Overlapping and distinct brain regions involved in estimating the spatial position of numerical and non-numerical magnitudes: An fMRI study

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Young children intuitively organize numbers on straight, horizontal lines from left to right before the onset of formal instruction

The number line estimation task is frequently used to measure children's numerical magnitude understanding. It is unclear whether the resulting straight, horizontal, left-to-right-oriented estimate patterns indicate task constraints or children's intuitive number-space mapping. Three- to six-year-old children (N = 72, Mage = 4.89, 56% girls, 94% German citizenship) were asked to explain the meaning of numbers to a teddy by laying out a rope and attaching cards showing non-symbolic numerosities (dots) to it. Most children intuitively created straight, horizontal, and left-to-right-oriented representations. Characteristics of the line correlated with age, mathematical competencies, and home numeracy. This demonstrates the usefulness of the number line estimation task for assessing how children intuitively map numbers onto space.

Inter-individual, hemispheric and sex variability of brain activations during numerosity processing

Numerosity perception is a fundamental and innate cognitive function shared by both humans and many animal species. Previous research has primarily focused on exploring the spatial and functional consistency of neural activations that were associated with the processing of numerosity information. However, the inter-individual variability of brain activations of numerosity perception remains unclear. In the present study, with a large-sample functional magnetic resonance imaging (fMRI) dataset (n = 460), we aimed to localize the functional regions related to numerosity perceptions and explore the inter-individual, hemispheric, and sex differences within these brain regions. Fifteen subject-specific activated regions, including the anterior intraparietal sulcus (aIPS), posterior intraparietal sulcus (pIPS), insula, inferior frontal gyrus (IFG), inferior temporal gyrus (ITG), premotor area (PM), middle occipital gyrus (MOG) and anterior cingulate cortex (ACC), were delineated in each individual and then used to create a functional probabilistic atlas to quantify individual variability in brain activations of numerosity processing. Though the activation percentages of most regions were higher than 60%, the intersections of most regions across individuals were considerably lower, falling below 50%, indicating substantial variations in brain activations related to numerosity processing among individuals. Furthermore, significant hemispheric and sex differences in activation location, extent, and magnitude were also found in these regions. Most activated regions in the right hemisphere had larger activation volumes and activation magnitudes, and were located more lateral and anterior than their counterparts in the left hemisphere. In addition, in most of these regions, males displayed stronger activations than females. Our findings demonstrate large inter-individual, hemispheric, and sex differences in brain activations related to numerosity processing, and our probabilistic atlas can serve as a robust functional and spatial reference for mapping the numerosity-related neural networks.

Engaging learners with games-Insights from functional near-infrared spectroscopy

The use of game elements in learning tasks is thought to facilitate emotional and behavioral responses as well as learner engagement. So far, however, little is known about the underlying neural mechanisms of game-based learning. In the current study, we added game elements to a number line estimation task assessing fraction understanding and compared brain activation patterns to a non-game-based task version. Forty-one participants performed both task versions in counterbalanced order while frontal brain activation patterns were assessed using near-infrared spectroscopy (within-subject, cross-sectional study design). Additionally, heart rate, subjective user experience, and task performance were recorded. Task performance, mood, flow experience, as well as heart rate did not differ between task versions. However, the game-based task-version was rated as more attractive, stimulating and novel compared to the non-game-based task version. Additionally, completing the game-based task version was associated with stronger activation in frontal brain areas generally involved in emotional and reward processing as well as attentional processes. These results provide new neurofunctional evidence substantiating that game elements in learning tasks seem to facilitate learning through emotional and cognitive engagement.

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 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.

