Specialization in the human brain: the case of numbers

A New Graphical Method for Displaying Two-Dimensional Echocardiography Results in Dogs: Comprehensive Analysis of Results of Diagnostic Imaging Organized in a BOX (CARDIOBOX)

INTRODUCTION AND OBJECTIVE

Rapid and efficient interpretation of echocardiographic findings is critical in clinical decision-making. This study aimed to design and validate a new graphical method, called CARDIOBOX, to represent echocardiographic findings in dogs.

METHODS

A prospective, observational, exploratory cohort study was conducted over three years. The design of CARDIOBOX was based on baseline values obtained from 802 healthy dogs and 2165 ill dogs. Using these data, a graph consisting of nine boxes was built to show the intervals of the different echocardiographic measurements. Validation of the method was performed by a survey of 55 veterinarians, who compared the use of CARDIOBOX with the use of numerical tables.

RESULTS

CARDIOBOX demonstrated significantly faster interpretability (p < 0.05) without reducing its effectiveness. In addition, the staff surveyed considered it easy to use and interpret.

CONCLUSIONS

The introduction of CARDIOBOX emerges as a resource that facilitates rapid and efficient interpretation of echocardiographic findings in dogs. This new graphical method is presented as a valuable tool for veterinary professionals in clinical decision-making in the field of veterinary cardiology.

Brain laterality of numbers and calculation: Complex networks and their development

This chapter reviews notions about the lateralization of numbers and calculation in the brain, including its developmental pattern. Such notions have changed dramatically in recent decades. What was once considered a function almost exclusively located in the left hemisphere has been found to be sustained by complex brain networks encompassing both hemispheres. Depending on the specific task, however, each hemisphere has its own role. Much of this progress was determined by the convergency of investigations conducted with different methods. Contrary to traditional wisdom, the right hemisphere is not involved in arithmetic just as far as generic spatial aspects are concerned. Very specific arithmetic functions like remembering the spatial templates for complex operations, or processing of zero in complex numbers, are indeed sustained in specific right-sided areas. The system used in the typical adult appears to be the result of a complex pattern of development. The numerical brain clearly evolved from less mature to more advanced brain networks because of growth and education. Children seem to be equipped with the ability to represent the number nonverbally from a very early age. The bilateral processing of number-related tasks is however a late acquisition.

The brain lateralization and development of math functions: progress since Sperry, 1974

In 1974, Roger Sperry, based on his seminal studies on the split-brain condition, concluded that math was almost exclusively sustained by the language dominant left hemisphere. The right hemisphere could perform additions up to sums less than 20, the only exception to a complete left hemisphere dominance. Studies on lateralized focal lesions came to a similar conclusion, except for written complex calculation, where spatial abilities are needed to display digits in the right location according to the specific requirements of calculation procedures. Fifty years later, the contribution of new theoretical and instrumental tools lead to a much more complex picture, whereby, while left hemisphere dominance for math in the right-handed is confirmed for most functions, several math related tasks seem to be carried out in the right hemisphere. The developmental trajectory in the lateralization of math functions has also been clarified. This corpus of knowledge is reviewed here. The right hemisphere does not simply offer its support when calculation requires generic space processing, but its role can be very specific. For example, the right parietal lobe seems to store the operation-specific spatial layout required for complex arithmetical procedures and areas like the right insula are necessary in parsing complex numbers containing zero. Evidence is found for a complex orchestration between the two hemispheres even for simple tasks: each hemisphere has its specific role, concurring to the correct result. As for development, data point to right dominance for basic numerical processes. The picture that emerges at school age is a bilateral pattern with a significantly greater involvement of the right-hemisphere, particularly in non-symbolic tasks. The intraparietal sulcus shows a left hemisphere preponderance in response to symbolic stimuli at this age.

Is Nonsymbolic Arithmetic Truly "Arithmetic"? Examining the Computational Capacity of the Approximate Number System in Young Children

Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like structure, like symbolic arithmetic. Children (n = 74 4- to -8-year-olds in Experiment 1; n = 52 7- to 8-year-olds in Experiment 2) first solved two nonsymbolic arithmetic problems. We then showed children two unequal sets of objects, and asked children which of the two derived solutions should be added to the smaller of the two sets to make them "about the same." We hypothesized that, if nonsymbolic arithmetic follows similar function rules to symbolic arithmetic, then children should be able to use the solutions of nonsymbolic computations as inputs into another nonsymbolic problem. Contrary to this hypothesis, we found that children were unable to reliably do so, suggesting that these solutions may not operate as independent representations that can be used inputs into other nonsymbolic computations. These results suggest that nonsymbolic and symbolic arithmetic computations are algorithmically distinct, which may limit the extent to which children can leverage nonsymbolic arithmetic intuitions to acquire formal mathematics knowledge.

