On the Relation between the Mental Number Line and Arithmetic Competencies

Left digit bias in children's and adults' paper-and-pencil number line estimation

Number line estimation tasks are frequently used to learn about numerical thinking, learning, and development. These tasks are often interpreted as though estimates are determined by overall magnitudes of target numerals, rather than specific instantiating digits. Yet estimates are strongly biased by leftmost digits. For example, numbers like "698" are placed too far to the left of numbers like "701" on a 0-1,000 line. This "left digit effect" or "left digit bias" has been investigated little in children, and only on electronic tasks. Here, we ask whether left digit bias appears in paper-and-pencil estimates, and whether it differs for paper-based versus computer-based tasks. In Study 1, 5- to 8-year-old children completed a 0-100 number line task on paper. In Study 2, 7- to 11-year-olds completed a 0-1,000 paper task. In Study 3, adults completed tasks on paper in both ranges. Large left digit effects were observed for children aged 8 years or older and adults, but we did not find evidence for left digit bias in younger children. Study 4 compared paper and computer tasks for adults and children aged 9-12 years. Strong left digit bias was observed in all conditions, with a larger effect for the paper-based task in children. Large left digit effects in number line estimation emerge regardless of task format, with a developmental trajectory broadly consistent with other studies. For children in the age range that reliably exhibits left digit bias (but not adults), paper-and-pencil number line estimation tasks elicit even greater bias than computer-based tasks.

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.

Understanding the complexities of mathematical cognition: A multi-level framework

Mathematics skills are associated with future employment, well-being, and quality of life. However, many adults and children fail to learn the mathematics skills they require. To improve this situation, we need to have a better understanding of the processes of learning and performing mathematics. Over the past two decades, there has been a substantial growth in psychological research focusing on mathematics. However, to make further progress, we need to pay greater attention to the nature of, and multiple elements involved in, mathematical cognition. Mathematics is not a single construct; rather, overall mathematics achievement is comprised of proficiency with specific components of mathematics (e.g., number fact knowledge, algebraic thinking), which in turn recruit basic mathematical processes (e.g., magnitude comparison, pattern recognition). General cognitive skills and different learning experiences influence the development of each component of mathematics as well as the links between them. Here, I propose and provide evidence for a framework that structures how these components of mathematics fit together. This framework allows us to make sense of the proliferation of empirical findings concerning influences on mathematical cognition and can guide the questions we ask, identifying where we are missing both research evidence and models of specific mechanisms.

A potential dissociation between perception and production version for bounded but not unbounded number line estimation

BACKGROUND

What, exactly, do number line estimation (NLE) tasks measure? Different versions of the task were observed to have different effects on performance.

METHOD

We investigated associations between the production (indicating the location) and perception version (indicating the number) of the bounded and unbounded NLE task and their relationship to arithmetic.

RESULTS

A stronger correlation was observed between the production and perception version of the unbounded than the bounded NLE task, indicating that both versions of the unbounded-but not the bounded-NLE task measure the same construct. Moreover, overall low but significant associations between NLE performance and arithmetic were only observed for the production version of the bounded NLE task.

CONCLUSION

These results substantiate that the production version of bounded NLE seems to rely on proportion judgment strategies, whereas both unbounded versions and the perception version of the bounded NLE task may rely more on magnitude estimation.

Longitudinal relationship between number line estimation and other mathematical abilities in Chinese preschool children

Many cross-sectional studies have shown that number line estimation is associated with other mathematical skills, but there has been limited longitudinal research. To systematically examine such associations longitudinally at the earliest stage of mathematical learning, the current study tested 40 Chinese preschoolers (mean age = 4.97 years, SD = 0.18) and followed them up 8 months later. For both waves of data collection, children were administered six tasks: number line estimation, dot counting, comparison of two dot arrays, comparison of triple dot arrays, symbolic number comparison, and simple addition. Results of two-wave cross-lagged panel analysis showed that (a) dot counting and non-symbolic numerical comparison at Time 1 had significant longitudinal associations with number line estimation at Time 2, (b) number line estimation had bidirectional associations with symbolic number comparison, and (c) number line estimation at Time 1 had a marginally significant longitudinal association with simple addition at Time 2. These results extend the small but accumulating literature on the longitudinal relations between number line estimation and other mathematical skills and specify the important role of number line estimation in the early development of mathematical skills.