Investigating the association between variability in sulcal pattern and academic achievement

Investigating how the brain may constrain academic achievement is not only relevant to understanding brain structure but also to providing insight into the origins of individual differences in these academic abilities. In this pre-registered study, we investigated whether the variability of sulcal patterns, a qualitative feature of the brain determined in-utero and not affected by brain maturation and learning, accounted for individual differences in reading and mathematics. Participants were 97 typically developing 10-year-olds. We examined (a) the association between the sulcal pattern of the IntraParietal Sulcus (IPS) and mathematical ability; (b) the association between the sulcal pattern of the Occipito Temporal Sulcus (OTS) and reading ability; and (c) the overlap and specificity of sulcal morphology of IPS and OTS and their associations with mathematics and reading. Despite its large sample, the present study was unable to replicate a previously observed relationship between the IPS sulcal pattern and mathematical ability and a previously observed association between the left posterior OTS sulcal pattern and reading. We found a weak association between right IPS sulcal morphology and symbolic number abilities and a weak association between left posterior OTS and reading. However, both these associations were the opposite of previous reports. We found no evidence for a possible overlap or specificity in the effect of sulcal morphology on mathematics and reading. Possible explanations for this weak association between sulcal morphology and academic achievement and suggestions for future research are discussed.

Bilateral intraparietal activation for number tasks in studies using an adaptation paradigm: A meta-analysis

Mathematical processing is important for professional successes. The Adaptation Paradigm has been widely used to study the brain underpinnings of mathematical processing. In this study, we aim at shedding light on an important component of mathematical processing, namely numerical cognition. To do so, we performed a meta-analysis using the Activation Likelihood Estimation method on studies that have employed the Adaptation Paradigm for examining numerical cognition. We found a bilateral Intraparietal Sulcus (IPS) activation in studies using both symbolic and non-symbolic stimuli formats. We also found a right lateralized brain activation for the non-symbolic condition and a left lateralized brain activation for the symbolic condition. These results imply that the Adaptation Paradigm likely targets numeric magnitude processing and confirms the potency of this paradigm to activate the Intraparietal Sulcus.

Order processing of number symbols is influenced by direction, but not format

This study probed the cognitive mechanisms that underlie order processing for number symbols, specifically the extent to which the direction and format in which number symbols are presented influence the processing of numerical order, as well as the extent to which the relationship between numerical order processing and mathematical achievement is specific to Arabic numerals or generalisable to other notational formats. Seventy adults who were bilingual in English and Chinese completed a Numerical Ordinality Task, using number sequences of various directional conditions (i.e., ascending, descending, mixed) and notational formats (i.e., Arabic numerals, English number words, and Chinese number words). Order processing was found to occur for ascending and descending number sequences (i.e., ordered but not non-ordered trials), with the overall pattern of data supporting the theoretical perspective that the strength and closeness of associations between items in the number sequence could underlie numerical order processing. However, order processing was found to be independent of the notational format in which the numerical stimuli were presented, suggesting that the psychological representations and processes associated with numerical order are abstract across different formats of number symbols. In addition, a relationship between the processing speed for numerical order judgements and mathematical achievement was observed for Arabic numerals and Chinese number words, and to a weaker extent, English number words. Together, our findings have started to uncover the cognitive mechanisms that could underlie order processing for different formats of number symbols, and raise new questions about the generalisability of these findings to other notational formats.

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.

Game-based learning environments affect frontal brain activity

Inclusion of game elements in learning environments to increase motivation and learning outcome is becoming increasingly popular. However, underlying mechanisms of game-based learning have not been studied sufficiently yet. In the present study, we investigated effects of game-based learning environments on a neurofunctional level. In particular, 59 healthy adults completed a game-based version (including game elements such as a narrative and virtual incentives) as well as a non-game-based version of a number line estimation task, to improve fractional knowledge, while their brain activity was monitored using near-infrared spectroscopy. Behavioral performance was comparable across the two versions, although there was a tendency that less errors were made in the game-based version. However, subjective user experience differed significantly between versions. Participants rated the game-based version as more attractive, novel, and stimulating but less efficient than the non-game-based version. Additionally, positive affect was reported to be higher while engaging in the game-based as compared to the non-game-based task version. Corroborating these user reports, we identified increased brain activation in areas associated with emotion and reward processing while playing the game-based version, which might be driven by rewarding elements of the game-based version. Moreover, frontal areas associated with attention were also more activated in the game-based version of the task. Hence, we observed converging evidence on a user experience and neurofunctional level indicating that the game-based version was more rewarding as well as emotionally and attentionally engaging. These results underscore the potential of game-based learning environments to promote more efficient learning by means of attention and reward up-tuning.