The Effect of Verbal Task Instruction on Spatial-Numerical Associations of Response Codes Effect Coding of Spatial-Numerical Associations: Evidence From Event-Related Potential

The spatial-numerical associations of response codes (SNARC) effect reveals that individuals can represent numbers spatially. In this study, event-related potential (ERP) technology was used to probe the effect of verbal-spatial task instructions on spatial-numerical association coding by using digit parity and magnitude judgment tasks, with the numbers 1–9 (except 5) and Chinese word labels (“left” and “right”) as experimental materials. The behavioral results of Experiment 1 showed that the SNARC effect was mainly based on verbal-spatial coding and appeared when the stimulus onset asynchrony (SOA) between the presentation of the verbal labels and the target digit was 0 ms. ERP results did not reveal any significant SNARC-related effects in either the N1 or P3 components. The behavioral results of Experiment 2 again showed that the SNARC effect was dominated by verbal-spatial coding. ERP results showed that significant effects related to verbal-spatial coding were found in both the early positive deflection of the stimulus-locked lateralized readiness potential (S-LRP) and the latency of the response-locked LRP (R-LRP). Hence, in this study, the nature of the spatial coding of the digit magnitudes was influenced by the processing of the word labels and affected both the response selection and response preparation stages.

Automatic integration of numerical formats examined with frequency-tagged EEG

How humans integrate and abstract numerical information across different formats is one of the most debated questions in human cognition. We addressed the neuronal signatures of the numerical integration using an EEG technique tagged at the frequency of visual stimulation. In an oddball design, participants were stimulated with standard sequences of numbers (< 5) depicted in single (digits, dots, number words) or mixed notation (dots-digits, number words-dots, digits-number words), presented at 10 Hz. Periodically, a deviant stimulus (> 5) was inserted at 1.25 Hz. We observed significant oddball amplitudes for all single notations, showing for the first time using this EEG technique, that the magnitude information is spontaneously and unintentionally abstracted, irrespectively of the numerical format. Significant amplitudes were also observed for digits-number words and number words-dots, but not for digits-dots, suggesting an automatic integration across some numerical formats. These results imply that direct and indirect neuro-cognitive links exist across the different numerical formats.

Symbols Are Special: An fMRI Adaptation Study of Symbolic, Nonsymbolic, and Non-Numerical Magnitude Processing in the Human Brain

How are different formats of magnitudes represented in the human brain? We used functional magnetic resonance imaging adaptation to isolate representations of symbols, quantities, and physical size in 45 adults. Results indicate that the neural correlates supporting the passive processing of number symbols are largely dissociable from those supporting quantities and physical size, anatomically and representationally. Anatomically, passive processing of quantities and size correlate with activation in the right intraparietal sulcus, whereas symbolic number processing, compared with quantity processing, correlates with activation in the left inferior parietal lobule. Representationally, neural patterns of activation supporting symbols are dissimilar from neural activation patterns supporting quantity and size in the bilateral parietal lobes. These findings challenge the longstanding notion that the culturally acquired ability to conceptualize symbolic numbers is represented using entirely the same brain systems that support the evolutionarily ancient system used to process quantities. Moreover, these data reveal that regions that support numerical magnitude processing are also important for the processing of non-numerical magnitudes. This discovery compels future investigations of the neural consequences of acquiring knowledge of symbolic numbers.