The relevance of basic numerical skills for fraction processing: Evidence from cross-sectional data

Recent research indicated that fraction understanding is an important predictor of later mathematical achievement. In the current study we investigated associations between basic numerical skills and students' fraction processing. We analyzed data of 939 German secondary school students (age range = 11.92 to 18.00 years) and evaluated the determinants of fraction processing considering basic numerical skills as predictors (i.e., number line estimation, basic arithmetic operations, non-symbolic magnitude comparison, etc.). Additionally, we controlled for general cognitive ability, grade level, and sex. We found that multiplication, subtraction, conceptual knowledge, number line estimation, and basic geometry were significantly associated with fraction processing beyond significant associations of general cognitive ability and sex. Moreover, relative weight analysis revealed that addition and approximate arithmetic should also be considered as relevant predictors for fraction processing. The current results provide food for thought that further research should focus on investigating whether recapitulating basic numerical content in secondary school mathematics education can be beneficial for acquiring more complex mathematical concepts such as fractions.

The number line estimation task is a valid tool for assessing mathematical achievement: A population-level study with 6484 Luxembourgish ninth-graders

The number line estimation task is an often-used measure of numerical magnitude understanding. The task also correlates substantially with broader measures of mathematical achievement. This raises the question of whether the task would be a useful component of mathematical achievement tests and instruments to diagnose dyscalculia or mathematical giftedness and whether a stand-alone version of the task can serve as a short screener for mathematical achievement. Previous studies on the relation between number line estimation accuracy and broader mathematical achievement were limited in that they used relatively small nonrepresentative samples and usually did not account for potentially confounding variables. To close this research gap, we report findings from a population-level study with nearly all Luxembourgish ninth-graders (N = 6484). We used multilevel regressions to test how a standardized mathematical achievement test relates to the accuracy in number line estimation on bounded number lines with whole numbers and fractions. We also investigated how these relations were moderated by classroom characteristics, person characteristics, and trial characteristics. Mathematical achievement and number line estimation accuracy were associated even after controlling for potentially confounding variables. Subpopulations of students showed meaningful differences in estimation accuracy, which can serve as benchmarks in future studies. Compared with the number line estimation task with whole numbers, the number line estimation task with fractions was more strongly related to mathematical achievement in students across the entire mathematical achievement spectrum. These results show that the number line estimation task is a valid and useful tool for diagnosing and monitoring mathematical achievement.

Development of the Mental Number Line Representation of Numbers 0–10 and Its Relationship to Mental Arithmetic

The internal representation of numbers on the mental number line (MNL) was demonstrated by performing the computerized version of the number-to-position (CNP) task on a touchscreen while restricting response time. We found that the estimation pattern is best fit by a sigmoid function, further denoted as the “sigmoidal model”. Two developmental leaps occurring during elementary school were recognized: (1) the division of the number line into two segments and (2) consistent use of different anchor points on the number line—the left endpoint in first grade, the right endpoint in second grade, and finally the midpoint in third grade. Additionally, when examining the differences between the breakpoints, we found that first graders demonstrated a breakpoint close to 6, which linearly decreased over the years until stabilizing close to 5. The relation between the ability to place individual numbers on a number line and performance of mental arithmetic showed that the consistent use of anchor points correlated significantly with faster responses in mental arithmetic.