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.

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.

Training nonsymbolic proportional reasoning in children and its effects on their symbolic math abilities

Our understanding of proportions can be both symbolic, as when doing calculations in school mathematics, or intuitive, as when folding a bed sheet in half. While an understanding of symbolic proportions is crucial for school mathematics, the cognitive foundations of this ability remain unclear. Here we implemented a computerized training game to test a causal link from intuitive (nonsymbolic) to symbolic proportional reasoning and other math abilities in 4th grade children. An experimental group was trained in nonsymbolic proportional reasoning (PR) with continuous extents, and an active control group was trained on a remarkably similar nonsymbolic magnitude comparison. We found that the experimental group improved at nonsymbolic PR across training sessions, showed near transfer to a paper-and-pencil nonsymbolic PR test, transfer to symbolic proportions, and far transfer to geometry. The active control group showed only a predicted far transfer to geometry. In a second experiment, these results were replicated with an independent cohort of children. Overall this study extends previous correlational evidence, suggesting a functional link between nonsymbolic PR on one hand and symbolic PR and geometry on the other.

Logical negation mapped onto the brain

High-level cognitive capacities that serve communication, reasoning, and calculation are essential for finding our way in the world. But whether and to what extent these complex behaviors share the same neuronal substrate are still unresolved questions. The present study separated the aspects of logic from language and numerosity-mental faculties whose distinctness has been debated for centuries-and identified a new cytoarchitectonic area as correlate for an operation involving logical negation. A novel experimental paradigm that was implemented here in an RT/fMRI study showed a single cluster of activity that pertains to logical negation. It was distinct from clusters that were activated by numerical comparison and from the traditional language regions. The localization of this cluster was described by a newly identified cytoarchitectonic area in the left anterior insula, ventro-medial to Broca's region. We provide evidence for the congruence between the histologically and functionally defined regions on multiple measures. Its position in the left anterior insula suggests that it functions as a mediator between language and reasoning areas.

Evidence for a visuospatial bias in decimal number comparison in adolescents and in adults

There is a close relation between spatial and numerical representations which can lead to interference as in Piaget's number conservation task or in the numerical Stroop task. Using a negative priming (NP) paradigm, we investigated whether the interference between spatial and numerical processing extends to more complex arithmetic processing by asking 12 year olds and adults to compare the magnitude of decimal numbers (i.e., the prime) and, subsequently, the length of two lines or the luminance of two circles (i.e., the probe). We found NP effects when participants compare length but not luminance. Our finding suggests that decimal comparison is impacted by a visuospatial bias due to the interference between the magnitude of the numbers to be compared and their physical length. We discuss the educational implications of these findings.

Functionally distinct contributions of parietal cortex to a numerical landmark task: An fMRI study

This study aimed at establishing the neural basis of magnitude processing of multiple numbers from working memory. We designed a numerical landmark task and embedded it in a fragmented trial event-related fMRI design, allowing to separate encoding from decision processing. An attentional localiser task not involving numbers allowed further functional specification. The results show that in a numerical landmark task the right anterior intraparietal sulcus is involved in number encoding while more posterior parietal regions, bilateral superior parietal lobule and right inferior parietal lobule, provide domain-general support in the form of constructing a working memory representation or orienting spatial attention within that mental representation during number comparison. The results are in line with earlier studies reporting a functional distinction between anterior and posterior parietal contributions to number processing and further specify their role at a functional level.

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.

Adolescents with Developmental Dyscalculia Do Not Have a Generalized Magnitude Deficit - Processing of Discrete and Continuous Magnitudes

The link between number and space has been discussed in the literature for some time, resulting in the theory that number, space and time might be part of a generalized magnitude system. To date, several behavioral and neuroimaging findings support the notion of a generalized magnitude system, although contradictory results showing a partial overlap or separate magnitude systems are also found. The possible existence of a generalized magnitude processing area leads to the question how individuals with developmental dyscalculia (DD), known for deficits in numerical-arithmetical abilities, process magnitudes. By means of neuropsychological tests and functional magnetic resonance imaging (fMRI) we aimed to examine the relationship between number and space in typical and atypical development. Participants were 16 adolescents with DD (14.1 years) and 14 typically developing (TD) peers (13.8 years). In the fMRI paradigm participants had to perform discrete (arrays of dots) and continuous magnitude (angles) comparisons as well as a mental rotation task. In the neuropsychological tests, adolescents with dyscalculia performed significantly worse in numerical and complex visuo-spatial tasks. However, they showed similar results to TD peers when making discrete and continuous magnitude decisions during the neuropsychological tests and the fMRI paradigm. A conjunction analysis of the fMRI data revealed commonly activated higher order visual (inferior and middle occipital gyrus) and parietal (inferior and superior parietal lobe) magnitude areas for the discrete and continuous magnitude tasks. Moreover, no differences were found when contrasting both magnitude processing conditions, favoring the possibility of a generalized magnitude system. Group comparisons further revealed that dyscalculic subjects showed increased activation in domain general regions, whilst TD peers activate domain specific areas to a greater extent. In conclusion, our results point to the existence of a generalized magnitude system in the occipito-parietal stream in typical development. The detailed investigation of spatial and numerical magnitude abilities in DD reveals that the deficits in number processing and arithmetic cannot be explained with a general magnitude deficiency. Our results further indicate that multiple neuro-cognitive components might contribute to the explanation of DD.

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.

Dyscalculia and the Calculating Brain

Dyscalculia, like dyslexia, affects some 5% of school-age children but has received much less investigative attention. In two thirds of affected children, dyscalculia is associated with another developmental disorder like dyslexia, attention-deficit disorder, anxiety disorder, visual and spatial disorder, or cultural deprivation. Infants, primates, some birds, and other animals are born with the innate ability, called subitizing, to tell at a glance whether small sets of scattered dots or other items differ by one or more item. This nonverbal approximate number system extends mostly to single digit sets as visual discrimination drops logarithmically to "many" with increasing numerosity (size effect) and crowding (distance effect). Preschoolers need several years and specific teaching to learn verbal names and visual symbols for numbers and school agers to understand their cardinality and ordinality and the invariance of their sequence (arithmetic number line) that enables calculation. This arithmetic linear line differs drastically from the nonlinear approximate number system mental number line that parallels the individual number-tuned neurons in the intraparietal sulcus in monkeys and overlying scalp distribution of discrete functional magnetic resonance imaging activations by number tasks in man. Calculation is a complex skill that activates both visual and spatial and visual and verbal networks. It is less strongly left lateralized than language, with approximate number system activation somewhat more right sided and exact number and arithmetic activation more left sided. Maturation and increasing number skill decrease associated widespread non-numerical brain activations that persist in some individuals with dyscalculia, which has no single, universal neurological cause or underlying mechanism in all affected individuals.

Brain Activation of Identity Switching in Multiple Identity Tracking Task

When different objects switch identities in the multiple identity tracking (MIT) task, viewers need to rebind objects’ identity and location, which requires attention. This rebinding helps people identify the regions targets are in (where they need to focus their attention) and inhibit unimportant regions (where distractors are). This study investigated the processing of attentional tracking after identity switching in an adapted MIT task. This experiment used three identity-switching conditions: a target-switching condition (where the target objects switched identities), a distractor-switching condition (where the distractor objects switched identities), and a no-switching condition. Compared to the distractor-switching condition, the target-switching condition elicited greater activation in the frontal eye fields (FEF), intraparietal sulcus (IPS), and visual cortex. Compared to the no-switching condition, the target-switching condition elicited greater activation in the FEF, inferior frontal gyrus (pars orbitalis) (IFG-Orb), IPS, visual cortex, middle temporal lobule, and anterior cingulate cortex. Finally, the distractor-switching condition showed greater activation in the IFG-Orb compared to the no-switching condition. These results suggest that, in the target-switching condition, the FEF and IPS (the dorsal attention network) might be involved in goal-driven attention to targets during attentional tracking. In addition, in the distractor-switching condition, the activation of the IFG-Orb may indicate salient change that pulls attention away automatically.

Developmental specialization of the left parietal cortex for the semantic representation of Arabic numerals: an fMR-adaptation study

The way the human brain constructs representations of numerical symbols is poorly understood. While increasing evidence from neuroimaging studies has indicated that the intraparietal sulcus (IPS) becomes increasingly specialized for symbolic numerical magnitude representation over developmental time, the extent to which these changes are associated with age-related differences in symbolic numerical magnitude representation or with developmental changes in non-numerical processes, such as response selection, remains to be uncovered. To address these outstanding questions we investigated developmental changes in the cortical representation of symbolic numerical magnitude in 6- to 14-year-old children using a passive functional magnetic resonance imaging adaptation design, thereby mitigating the influence of response selection. A single-digit Arabic numeral was repeatedly presented on a computer screen and interspersed with the presentation of novel digits deviating as a function of numerical ratio (smaller/larger number). Results demonstrated a correlation between age and numerical ratio in the left IPS, suggesting an age-related increase in the extent to which numerical symbols are represented in the left IPS. Brain activation of the right IPS was modulated by numerical ratio but did not correlate with age, indicating hemispheric differences in IPS engagement during the development of symbolic numerical representation.

How number line estimation skills relate to neural activations in single digit subtraction problems

The Number Line (NL) task requires judging the relative numerical magnitude of a number and estimating its value spatially on a continuous line. Children's skill on this task has been shown to correlate with and predict future mathematical competence. Neurofunctionally, this task has been shown to rely on brain regions involved in numerical processing. However, there is no direct evidence that performance on the NL task is related to brain areas recruited during arithmetical processing and that these areas are domain-specific to numerical processing. In this study, we test whether 8- to 14-year-old's behavioral performance on the NL task is related to fMRI activation during small and large single-digit subtraction problems. Domain-specific areas for numerical processing were independently localized through a numerosity judgment task. Results show a direct relation between NL estimation performance and the amount of the activation in key areas for arithmetical processing. Better NL estimators showed a larger problem size effect than poorer NL estimators in numerical magnitude (i.e., intraparietal sulcus) and visuospatial areas (i.e., posterior superior parietal lobules), marked by less activation for small problems. In addition, the direction of the activation with problem size within the IPS was associated with differences in accuracies for small subtraction problems. This study is the first to show that performance in the NL task, i.e. estimating the spatial position of a number on an interval, correlates with brain activity observed during single-digit subtraction problem in regions thought to be involved in numerical magnitude and spatial processes.

Comparing apples and pears in studies on magnitude estimations

The present article is concerned with studies on magnitude estimations that strived to uncover the underlying mental representation(s) of magnitudes. We point out a number of methodological differences and shortcomings that make it difficult drawing general conclusions. To solve this problem, we propose a taxonomy by which those studies could be classified, taking into account central methodological aspects of magnitude estimation tasks. Finally, we suggest perspectives for future research on magnitude estimations, which might abandon the hunt for the mathematical model that explains estimations best and turn, instead, to investigate the underlying principles of estimations (e.g., strategies) and ways of their improvement.