Emerging neurodevelopmental perspectives on mathematical learning

Strong foundational skills in mathematical problem solving, acquired in early childhood, are critical not only for success in the science, technology, engineering, and mathematical (STEM) fields but also for quantitative reasoning in everyday life. The acquisition of mathematical skills relies on protracted interactive specialization of functional brain networks across development. Using a systems neuroscience approach, this review synthesizes emerging perspectives on neurodevelopmental pathways of mathematical learning, highlighting the functional brain architecture that supports these processes and sources of heterogeneity in mathematical skill acquisition. We identify the core neural building blocks of numerical cognition, anchored in the posterior parietal and ventral temporal-occipital cortices, and describe how memory and cognitive control systems, anchored in the medial temporal lobe and prefrontal cortex, help scaffold mathematical skill development. We highlight how interactive specialization of functional circuits influences mathematical learning across different stages of development. Functional and structural brain integrity and plasticity associated with math learning can be examined using an individual differences approach to better understand sources of heterogeneity in learning, including cognitive, affective, motivational, and sociocultural factors. Our review emphasizes the dynamic role of neurodevelopmental processes in mathematical learning and cognitive development more generally.

Tactile to visual number priming in the left intraparietal cortex of sighted Braille readers

Numbers can be presented in different notations and sensory modalities. It is currently debated to what extent these formats overlap onto a single representation. We asked whether such an overlap exists between symbolic numbers represented in two sensory modalities: Arabic digits and Braille numbers. A unique group of sighted Braille readers underwent extensive Braille reading training and was tested in an fMRI repetition-suppression paradigm with tactile Braille digit primes and visual Arabic digit targets. Our results reveal cross-modal priming: compared to repetition of two different quantities (e.g., Braille “5” and Arabic “2”), repetition of the same quantity presented in two modalities (e.g., Braille “5” and Arabic “5”) led to a reduction of activation in several sub-regions of the Intraparietal Sulcus (IPS), a key cortical region for magnitude processing. Thus, in sighted Braille readers, the representations of numbers read by sight and by touch overlap to a degree sufficient to cause repetition suppression. This effect was modulated by the numerical prime-probe distance. Altogether this indicates that the left parietal cortex hosts neural assemblies that are sensitive to numerical information from different notations (number words or Arabic digits) and modalities (tactile and visual).

Shared Numerosity Representations Across Formats and Tasks Revealed with 7 Tesla fMRI: Decoding, Generalization, and Individual Differences in Behavior

Debate continues on whether encoding of symbolic number is grounded in nonsymbolic numerical magnitudes. Nevertheless, fluency of perceiving both number formats, and translating between them, predicts math skills across the life span. Therefore, this study asked if numbers share cortical activation patterns across formats and tasks, and whether neural response to number predicts math-related behaviors. We analyzed patterns of neural activation using 7 Tesla functional magnetic resonance imaging in a sample of 39 healthy adults. Discrimination was successful between numerosities 2, 4, 6, and 8 dots and generalized to activation patterns of the same numerosities represented as Arabic digits in the bilateral parietal lobes and left inferior frontal gyrus (IFG) (and vice versa). This indicates that numerosity-specific neural resources are shared between formats. Generalization was also successful across tasks where participants either identified or compared numerosities in bilateral parietal lobes and IFG. Individual differences in decoding did not relate to performance on a number comparison task completed outside of the scanner, but generalization between formats and across tasks negatively related to math achievement in the parietal lobes. Together, these findings suggest that individual differences in representational specificity within format and task contexts relate to mathematical expertise.

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

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

Numerical cognition: A meta-analysis of neuroimaging, transcranial magnetic stimulation and brain-damaged patients studies

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We review neuroimaging, TMS, and patients studies on numerical cognition.

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We focused on the predictions derived from the Triple Code Model (TCM).

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Our findings generally agree with TCM predictions.

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Our results open avenues for the study of the neural bases of numerical cognition.

This article offers the first comprehensive review examining the neurocognitive bases of numerical cognition from neuroimaging, Transcranial Magnetic Stimulation (TMS) and brain-damaged patients studies. We focused on the predictions derived from the Triple Code Model (TCM), particularly the assumption that the representation of numerical quantities rests on a single format-independent representation (i.e., the analogical code) involving both intraparietal sulci (IPS). To do so, we conducted a meta-analysis based on 28 neuroimaging, 12 TMS and 12 brain-damaged patients studies, including arithmetic and magnitude tasks in symbolic and non-symbolic formats. Our findings generally agree with the TCM predictions indicating that both IPS are engaged in all tasks. Nonetheless, the results of brain-damaged patients studies conflicted with neuroimaging and TMS studies, suggesting a right hemisphere lateralization for non-symbolic formats. Our findings also led us to discuss the involvement of brain regions other than IPS in the processing of the analogical code as well as the neural substrate of other codes underlying numerical cognition (i.e., the auditory-verbal code).