Emerging School Readiness Profiles: Motor Skills Matter for Cognitive- and Non-cognitive First Grade School Outcomes

A promising approach for studying school readiness involves a person-centered approach, aimed at exploring how functioning in diverse developmental domains conjointly affects children’s school outcomes. Currently, however, a systematic understanding lacks of how motor skills, in conjunction with other school readiness skills, affect a child’s school outcomes. Additionally, little is known about longitudinal associations of school readiness with non-academic (e.g., socioemotional) school outcomes. Therefore, we examined the school readiness skills of a sample of Dutch children (N = 91) with a mean age of 3 years and 4 months (46% girls). We used a multi-informant test battery to assess children’s school readiness in terms of executive functions (EFs), language and emergent literacy, motor skills, and socioemotional behavior. During the spring term of a child’s first grade year, we collected academic and non-academic (i.e., EFs, motor skills, socioemotional- and classroom behavior, and creative thinking) school outcomes. A latent profile analysis revealed four distinct profiles. Children in the “Parent Positive” (29%) profile were rated positively by their parents, and performed variably on motor and language/emergent literacy skills tests. The second profile–“Multiple Strengths” (13%)–consisted of children showing strengths in multiple domains, especially with respect to motor skills. Children from the third profile–“Average Performers” (50%)–did not show any distinct strengths or weaknesses, rather displayed school readiness skill levels close to, or just below the sample mean. Finally, the “Parental Concern” (8%) profile was characterized by high levels of parental concerns, while displaying slightly above average performance on specific motor and language skills. Motor skills clearly distinguished between profiles, next to parent-rated EFs and socioemotional behavior, and to a lesser extent emergent literacy skills. School readiness profiles were found to differ in mean scores on first grade academic achievement, parent- and teacher-rated EFs, motor skills, parent-rated socioemotional functioning, and pre-requisite learning skills. The pattern of mean differences was complex, suggesting that profiles could not be ranked from low to high in terms of school outcomes. Longitudinal studies are needed to disentangle the interaction between emerging school readiness of the child and the surrounding context.

The association of basic numerical abilities and math achievement: The mediating role of visuospatial and arithmetical abilities

Basic numerical abilities such as number line estimation have been observed repeatedly to be associated with mathematical achievement. Recently, it was argued that the association between basic numerical abilities and mathematical achievement is fully mediated by visuospatial abilities. However, arithmetical abilities have not yet been considered as influencing this association, even though solution strategies in number line estimation as well as mathematical achievement often involve arithmetical procedures. Therefore, we investigated the mediating role of arithmetical and visuospatial abilities on the association between number line estimation and mathematical achievement in a sample of n = 599 German elementary school students. The results indicated that arithmetical abilities as well as visuospatial abilities mediated the association between number line estimation and mathematical achievement. However, neither visuospatial nor arithmetical abilities fully mediated the association between number line estimation and mathematical achievement when considered in isolation. This substantiates the relevance of the intertwined development of visuospatial and arithmetical abilities as well as basic numerical abilities such as number line estimation (i.e., the combination of domain-specific numerical and domain-general abilities) driving mathematical achievement.

The new unbounded number line estimation task: A systematic literature review

In 2011, Cohen and Blanc-Goldhammer introduced a new unbounded version of the number line estimation task which they argued to overcome limitations of its 'traditional' bounded counterpart. In line with this proposition, there is increasing evidence that the unbounded number line estimation task may indeed reflect a purer measure of the underlying representation of number magnitude. However, only few studies used this task version yet. In the present literature review, we aimed at evaluating all studies employing the unbounded task version so far to provide an overview of the current state of research. To identify all relevant articles and to evaluate the validity of the task, we conducted a systematic literature search in different databases following the PRISMA guidelines in May 2021. Methodological differences and commonalities of the 16 studies that met the inclusion criteria are discussed here. Our evaluation indicated considerable differences between studies with respect to the number range covered or methodological features such as display size. Additionally, five studies observed evidence for estimation biases in the unbounded task as well. Nevertheless, this review also substantiated the claim that the unbounded task version might indeed be a more valid and purer measure of the mental representation of number magnitude as the results of 14 studies confirmed this hypothesis.

Operational momentum during children's approximate arithmetic relates to symbolic math skills and space-magnitude association

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space-magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space-magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.

