However, findings are inconsistent as to which home numeracy activities are related to which mathematical skills. The present study disentangled between various mathematical abilities that were previously masked by the use of composite scores of mathematical achievement. Our aim was to shed light on the specific associations between home numeracy and various mathematical abilities. The relationships between kindergartners' home numeracy activities, their basic number processing and calculation skills were investigated. Participants were 128 kindergartners (Mage = 5.43 years, SD = 0.29, range: 4.88-6.02 years) and their parents. The children completed non-symbolic and symbolic comparison tasks, non-symbolic and symbolic number line estimation tasks, mapping tasks (enumeration and connecting), and two calculation tasks. Their parents completed a home numeracy questionnaire. Results indicated small but significant associations between formal home numeracy activities that involved more explicit teaching efforts (i.e., identifying numerals, counting) and children's enumeration skills. There was no correlation between formal home numeracy activities and non-symbolic number processing. Informal home numeracy activities that involved more implicit teaching attempts, such as "playing games" and "using numbers in daily life," were (weakly) correlated with calculation and symbolic number line estimation, respectively. The present findings suggest that disentangling between various basic number processing and calculation skills in children might unravel specific relations with both formal and informal home numeracy activities. This might explain earlier reported contradictory findings on the association between home numeracy and mathematical abilities.

About why there is a shift from cardinal to ordinal processing in the association with arithmetic between first and second grade

Digit comparison is strongly related to individual differences in children's arithmetic ability. Why this is the case, however, remains unclear to date. Therefore, we investigated the relative contribution of three possible cognitive mechanisms in first and second graders' digit comparison performance: digit identification, digit-number word matching and digit ordering ability. Furthermore, we examined whether these components could account for the well-established relation between digit comparison performance and arithmetic. As expected, all candidate predictors were related to digit comparison in both age groups. Moreover, in first graders, digit ordering and in second graders both digit identification and digit ordering explained unique variance in digit comparison performance. However, when entering these unique predictors of digit comparison into a mediation model with digit comparison as predictor and arithmetic as outcome, we observed that whereas in second graders digit ordering was a full mediator, in first graders this was not the case. For them, the reverse was true and digit comparison fully mediated the relation between digit ordering and arithmetic. These results suggest that between first and second grade, there is a shift in the predictive value for arithmetic from cardinal processing and procedural knowledge to ordinal processing and retrieving declarative knowledge from memory; a process which is possibly due to a change in arithmetic strategies at that age. A video abstract of this article can be viewed at: https://youtu.be/dDB0IGi2Hf8.

A "sense of magnitude" requires a new alternative for learning numerical symbols

Leibovich et al. proposed that the processing of numerosities is based primarily on a "sense of magnitude." The consequences of this proposal for how numerical symbols acquire their meaning are, however, neglected. We argue that symbols cannot be learned by associating them with a system that is not yet able to derive discrete numbers accurately because of immature cognitive control.

Developmental trajectories of children's symbolic numerical magnitude processing skills and associated cognitive competencies

Although symbolic numerical magnitude processing skills are key for learning arithmetic, their developmental trajectories remain unknown. Therefore, we delineated during the first 3years of primary education (5-8years of age) groups with distinguishable developmental trajectories of symbolic numerical magnitude processing skills using a model-based clustering approach. Three clusters were identified and were labeled as inaccurate, accurate but slow, and accurate and fast. The clusters did not differ in age, sex, socioeconomic status, or IQ. We also tested whether these clusters differed in domain-specific (nonsymbolic magnitude processing and digit identification) and domain-general (visuospatial short-term memory, verbal working memory, and processing speed) cognitive competencies that might contribute to children's ability to (efficiently) process the numerical meaning of Arabic numerical symbols. We observed minor differences between clusters in these cognitive competencies except for verbal working memory for which no differences were observed. Follow-up analyses further revealed that the above-mentioned cognitive competencies did not merely account for the cluster differences in children's development of symbolic numerical magnitude processing skills, suggesting that other factors account for these individual differences. On the other hand, the three trajectories of symbolic numerical magnitude processing revealed remarkable and stable differences in children's arithmetic fact retrieval, which stresses the importance of symbolic numerical magnitude processing for learning arithmetic.