The brain-structural correlates of mathematical expertise

Studies in several domains of expertise have established that experience-dependent plasticity brings about both functional and anatomical changes. However, little is known about how such changes come to shape the brain in the case of expertise acquired by professional mathematicians. Here, we aimed to identify cognitive and brain-structural (grey and white matter) characteristics of mathematicians as compared to non-mathematicians. Mathematicians and non-mathematician academics from the University of Oxford underwent structural and diffusion MRI scans, and were tested on a cognitive battery assessing working memory, attention, IQ, numerical and social skills. At the behavioural level, mathematical expertise was associated with better performance in domain-general and domain-specific dimensions. At the grey matter level, in a whole-brain analysis, behavioural performance correlated with grey matter density in left superior frontal gyrus - positively for mathematicians but negatively for non-mathematicians; in a region of interest analysis, we found in mathematicians higher grey matter density in the right superior parietal lobule, but lower grey matter density in the right intraparietal sulcus and in the left inferior frontal gyrus. In terms of white matter, there were no significant group differences in fractional anisotropy or mean diffusivity. These results reveal new insights into the relationship between mathematical expertise and grey matter metrics in brain regions previously implicated in numerical cognition, as well as in regions that have so far received less attention in this field. Further studies, based on longitudinal designs and cognitive training, could examine the conjecture that such cross-sectional findings arise from a bidirectional link between experience and structural brain changes that is itself subject to change across the lifespan.

Shared and distinct neural circuitry for nonsymbolic and symbolic double-digit addition

Symbolic arithmetic is a complex, uniquely human ability that is acquired through direct instruction. In contrast, the capacity to mentally add and subtract nonsymbolic quantities such as dot arrays emerges without instruction and can be seen in human infants and nonhuman animals. One possibility is that the mental manipulation of nonsymbolic arrays provides a critical scaffold for developing symbolic arithmetic abilities. To explore this hypothesis, we examined whether there is a shared neural basis for nonsymbolic and symbolic double-digit addition. In parallel, we asked whether there are brain regions that are associated with nonsymbolic and symbolic addition independently. First, relative to visually matched control tasks, we found that both nonsymbolic and symbolic addition elicited greater neural signal in the bilateral intraparietal sulcus (IPS), bilateral inferior temporal gyrus, and the right superior parietal lobule. Subsequent representational similarity analyses revealed that the neural similarity between nonsymbolic and symbolic addition was stronger relative to the similarity between each addition condition and its visually matched control task, but only in the bilateral IPS. These findings suggest that the IPS is involved in arithmetic calculation independent of stimulus format.

Mechanisms of interactive specialization and emergence of functional brain circuits supporting cognitive development in children

Cognitive development is thought to depend on the refinement and specialization of functional circuits over time, yet little is known about how this process unfolds over the course of childhood. Here we investigated growth trajectories of functional brain circuits and tested an interactive specialization model of neurocognitive development which posits that the refinement of task-related functional networks is driven by a shared history of co-activation between cortical regions. We tested this model in a longitudinal cohort of 30 children with behavioral and task-related functional brain imaging data at multiple time points spanning childhood and adolescence, focusing on the maturation of parietal circuits associated with numerical problem solving and learning. Hierarchical linear modeling revealed selective strengthening as well as weakening of functional brain circuits. Connectivity between parietal and prefrontal cortex decreased over time, while connectivity within posterior brain regions, including intra-hemispheric and inter-hemispheric parietal connectivity, as well as parietal connectivity with ventral temporal occipital cortex regions implicated in quantity manipulation and numerical symbol recognition, increased over time. Our study provides insights into the longitudinal maturation of functional circuits in the human brain and the mechanisms by which interactive specialization shapes children's cognitive development and learning.