Relations among spatial skills, number line estimation, and exact and approximate calculation in young children

Decades of research have established that spatial skills correlate with numerical skills. However, because both spatial and numerical skills are multidimensional, we sought to determine how specific spatial skills relate to specific numeracy skills. We used a cohort-sequential design, assessing a large diverse sample of students (N = 612, initially in pre-kindergarten [pre-K]-3rd grade, 4-9 years of age) at four time points spanning 2 years. We examined how initial levels of five spatial skills (visuospatial working memory [VSWM], mental transformation, mental rotation, proportional reasoning, and analog magnitude system [AMS] acuity) related to initial levels and growth rates in exact and approximate calculation skills, and we further investigated number line estimation as a potential mediator. We found unique patterns of relations between spatial skills and numeracy. Initial levels of mental rotation, proportional reasoning, and AMS acuity related to initial levels of exact calculation skill; initial levels of AMS acuity related to initial levels of approximate calculation; and initial levels of proportional reasoning related to initial levels of number line estimation. VSWM and mental transformation did not relate to numeracy skills after controlling for other spatial skills. Initial levels of number line estimation related to both exact and approximate calculation after controlling for spatial skills. Notably, neither spatial skills nor number line estimation predicted growth in exact or approximate calculation skills. These results indicate that there is specificity in the time-invariant relations between spatial skills and numeracy, and they suggest that researchers and educators should treat spatial skills and numeracy as multidimensional constructs with complex and unique interrelations.

The influence of children's mathematical competence on performance in mental number line, time knowledge and time perception

A growing body of research suggests that space, time and number are represented within a common system. Other studies have shown this relationship is related to the mathematical competency. Here we examined the influence of the mathematical capacities of 8-12 years old children, grouped into high (n = 63) and low (n = 58) on performance in mental number line, time knowledge and time perception. The results revealed that mathematical competency influences mental number line and time knowledge, but with regard to time perception the effects were only observed in time production task. In addition, the results of correlation analysis revealed interaction between time knowledge, time production (but not reproduction) and mental number line. Finally, the findings are discussed within the framework of the recent theories regarding representation of space, time and number.

Number line development of Chilean children from preschool to the end of kindergarten

Children's performance on number line tasks reflects their developing number system knowledge. Before 5 years of age, most children perform poorly on even the simplest number lines (i.e., 0-10). Our goal was to understand how number line skills develop before formal schooling. Chilean preschoolers attempted a 0-10 number line task three times over 2 years: at the beginning of pre-kindergarten (M = 4:7 [years;months]; Time 1), at the end of pre-kindergarten (M = 5:0; Time 2), and at the end of kindergarten (M = 5:10; Time 3). We used latent class analysis to group children according to their patterns of performance across number targets. At Time 1, 86% of children had error patterns indicating that they randomly placed estimates on the line. At Time 2, 56% of children continued to respond randomly. At Time 3, 56% of children showed competent performance across the number line, 23% were accurate only near the endpoints, and 21% were accurate only for low target numbers near the origin. Latent transition analyses showed that the transition from less to more proficient estimation classes was predicted by children's number identification skills. Thus, number line performance changed dramatically from 4 to 6 years of age as children began to develop the cognitive and numerical skills necessary to accurately estimate numbers on a number line.

Longitudinal Performance in Basic Numerical Skills Mediates the Relationship Between Socio-Economic Status and Mathematics Anxiety: Evidence From Chile