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.

Evaluating the Effect of Labeled Benchmarks on Children's Number Line Estimation Performance and Strategy Use

Some authors argue that age-related improvements in number line estimation (NLE) performance result from changes in strategy use. More specifically, children's strategy use develops from only using the origin of the number line, to using the origin and the endpoint, to eventually also relying on the midpoint of the number line. Recently, Peeters et al. (unpublished) investigated whether the provision of additional unlabeled benchmarks at 25, 50, and 75% of the number line, positively affects third and fifth graders' NLE performance and benchmark-based strategy use. It was found that only the older children benefitted from the presence of these benchmarks at the quartiles of the number line (i.e., 25 and 75%), as they made more use of these benchmarks, leading to more accurate estimates. A possible explanation for this lack of improvement in third graders might be their inability to correctly link the presented benchmarks with their corresponding numerical values. In the present study, we investigated whether labeling these benchmarks with their corresponding numerical values, would have a positive effect on younger children's NLE performance and quartile-based strategy use as well. Third and sixth graders were assigned to one of three conditions: (a) a control condition with an empty number line bounded by 0 at the origin and 1,000 at the endpoint, (b) an unlabeled condition with three additional external benchmarks without numerical labels at 25, 50, and 75% of the number line, and (c) a labeled condition in which these benchmarks were labeled with 250, 500, and 750, respectively. Results indicated that labeling the benchmarks has a positive effect on third graders' NLE performance and quartile-based strategy use, whereas sixth graders already benefited from the mere provision of unlabeled benchmarks. These findings imply that children's benchmark-based strategy use can be stimulated by adding additional externally provided benchmarks on the number line, but that, depending on children's age and familiarity with the number range, these additional external benchmarks might need to be labeled.

Deficits in Approximate Number System Acuity and Mathematical Abilities in 6.5-Year-Old Children Born Extremely Preterm

Preterm children are at increased risk for poor academic achievement, especially in math. In the present study, we examined whether preterm children differ from term-born children in their intuitive sense of number that relies on an unlearned, approximate number system (ANS) and whether there is a link between preterm children’s ANS acuity and their math abilities. To this end, 6.5-year-old extremely preterm (i.e., <27 weeks gestation, n = 82) and term-born children (n = 89) completed a non-symbolic number comparison (ANS acuity) task and a standardized math test. We found that extremely preterm children had significantly lower ANS acuity than term-born children and that these differences could not be fully explained by differences in verbal IQ, perceptual reasoning skills, working memory, or attention. Differences in ANS acuity persisted even when demands on visuo-spatial skills and attention were reduced in the ANS task. Finally, we found that ANS acuity and math ability are linked in extremely preterm children, similar to previous results from term-born children. These results suggest that deficits in the ANS may be at least partly responsible for the deficits in math abilities often observed in extremely preterm children.

Approximate number sense correlates with math performance in gifted adolescents

Nonhuman animals, human infants, and human adults all share an Approximate Number System (ANS) that allows them to imprecisely represent number without counting. Among humans, people differ in the precision of their ANS representations, and these individual differences have been shown to correlate with symbolic mathematics performance in both children and adults. For example, children with specific math impairment (dyscalculia) have notably poor ANS precision. However, it remains unknown whether ANS precision contributes to individual differences only in populations of people with lower or average mathematical abilities, or whether this link also is present in people who excel in math. Here we tested non-symbolic numerical approximation in 13- to 16-year old gifted children enrolled in a program for talented adolescents (the Center for Talented Youth). We found that in this high achieving population, ANS precision significantly correlated with performance on the symbolic math portion of two common standardized tests (SAT and ACT) that typically are administered to much older students. This relationship was robust even when controlling for age, verbal performance, and reaction times in the approximate number task. These results suggest that the Approximate Number System is linked to symbolic math performance even at the top levels of math performance.

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.

Beyond comparison: The influence of physical size on number estimation is modulated by notation, range and spatial arrangement

Can physical size affect number estimation? Previous studies have shown that physical size influences non-symbolic numerosity in comparison tasks (e.g. which of two dots is larger). The current study investigated the conditions under which physical size can affect numerosity estimation. We employed a line mapping task in order to avoid the context of comparison and the need to provide a verbal label to estimate a quantity. Adult participants were briefly presented with the digits 2-8 or groups of 2-8 dots in 3 different physical sizes and were asked to estimate the position of a presented numerosity on a vertical line from 0 to 10. Physical size affected number estimation only above the subitizing range (i.e., >4) and only for non-symbolic numbers (e.g. dot arrays). Presenting non-symbolic numbers as canonical arrangements (like on a game die) reduced the effect of the physical size in the counting range (5-9). Accordingly, we suggest that the effect of task-irrelevant physical size on performance is modulated by the ability of participants to provide an accurate estimate of number: when the estimated number is easier to perceive (i.e., subitizing range or canonical arrangements), the influence of the physical size is smaller compared to when it is more difficult to give an accurate estimate of number (i.e., counting range, random arrangement). By doing so, we describe the factors that modulate the effect of physical size on number processing and provide another example of the important role continuous properties, such as physical size, play in non-symbolic number processing.

Electrophysiological dynamic brain connectivity during symbolic magnitude comparison in children with different mathematics achievement levels

Children with mathematical difficulties usually have an impaired ability to process symbolic representations. Functional MRI methods have suggested that early frontoparietal connectivity can predict mathematic achievements; however, the study of brain connectivity during numerical processing remains unexplored. With the aim of evaluating this in children with different math proficiencies, we selected a sample of 40 children divided into two groups [high achievement (HA) and low achievement (LA)] according to their arithmetic scores in the Wide Range Achievement Test, 4th ed.. Participants performed a symbolic magnitude comparison task (i.e. determining which of two numbers is numerically larger), with simultaneous electrophysiological recording. Partial directed coherence and graph theory methods were used to estimate and depict frontoparietal connectivity in both groups. The behavioral measures showed that children with LA performed significantly slower and less accurately than their peers in the HA group. Significantly higher frontocentral connectivity was found in LA compared with HA; however, when the connectivity analysis was restricted to parietal locations, no relevant group differences were observed. These findings seem to support the notion that LA children require greater memory and attentional efforts to meet task demands, probably affecting early stages of symbolic comparison.

Mental Numerosity Line in the Human's Approximate Number System

Previous studies have demonstrated existence of a mental line for symbolic numbers (e.g., Arabic digits). For nonsymbolic number systems, however, it remains unresolved whether a spontaneous spatial layout of numerosity exists. The current experiment investigated whether SNARC-like (Spatial-Numerical Association of Response Codes) effects exist in approximate processing of numerosity, as well as of size and density. Participants were asked to judge whether two serially presented stimuli (i.e., dot arrays, pentagons) were the same regarding numbers of dots, sizes of the pentagon, or densities of dots. Importantly, two confounds that were overlooked by most previous studies were controlled in this study: no ordered numerosity was presented, and only numerosity in the approximate number system (beyond the subitizing range) was used. The results demonstrated that there was a SNARC-like effect only in the numerosity-matching task. The results suggest that numerosity could be spontaneously aligned to a left-to-right oriented mental line according to magnitude information in human's approximate number system.

Benchmark-based strategies in whole number line estimation

In this study, we used verbal protocols to identify whether adults spontaneously apply quartile-based strategies or whether they need additional external support to use these strategies when solving a 0-1,000 number line estimation (NLE) task. Participants were assigned to one of three conditions based on the number of external benchmarks provided on the number line. In the bounded condition only the origin and endpoint were indicated, the mid-point condition included an additional external benchmark at 50%, and in the quartile condition three additional external benchmarks at 25%, 50%, and 75% were specified. Firstly, participants in the bounded condition reported to spontaneously apply quartile-based strategies to calibrate their estimates. Moreover, participants frequently relied on the external benchmarks for creating internal benchmarks at the mid-point, quartiles, and even octiles of the number line. Secondly, overall estimation accuracy improved as the number of external benchmarks increased, and target numbers close to external benchmarks were estimated more accurately and with less variability. Thirdly, the use of a larger variety in benchmark-based strategies was positively related to NLE accuracy. In summary, this study provides evidence that the NLE task induces more sophisticated strategy use in participants than initially anticipated.

Relation between Approximate Number System Acuity and Mathematical Achievement: The Influence of Fluency

Previous studies have observed inconsistent relations between the acuity of the Approximate Number System (ANS) and mathematical achievement. In this paper, we hypothesize that the relation between ANS acuity and mathematical achievement is influenced by fluency; that is, the mathematical achievement test covering a greater expanse of mathematical fluency may better reflect the relation between ANS acuity and mathematics skills. We explored three types of mathematical achievement tests utilized in this study: Subtraction, graded, and semester-final examination. The subtraction test was designed to measure the mathematical fluency. The graded test was more fluency-based than the semester-final examination, but both involved the same mathematical knowledge from the class curriculum. A total of 219 fifth graders from primary schools were asked to perform all three tests, then given a numerosity comparison task, a visual form perception task (figure matching), and a series of other tasks to assess general cognitive processes (mental rotation, non-verbal matrix reasoning, and choice reaction time). The findings were consistent with our expectations. The relation between ANS acuity and mathematical achievement was particularly clearly reflected in the participants' performance on the visual form perception task, which supports the domain-general explanations for the underlying mechanisms of the relation between ANS acuity and math achievement.

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).

Improving Preschoolers' Arithmetic through Number Magnitude Training: The Impact of Non-Symbolic and Symbolic Training

The numerical cognition literature offers two views to explain numerical and arithmetical development. The unique-representation view considers the approximate number system (ANS) to represent the magnitude of both symbolic and non-symbolic numbers and to be the basis of numerical learning. In contrast, the dual-representation view suggests that symbolic and non-symbolic skills rely on different magnitude representations and that it is the ability to build an exact representation of symbolic numbers that underlies math learning. Support for these hypotheses has come mainly from correlative studies with inconsistent results. In this study, we developed two training programs aiming at enhancing the magnitude processing of either non-symbolic numbers or symbolic numbers and compared their effects on arithmetic skills. Fifty-six preschoolers were randomly assigned to one of three 10-session-training conditions: (1) non-symbolic training (2) symbolic training and (3) control training working on story understanding. Both numerical training conditions were significantly more efficient than the control condition in improving magnitude processing. Moreover, symbolic training led to a significantly larger improvement in arithmetic than did non-symbolic training and the control condition. These results support the dual-representation view.

Sensory-integration system rather than approximate number system underlies numerosity processing: A critical review

It is widely accepted that human and nonhuman species possess a specialized system to process large approximate numerosities. The theory of an evolutionarily ancient approximate number system (ANS) has received converging support from developmental studies, comparative experiments, neuroimaging, and computational modelling, and it is one of the most dominant and influential theories in numerical cognition. The existence of an ANS system is significant, as it is believed to be the building block of numerical development in general. The acuity of the ANS is related to future arithmetic achievements, and intervention strategies therefore aim to improve the ANS. Here we critically review current evidence supporting the existence of an ANS. We show that important shortcomings and confounds exist in the empirical studies on human and non-human animals as well as the logic used to build computational models that support the ANS theory. We conclude that rather than taking the ANS theory for granted, a more comprehensive explanation might be provided by a sensory-integration system that compares or estimates large approximate numerosities by integrating the different sensory cues comprising number stimuli.

Effects of Non-Symbolic Approximate Number Practice on Symbolic Numerical Abilities in Pakistani Children

Current theories of numerical cognition posit that uniquely human symbolic number abilities connect to an early developing cognitive system for representing approximate numerical magnitudes, the approximate number system (ANS). In support of this proposal, recent laboratory-based training experiments with U.S. children show enhanced performance on symbolic addition after brief practice comparing or adding arrays of dots without counting: tasks that engage the ANS. Here we explore the nature and generality of this effect through two brief training experiments. In Experiment 1, elementary school children in Pakistan practiced either a non-symbolic numerical addition task or a line-length addition task with no numerical content, and then were tested on symbolic addition. After training, children in the numerical training group completed the symbolic addition test faster than children in the line length training group, suggesting a causal role of brief, non-symbolic numerical training on exact, symbolic addition. These findings replicate and extend the core findings of a recent U.S. laboratory-based study to non-Western children tested in a school setting, attesting to the robustness and generalizability of the observed training effects. Experiment 2 tested whether ANS training would also enhance the consistency of performance on a symbolic number line task. Over several analyses of the data there was some evidence that approximate number training enhanced symbolic number line placements relative to control conditions. Together, the findings suggest that engagement of the ANS through brief training procedures enhances children's immediate attention to number and engagement with symbolic number tasks.

The Symbol Grounding Problem Revisited: A Thorough Evaluation of the ANS Mapping Account and the Proposal of an Alternative Account Based on Symbol-Symbol Associations

Recently, a lot of studies in the domain of numerical cognition have been published demonstrating a robust association between numerical symbol processing and individual differences in mathematics achievement. Because numerical symbols are so important for mathematics achievement, many researchers want to provide an answer on the ‘symbol grounding problem,’ i.e., how does a symbol acquires its numerical meaning? The most popular account, the approximate number system (ANS) mapping account, assumes that a symbol acquires its numerical meaning by being mapped on a non-verbal and ANS. Here, we critically evaluate four arguments that are supposed to support this account, i.e., (1) there is an evolutionary system for approximate number processing, (2) non-symbolic and symbolic number processing show the same behavioral effects, (3) non-symbolic and symbolic numbers activate the same brain regions which are also involved in more advanced calculation and (4) non-symbolic comparison is related to the performance on symbolic mathematics achievement tasks. Based on this evaluation, we conclude that all of these arguments and consequently also the mapping account are questionable. Next we explored less popular alternative, where small numerical symbols are initially mapped on a precise representation and then, in combination with increasing knowledge of the counting list result in an independent and exact symbolic system based on order relations between symbols. We evaluate this account by reviewing evidence on order judgment tasks following the same four arguments. Although further research is necessary, the available evidence so far suggests that this symbol–symbol association account should be considered as a worthy alternative of how symbols acquire their meaning.

A Systematic Investigation of Accuracy and Response Time Based Measures Used to Index ANS Acuity

The approximate number system (ANS) was proposed to be a building block for later mathematical abilities. Several measures have been used interchangeably to assess ANS acuity. Some of these measures were based on accuracy data, whereas others relied on response time (RT) data or combined accuracy and RT data. Previous studies challenged the view that all these measures can be used interchangeably, because low correlations between some of the measures had been observed. These low correlations might be due to poor reliability of some of the measures, since the majority of these measures are mathematically related. Here we systematically investigated the relationship between common ANS measures while avoiding the potential confound of poor reliability. Our first experiment revealed high correlations between all accuracy based measures supporting the assumption that all of them can be used interchangeably. In contrast, not all RT based measures were highly correlated. Additionally, our results revealed a speed-accuracy trade-off. Thus, accuracy and RT based measures provided conflicting conclusions regarding ANS acuity. Therefore, we investigated in two further experiments which type of measure (accuracy or RT) is more informative about the underlying ANS acuity, depending on participants’ preferences for accuracy or speed. To this end, we manipulated participants’ preferences for accuracy or speed both explicitly using different task instructions and implicitly varying presentation duration. Accuracy based measures were more informative about the underlying ANS acuity than RT based measures. Moreover, the influence of the underlying representations on accuracy data was more pronounced when participants preferred accuracy over speed after the accuracy instruction as well as for long or unlimited presentation durations. Implications regarding the diffusion model as a theoretical framework of dot comparison as well as regarding the relationship between ANS acuity and math performance are discussed.