Effects of adaptation on numerosity decoding in the human brain

Psychophysical studies have shown that numerosity is a sensory attribute susceptible to adaptation. Neuroimaging studies have reported that, at least for relatively low numbers, numerosity can be accurately discriminated in the intra-parietal sulcus. Here we developed a novel rapid adaptation paradigm where adapting and test stimuli are separated by pauses sufficient to dissociate their BOLD activity. We used multivariate pattern recognition to classify brain activity evoked by non-symbolic numbers over a wide range (20-80), both before and after psychophysical adaptation to the highest numerosity. Adaptation caused underestimation of all lower numerosities, and decreased slightly the average BOLD responses in V1 and IPS. Using support vector machine, we showed that the BOLD response of IPS, but not in V1, classified numerosity well, both when tested before and after adaptation. However, there was no transfer from training pre-adaptation responses to testing post-adaptation, and vice versa, indicating that adaptation changes the neuronal representation of the numerosity. Interestingly, decoding was more accurate after adaptation, and the amount of improvement correlated with the amount of perceptual underestimation of numerosity across subjects. These results suggest that numerosity adaptation acts directly on IPS, rather than indirectly via other low-level stimulus parameters analysis, and that adaptation improves the capacity to discriminate numerosity.

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.

Using cognitive training studies to unravel the mechanisms by which the approximate number system supports symbolic math ability

A picture is emerging that preverbal nonsymbolic numerical representations derived from the approximate number system (ANS) play an important role in mathematical development and sustained mathematical thinking. Functional imaging studies are revealing developmental trends in how the brain represents number. We propose that combining behavioral and neuroimaging techniques with cognitive training approaches will help identify the fundamental relationship between the ANS and symbolic mathematics. Understanding this relationship should ultimately benefit educators by providing ways to harness the ANS and hopefully improve math readiness in young children.

Evidence for distinct magnitude systems for symbolic and non-symbolic number

Cognitive models of magnitude representation are mostly based on the results of studies that use a magnitude comparison task. These studies show similar distance or ratio effects in symbolic (Arabic numerals) and non-symbolic (dot arrays) variants of the comparison task, suggesting a common abstract magnitude representation system for processing both symbolic and non-symbolic numerosities. Recently, however, it has been questioned whether the comparison task really indexes a magnitude representation. Alternatively, it has been hypothesized that there might be different representations of magnitude: an exact representation for symbolic magnitudes and an approximate representation for non-symbolic numerosities. To address the question whether distinct magnitude systems exist, we used an audio-visual matching paradigm in two experiments to explore the relationship between symbolic and non-symbolic magnitude processing. In Experiment 1, participants had to match visually and auditory presented numerical stimuli in different formats (digits, number words, dot arrays, tone sequences). In Experiment 2, they were instructed only to match the stimuli after processing the magnitude first. The data of our experiments show different results for non-symbolic and symbolic number and are difficult to reconcile with the existence of one abstract magnitude representation. Rather, they suggest the existence of two different systems for processing magnitude, i.e., an exact symbolic system next to an approximate non-symbolic system.

Aging and the number sense: preserved basic non-symbolic numerical processing and enhanced basic symbolic processing

Aging often leads to general cognitive decline in domains such as memory and attention. The effect of aging on numerical cognition, particularly on foundational numerical skills known as the number sense, is not well-known. Early research focused on the effect of aging on arithmetic. Recent studies have begun to investigate the impact of healthy aging on basic numerical skills, but focused on non-symbolic quantity discrimination alone. Moreover, contradictory findings have emerged. The current study aimed to further investigate the impact of aging on basic non-symbolic and symbolic numerical skills. A group of 25 younger (18-25) and 25 older adults (60-77) participated in non-symbolic and symbolic numerical comparison tasks. Mathematical and spelling abilities were also measured. Results showed that aging had no effect on foundational non-symbolic numerical skills, as both groups performed similarly [RTs, accuracy and Weber fractions (w)]. All participants showed decreased non-symbolic acuity (accuracy and w) in trials requiring inhibition. However, aging appears to be associated with a greater decline in discrimination speed in such trials. Furthermore, aging seems to have a positive impact on mathematical ability and basic symbolic numerical processing, as older participants attained significantly higher mathematical achievement scores, and performed significantly better on the symbolic comparison task than younger participants. The findings suggest that aging and its lifetime exposure to numbers may lead to better mathematical achievement and stronger basic symbolic numerical skills. Our results further support the observation that basic non-symbolic numerical skills are resilient to aging, but that aging may exacerbate poorer performance on trials requiring inhibitory processes. These findings lend further support to the notion that preserved basic numerical skills in aging may reflect the preservation of an innate, primitive, and embedded number sense.