Socio-economic status (SES) and mathematical performance seem to be risk factors of mathematics anxiety (MA) in both children and adults. However, there is little evidence about how exactly these three constructs are related, especially during early stages of mathematical learning. In the present study, we assessed longitudinal performance in symbolic and non-symbolic basic numerical skills in pre-school and second grade students, as well as MA in second grade students. Participants were 451 children (average pre-school age = 5 years, 6 months) from 12 schools in Chile, which differed in school vulnerability index (SVI), an indicator of SES. We tested an explanatory model of MA that included SES and longitudinal performance in basic numerical skills as predictors. The results showed a direct effect of SES on MA and a mediating effect of performance in symbolic and non-symbolic comparison tasks in pre-school. However, in second grade, only performance in symbolic comparison significantly mediated the SES-MA relationship. These findings suggest that performance in non-symbolic comparison plays an important role in explaining MA at initial stages, but that its influence is no longer significant by the time children reach formal instruction in second grade. By contrast, as children's formal educational experience in mathematics increases, MA becomes linked primarily to symbolic numerical tasks. In sum, SES affects MA and this is due in part to the effect of SES on the development of numerical learning in pre-school, which in turn has an impact on subsequent, more complex learning, ultimately leading to differences in MA. We discuss the implications of these findings for preventing and acting upon the emergence of MA.

Children's neural activity during number line estimations assessed by functional near-infrared spectroscopy (fNIRS)

Number line estimation (NLE) is an educational task in which children estimate the location of a value (e.g., 25) on a blank line that represents a numerical range (e.g., 0-100). NLE performance is a strong predictor of success in mathematics, and error patterns on this task help provide a glimpse into how children may represent number internally. However, a missing and fundamental element of this puzzle is the identification of neural correlates of NLE in children. That is, understanding possible neural signatures related to NLE performance will provide valuable insight into the cognitive processes that underlie children's development of NLE ability. Using functional near-infrared spectroscopy (fNIRS), we provide the first investigation of concurrent behavioral and cortical signatures of NLE performance in children. Specifically, our results highlight significant fronto-parietal changes in cortical activation in response to increases in NLE scale (e.g., 0-100 vs. 0-100,000). Furthermore, our results demonstrate that NLE performance feedback (auditory, visual, or audiovisual), as well as children's grade (2nd vs. 3rd) influence cortical responding during an NLE task.

Different neural activations for an approaching friend versus stranger: Linking personal space to numerical cognition

INTRODUCTION

Typically, humans place themselves at a preferred distance from others. This distance is known to characterize human spatial behavior. Here, we focused on neurocognitive conditions that may affect interpersonal distances. The current study investigated whether neurocognitive deficiencies in numerical and spatial knowledge may affect social perception and modulate personal space.

METHOD

In an event-related potential (ERP) study, university students with developmental dyscalculia (DD) and typically developing control participants were given a computerized version of the comfortable interpersonal distance task, in which participants were instructed to press the spacebar when they began to feel uncomfortable by the approach of a virtual protagonist.

RESULTS

Results showed that students with deficiencies in numerical and spatial skills (i.e., DD) demonstrated reduced variability in their preferred distance from an approaching friend. Importantly, DD showed decreased amplitude of the N1 wave in the friend condition.

CONCLUSION

These results suggest that people coping with deficiencies in spatial cognition have a less efficient allocation of spatial attention in the service of processing personal distances. Accordingly, the study highlights the fundamental role of spatial neurocognition in organizing social space.

Time Reading in Middle and Secondary School Students: The Influence of Basic-Numerical Abilities

Time reading skills are central for the management of personal and professional life. However, little is known about the differential influence of basic numerical abilities on analog and digital time reading in general and in middle and secondary school students in particular. The present study investigated the influence of basic numerical skills separately for analog and digital time reading in N = 709 students from 5th to 8th grade. The present findings suggest that the development of time reading skills is not completed by the end of primary school. Results indicated that aspects of magnitude manipulation and arithmetic fact knowledge predicted analog time reading significantly over and above the influence of age. Furthermore, results showed that spatial representations of number magnitude, magnitude manipulation, arithmetic fact knowledge, and conceptual knowledge were significant predictors of digital time reading beyond general cognitive ability and sex. To the best of our knowledge, the present study is the first to show differential effects of basic numerical abilities on analog and digital time reading skills in middle and secondary school students. As time readings skills are crucial for everyday life, these results are highly relevant to better understand basic numerical processes underlying time reading.

The spatial-numerical association of response codes effect and math skills: why related?

Evidence from multiple studies conducted in the past few decades converges on the conclusion that numerical properties can be associated with specific directions in space. Such spatial-numerical associations (SNAs), as a signature of elementary number processing, seem to be a likely correlate of math skills. Nevertheless, almost three decades of research on the spatial-numerical association of response codes (SNARC) effect, the hallmark of SNAs, has not provided conclusive results on whether there is a relation with math skills. Here, going beyond reviewing the existing literature on the topic, we try to answer a more fundamental question about why the SNARC effect should (and should not) be related to math skills. We propose a multiroute model framework for a SNARC-math skills relationship. We conclude that the relationship is not straightforward and that several other factors should be considered, which under certain circumstances or in certain groups can cause effects of opposite directions. The model can account for conflicting results, and thus may be helpful for deriving predictions in future studies.

Gender differences in the development of semantic and spatial processing of numbers

This study recruited kindergarteners and first graders to investigate gender and grade differences in semantic and spatial processing of number magnitude. Results based on the Bayesian statistics showed that (1) there was extreme evidence in favour of grade differences in both semantic processing and spatial processing; (2) there were no gender differences in semantic processing; and (3) boys developed earlier than girls in spatial processing of numbers, especially for the more difficult task. These results are discussed in terms of gender differences in cognitive mechanisms underlying semantic and spatial processing of number magnitude.

Fixated in unfamiliar territory: Mapping estimates across typical and atypical number lines

Adults (N = 72) estimated the location of target numbers on number lines that varied in numerical range (i.e., typical range 0-10,000 or atypical range 0-7,000) and spatial orientation (i.e., the 0 endpoint on the left [traditional] or on the right [reversed]). Eye-tracking data were used to assess strategy use. Participants made meaningful first fixations on the line, with fixations occurring around the origin for low target numbers and around the midpoint and endpoint for high target numbers. On traditional direction number lines, participants used left-to-right scanning and showed a leftward bias; these effects were reduced for the reverse direction number lines. Participants made fixations around the midpoint for both ranges but were less accurate when estimating target numbers around the midpoint on the 7,000-range number line. Thus, participants are using the internal benchmark (i.e., midpoint) to guide estimates on atypical range number lines, but they have difficulty calculating the midpoint, leading to less accurate estimates. In summary, both range and direction influenced strategy use and accuracy, suggesting that both numerical and spatial processes influence number line estimation.

Malleability of mappings between Arabic numerals and approximate quantities: Factors underlying individual differences and the relation to math

Humans tend to be inaccurate and inconsistent when estimating a large number of objects. Furthermore, we modify our estimates when feedback or a reference array is provided, indicating that the mappings between perceived numerosity and their corresponding numerals are largely malleable in response to calibration. However, there is great variability in response to calibration across individuals. Using uncalibrated and calibrated numerosity estimation conditions, the current study explored the factors underlying individual differences in the extent and nature of the malleability of numerosity estimation performance as a result of calibration in a sample of 71 undergraduate students. We found that individual differences in performance were reliable across conditions, and participants' responses to calibration varied greatly. Participants who were less consistent or had more proportionally spaced (i.e., linear) estimates before calibration tended to shift the distributions of their estimates to a greater extent. Higher calculation competence also predicted an increase in how linear participants' estimates were after calibration. Moreover, the effect of calibration was not continuous across numerosities within participants. This suggests that the mechanisms underlying numeral-numerosity mappings may be less systematic than previously thought and likely depend on cognitive mechanisms beyond representation of numerosities. Taken together, the mappings between numerosities and numerical symbols may not be stable and direct, but transient and mediated by task-related (e.g., strategic) mechanisms. Rather than estimation skills being foundational for math competence, math competence may also influence estimation skills. Therefore, numerosity estimation tasks are not a pure measure of number representations.

A Mathematical Model of How People Solve Most Variants of the Number-Line Task

Current understanding of the development of quantity representations is based primarily on performance in the number-line task. We posit that the data from number-line tasks reflect the observer's underlying representation of quantity, together with the cognitive strategies and skills required to equate line length and quantity. Here, we specify a unified theory linking the underlying psychological representation of quantity and the associated strategies in four variations of the number-line task: the production and estimation variations of the bounded and unbounded number-line tasks. Comparison of performance in the bounded and unbounded number-line tasks provides a unique and direct way to assess the role of strategy in number-line completion. Each task produces a distinct pattern of data, yet each pattern is hypothesized to arise, at least in part, from the same underlying psychological representation of quantity. Our model predicts that the estimated biases from each task should be equivalent if the different completion strategies are modeled appropriately and no other influences are at play. We test this equivalence hypothesis in two experiments. The data reveal all variations of the number-line task produce equivalent biases except for one: the estimation variation of the bounded number-line task. We discuss the important implications of these findings.

On the linear representation of numbers: evidence from a new two-numbers-to-two positions task

In the number-to-position methodology, a number is presented on each trial and the observer places it on a straight line in a position that corresponds to its felt subjective magnitude. In the novel modification introduced in this study, the two-numbers-to-two-positions method, a pair of numbers rather than a single number is presented on each trial and the observer places them in appropriate positions on the same line. Responses in this method indicate not only the subjective magnitude of each single number but, simultaneously, provide a direct estimation of their subjective numerical distance. The results of four experiments provide strong evidence for a linear representation of numbers and, commensurately, for the linear representation of numerical distances. We attribute earlier results that indicate a logarithmic representation to the ordered nature of numbers and to the task used and not to a truly non-linear underlying representation.

The SNARC effect is associated with worse mathematical intelligence and poorer time estimation

Interactions between the ways we process space, numbers and time may arise from shared and innate generic magnitude representations. Alternatively or concurrently, such interactions could be due to the use of physical magnitudes, like spatial extent, as metaphors for more abstract ones, like number and duration. That numbers might be spatially represented along a mental number line is suggested by the SNARC effect: faster left-side responses to small single digits, like 1 or 2, and faster right-side responses to large ones, like 8 or 9. Previously, we found that time estimation predicts mathematical intelligence and speculated that it may predict spatial ability too. Here, addressing this issue, we test-on a relatively large sample of adults and entirely within subjects-the relationships between (a) time: proficiency at producing and evaluating durations shorter than one second, (b) space: the ability to mentally rotate objects, (c) numbers: mathematical reasoning skills, and (d) space-number associations: the SNARC effect. Better time estimation was linked to greater mathematical intelligence and better spatial skills. Strikingly, however, stronger associations between space and numbers predicted worse mathematical intelligence and poorer time estimation.

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.

Children's Understanding of the Natural Numbers' Structure

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between "5" and "10" is larger than the distance between "75" and "80." This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, ; Siegler & Opfer, ). However, several investigators have questioned this argument (e.g., Barth & Paladino, ; Cantlon, Cordes, Libertus, & Brannon, ; Cohen & Blanc-Goldhammer, ). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as "5" or "80"). In Experiments 1 and 2, when 5- and 6-year-olds choose between number lines in a forced-choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1-100 are arranged. The bead placement of 4- and 5-year-olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number-line task may depend on strategies specific to the task.

On the genesis of spatial-numerical associations: Evolutionary and cultural factors co-construct the mental number line

Mapping numbers onto space is a common cognitive representation that has been explored in both behavioral and neuroimaging contexts. Empirical work probing the diverse nature of these spatial-numerical associations (SNAs) has led researchers to question 1) how the human brain links numbers with space, and 2) whether this link is biologically vs. culturally determined. We review the existing literature on the development of SNAs and situate that empirical work within cognitive and neuroscientific theoretical frameworks. We propose that an evolutionarily-ancient frontal-parietal circuit broadly tuned to multiple magnitude dimensions provides the phylogenetic substrate for SNAs, while enculturation and sensorimotor experience shape their specific profiles. We then use this perspective to discuss educational implications and highlight promising avenues for future research.

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.

Implicit and Explicit Number-Space Associations Differentially Relate to Interference Control in Young Adults With ADHD

Behavioral evidence for the link between numerical and spatial representations comes from the spatial-numerical association of response codes (SNARC) effect, consisting in faster reaction times to small/large numbers with the left/right hand respectively. The SNARC effect is, however, characterized by considerable intra- and inter-individual variability. It depends not only on the explicit or implicit nature of the numerical task, but also relates to interference control. To determine whether the prevalence of the latter relation in the elderly could be ascribed to younger individuals' ceiling performances on executive control tasks, we determined whether the SNARC effect related to Stroop and/or Flanker effects in 26 young adults with ADHD. We observed a divergent pattern of correlation depending on the type of numerical task used to assess the SNARC effect and the type of interference control measure involved in number-space associations. Namely, stronger number-space associations during parity judgments involving implicit magnitude processing related to weaker interference control in the Stroop but not Flanker task. Conversely, stronger number-space associations during explicit magnitude classifications tended to be associated with better interference control in the Flanker but not Stroop paradigm. The association of stronger parity and magnitude SNARC effects with weaker and better interference control respectively indicates that different mechanisms underlie these relations. Activation of the magnitude-associated spatial code is irrelevant and potentially interferes with parity judgments, but in contrast assists explicit magnitude classifications. Altogether, the present study confirms the contribution of interference control to number-space associations also in young adults. It suggests that magnitude-associated spatial codes in implicit and explicit tasks are monitored by different interference control mechanisms, thereby explaining task-related intra-individual differences in number-space associations.

Distinct Representations of Magnitude and Spatial Position within Parietal Cortex during Number-Space Mapping

Mapping numbers onto space is foundational to mathematical cognition. These cognitive operations are often conceptualized in the context of a "mental number line" and involve multiple brain regions in or near the intraparietal sulcus (IPS) that have been implicated both in numeral and spatial cognition. Here we examine possible differentiation of function within these brain areas in relating numbers to spatial positions. By isolating the planning phase of a number line task and introducing spatiotopic mapping tools from fMRI into mental number line task research, we are able to focus our analysis on the neural activity of areas in anterior IPS (aIPS) previously associated with number processing and on spatiotopically organized areas in and around posterior IPS (pIPS), while participants prepare to place a number on a number line. Our results support the view that the nonpositional magnitude of a numerical symbol is coded in aIPS, whereas the position of a number in space is coded in posterior areas of IPS. By focusing on the planning phase, we are able to isolate activation related to the cognitive, rather than the sensory-motor, aspects of the task. Also, to allow the separation of spatial position from magnitude, we tested both a standard positive number line (0 to 100) and a zero-centered mixed number line (-100 to 100). We found evidence of a functional dissociation between aIPS and pIPS: Activity in aIPS was associated with a landmark distance effect not modulated by spatial position, whereas activity in pIPS revealed a contralateral preference effect.

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.

Place-value understanding in number line estimation predicts future arithmetic performance

In multi-digit numbers, the value of each digit is determined by its position within the digit string. Children's understanding of this place-value structure constitutes a building block for later arithmetic skills. We investigated whether a number line estimation task can provide an assessment of place-value understanding in first grade. We hypothesized that estimating the position of two-digit numbers requires place-value understanding. Therefore, we fitted a linear function to children's estimates of two-digit numbers and considered the resulting slope as a measure of children's place-value understanding. We observed a significant correlation between this slope and children's performance in a transcoding task known to require place-value understanding. Additionally, the slope for two-digit numbers assessed at the beginning of grade 1 predicted children's arithmetic performance at the end of grade 1. These results indicate that the number line estimation task may indeed constitute a valid measure for first-graders' place-value understanding. Moreover, these findings are hard to reconcile with the view that number line estimation directly assesses a spatial representation of numbers. Instead, our results suggest that numerical processes involved in performing the task (such as place-value understanding) may drive the association between number line estimation and arithmetic performance.