Cognitive factors affecting children's nonsymbolic and symbolic magnitude judgment abilities: A latent profile analysis

Early math abilities are claimed to be linked to magnitude representation ability. Some claim that nonsymbolic magnitude abilities scaffold the acquisition of symbolic (Arabic number) magnitude abilities and influence math ability. Others claim that symbolic magnitude abilities, and ipso facto math abilities, are independent of nonsymbolic abilities and instead depend on the ability to process number symbols (e.g., 2, 7). Currently, the issue of whether symbolic abilities are or are not related to nonsymbolic abilities, and the cognitive factors associated with nonsymbolic-symbolic relationships, remains unresolved. We suggest that different nonsymbolic-symbolic relationships reside within the general magnitude ability distribution and that different cognitive abilities are likely associated with these different relationships. We further suggest that the different nonsymbolic-symbolic relationships and cognitive abilities in combination differentially predict math abilities. To test these claims, we used latent profile analysis to identify nonsymbolic-symbolic judgment patterns of 124, 5- to 7-year-olds. We also assessed four cognitive factors (visuospatial working memory [VSWM], naming numbers, nonverbal IQ, and basic reaction time [RT]) and two math abilities (number transcoding and single-digit addition abilities). Four nonsymbolic-symbolic ability profiles were identified. Naming numbers, VSWM, and basic RT abilities were differentially associated with the different ability profiles and in combination differentially predicted math abilities. Findings show that different patterns of nonsymbolic-symbolic magnitude abilities can be identified and suggest that an adequate account of math development should specify the inter-relationship between cognitive factors and nonsymbolic-symbolic ability patterns.

The Contribution of Numerical Magnitude Comparison and Phonological Processing to Individual Differences in Fourth Graders' Multiplication Fact Ability

Although numerical magnitude processing has been related to individual differences in arithmetic, its role in children’s multiplication performance remains largely unknown. On the other hand, studies have indicated that phonological awareness is an important correlate of individual differences in children’s multiplication performance, but the involvement of phonological memory, another important phonological processing skill, has not been studied in much detail. Furthermore, knowledge about the relative contribution of above mentioned processes to the specific arithmetic operation of multiplication in children is lacking. The present study therefore investigated for the first time the unique contributions of numerical magnitude comparison and phonological processing in explaining individual differences in 63 fourth graders’ multiplication fact ability (mean age = 9.6 years, SD = .67). The results showed that children’s multiplication fact competency correlated significantly with symbolic and nonsymbolic magnitude comparison as well as with phonological short-term memory. A hierarchical regression analysis revealed that, after controlling for intellectual ability and general reaction time, both symbolic and nonsymbolic magnitude comparison and phonological short-term memory accounted for unique variance in multiplication fact performance. The ability to compare symbolic magnitudes was found to contribute the most, indicating that the access to numerical magnitudes by means of Arabic digits is a key factor in explaining individual differences in children’s multiplication fact ability.

The precision of mapping between number words and the approximate number system predicts children's formal math abilities

Children can represent number in at least two ways: by using their non-verbal, intuitive approximate number system (ANS) and by using words and symbols to count and represent numbers exactly. Furthermore, by the time they are 5years old, children can map between the ANS and number words, as evidenced by their ability to verbally estimate numbers of items without counting. How does the quality of the mapping between approximate and exact numbers relate to children's math abilities? The role of the ANS-number word mapping in math competence remains controversial for at least two reasons. First, previous work has not examined the relation between verbal estimation and distinct subtypes of math abilities. Second, previous work has not addressed how distinct components of verbal estimation-mapping accuracy and variability-might each relate to math performance. Here, we addressed these gaps by measuring individual differences in ANS precision, verbal number estimation, and formal and informal math abilities in 5- to 7-year-old children. We found that verbal estimation variability, but not estimation accuracy, predicted formal math abilities, even when controlling for age, expressive vocabulary, and ANS precision, and that it mediated the link between ANS precision and overall math ability. These findings suggest that variability in the ANS-number word mapping may be especially important for formal math abilities.