An alternative to domain-general or domain-specific frameworks for theorizing about human evolution and ontogenesis

This paper maintains that neither a domain-general nor a domain-specific framework is appropriate for furthering our understanding of human evolution and ontogenesis. Rather, as we learn increasingly more about the dynamics of gene-environment interaction and gene expression, theorists should consider a third alternative: a domain-relevant approach, which argues that the infant brain comes equipped with biases that are relevant to, but not initially specific to, processing different kinds of input. The hypothesis developed here is that domain-specific core knowledge/specialized functions do not constitute the start state; rather, functional specialization emerges progressively through neuronal competition over developmental time. Thus, the existence of category-specific deficits in brain-damaged adults cannot be used to bolster claims that category-specific or domain-specific modules underpin early development, because neural specificity in the adult brain is likely to have been the emergent property over time of a developing, self-structuring system in interaction with the environment.

Qualitatively different coding of symbolic and nonsymbolic numbers in the human brain

Are symbolic and nonsymbolic numbers coded differently in the brain? Neuronal data indicate that overlap in numerical tuning curves is a hallmark of the approximate, analogue nature of nonsymbolic number representation. Consequently, patterns of fMRI activity should be more correlated when the representational overlap between two numbers is relatively high. In bilateral intraparietal sulci (IPS), for nonsymbolic numbers, the pattern of voxelwise correlations between pairs of numbers mirrored the amount of overlap in their tuning curves under the assumption of approximate, analogue coding. In contrast, symbolic numbers showed a flat field of modest correlations more consistent with discrete, categorical representation (no systematic overlap between numbers). Directly correlating activity patterns for a given number across formats (e.g., the numeral "6" with six dots) showed no evidence of shared symbolic and nonsymbolic number-specific representations. Overall (univariate) activity in bilateral IPS was well fit by the log of the number being processed for both nonsymbolic and symbolic numbers. IPS activity is thus sensitive to numerosity regardless of format; however, the nature in which symbolic and nonsymbolic numbers are encoded is fundamentally different.

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.

Dissociated neural correlates of quantity processing of quantifiers, numbers, and numerosities

Quantities can be represented using either mathematical language (i.e., numbers) or natural language (i.e., quantifiers). Previous studies have shown that numerical processing elicits greater activation in the brain regions around the intraparietal sulcus (IPS) relative to other semantic processes. However, little research has been conducted to investigate whether the IPS is also critical for the semantic processing of quantifiers in natural language. In this study, 20 adults were scanned with functional magnetic resonance imaging while they performed semantic distance judgment involving six types of materials (i.e., frequency adverbs, quantity pronouns and nouns, animal names, Arabic digits, number words, and dot arrays). Conjunction analyses of brain activation showed that numbers and dot arrays elicited greater activation in the right IPS than did words (i.e., animal names) or quantifiers (i.e., frequency adverbs and quantity pronouns and nouns). Quantifiers elicited more activation in left middle temporal gyrus and inferior frontal gyrus than did numbers and dot arrays. No differences were found between quantifiers and animal names. These findings suggest that, although quantity processing for numbers and dot arrays typically relies on the right IPS region, quantity processing for quantifiers typically relies on brain regions for general semantic processing. Thus, the IPS does not appear to be the only brain region for quantity processing.

A unitary or multiple representations of numerical magnitude? - the case of structure in symbolic and non-symbolic quantities

Currently, there is a controversial debate on whether there is an abstract representation of number magnitude, multiple different ones, or multiple different ones that project onto a unitary representation. The current study aimed at evaluating this issue by means of a magnitude comparison task involving Arabic numbers and structured as well as unstructured non-symbolic patterns of squares. In particular, we were interested whether a specific numerical effect, the unit-decade compatibility effect reflecting decomposed processing of tens and units complying with the place-value structure of the Arabic number system, is affected by input notation. Indeed, a reliable unit-decade compatibility effect was observed in the symbolic-digital notation condition but was absent for unstructured non-symbolic notation. However, for structured non-symbolic notation a - albeit negative - compatibility effect was observed as well. Theses results are hard to reconcile with the notion of an abstract representation of number magnitude. Instead, our data support the existence of multiple representations of numerical magnitude. In addition, the current data indicate that it may not be a question of symbolic vs. non-symbolic notation only but also an issue of the structuring of the input notation. While unstructured non-symbolic quantities seemed to be processed holistically we found evidence suggesting at least partially decomposed processing not only for symbolic Arabic numbers but also for structured non-symbolic quantities.

Impact of high mathematics education on the number sense

In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS.