Origins of mathematical intuitions: the case of arithmetic

Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.

Approximate number and approximate time discrimination each correlate with school math abilities in young children

What is the relationship between our intuitive sense of number (e.g., when estimating how many marbles are in a jar), and our intuitive sense of other quantities, including time (e.g., when estimating how long it has been since we last ate breakfast)? Recent work in cognitive, developmental, comparative psychology, and computational neuroscience has suggested that our representations of approximate number, time, and spatial extent are fundamentally linked and constitute a "generalized magnitude system". But, the shared behavioral and neural signatures between number, time, and space may alternatively be due to similar encoding and decision-making processes, rather than due to shared domain-general representations. In this study, we investigate the relationship between approximate number and time in a large sample of 6-8 year-old children in Uruguay by examining how individual differences in the precision of number and time estimation correlate with school mathematics performance. Over four testing days, each child completed an approximate number discrimination task, an approximate time discrimination task, a digit span task, and a large battery of symbolic math tests. We replicate previous reports showing that symbolic math abilities correlate with approximate number precision and extend those findings by showing that math abilities also correlate with approximate time precision. But, contrary to approximate number and time sharing common representations, we find that each of these dimensions uniquely correlates with formal math: approximate number correlates more strongly with formal math compared to time and continues to correlate with math even when precision in time and individual differences in working memory are controlled for. These results suggest that there are important differences in the mental representations of approximate number and approximate time and further clarify the relationship between quantity representations and mathematics.

The influence of math anxiety on symbolic and non-symbolic magnitude processing

Deficits in basic numerical abilities have been investigated repeatedly as potential risk factors of math anxiety. Previous research suggested that also a deficient approximate number system (ANS), which is discussed as being the foundation for later math abilities, underlies math anxiety. However, these studies examined this hypothesis by investigating ANS acuity using a symbolic number comparison task. Recent evidence questions the view that ANS acuity can be assessed using a symbolic number comparison task. To investigate whether there is an association between math anxiety and ANS acuity, we employed both a symbolic number comparison task and a non-symbolic dot comparison task, which is currently the standard task to assess ANS acuity. We replicated previous findings regarding the association between math anxiety and the symbolic distance effect for response times. High math anxious individuals showed a larger distance effect than less math anxious individuals. However, our results revealed no association between math anxiety and ANS acuity assessed using a non-symbolic dot comparison task. Thus, our results did not provide evidence for the hypothesis that a deficient ANS underlies math anxiety. Therefore, we propose that a deficient ANS does not constitute a risk factor for the development of math anxiety. Moreover, our results suggest that previous interpretations regarding the interaction of math anxiety and the symbolic distance effect have to be updated. We suggest that impaired number comparison processes in high math anxious individuals might account for the results rather than deficient ANS representations. Finally, impaired number comparison processes might constitute a risk factor for the development of math anxiety. Implications for current models regarding the origins of math anxiety are discussed.

Number As a Primary Perceptual Attribute: A Review

Although humans are the only species to possess language-driven abstract mathematical capacities, we share with many other animals a nonverbal capacity for estimating quantities or numerosity. For some time, researchers have clearly differentiated between small numbers of items--less than about four--referred to as the subitizing range, and larger numbers, where counting or estimation is required. In this review, we examine more recent evidence suggesting a further division, between sets of items greater than the subitizing range, but sparse enough to be individuated as single items; and densely packed stimuli, where they crowd each other into what is better considered as a texture. These two different regimes are psychophysically discriminable in that they follow distinct psychophysical laws and show different dependencies on eccentricity and on luminance levels. But provided the elements are not too crowded (less than about two items per square degree in central vision, less in the periphery), there is little evidence that estimation of numerosity depends on mechanisms responsive to texture. The distinction is important, as the ability to discriminate numerosity, but not texture, correlates with formal maths skills.

A Map for Social Navigation in the Human Brain

Deciphering the neural mechanisms of social behavior has propelled the growth of social neuroscience. The exact computations of the social brain, however, remain elusive. Here we investigated how the human brain tracks ongoing changes in social relationships using functional neuroimaging. Participants were lead characters in a role-playing game in which they were to find a new home and a job through interactions with virtual cartoon characters. We found that a two-dimensional geometric model of social relationships, a "social space" framed by power and affiliation, predicted hippocampal activity. Moreover, participants who reported better social skills showed stronger covariance between hippocampal activity and "movement" through "social space." The results suggest that the hippocampus is crucial for social cognition, and imply that beyond framing physical locations, the hippocampus computes a more general, inclusive, abstract, and multidimensional cognitive map consistent with its role in episodic memory.

Neural correlates of merging number words

Complex number words (e.g., "twenty two") are formed by merging together several simple number words (e.g., "twenty" and "two"). In the present study, we explored the neural correlates of this operation and investigated to what extent it engages brain areas involved processing numerical quantity and linguistic syntactic structure. Participants speaking two typologically distinct languages, French and Chinese, were required to read aloud sequences of simple number words while their cerebral activity was recorded by functional magnetic resonance imaging. Each number word could either be merged with the previous ones (e.g., 'twenty three') or not (e.g., 'three twenty'), thus forming four levels ranging from lists of number words to complex numerals. When a number word could be merged with the preceding ones, it was named faster than when it could not. Neuroimaging results showed that the number of merges correlated with activation in the left inferior frontal gyrus and in the left inferior parietal lobule. Consistent findings across Chinese and French participants suggest that these regions serve as the neural bases for forming complex number words in different languages.

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.

Link between cognitive neuroscience and education: the case of clinical assessment of developmental dyscalculia

In recent years, cognitive neuroscience research has identified several biological and cognitive features of number processing deficits that may now make it possible to diagnose mental or educational impairments in arithmetic, even earlier and more precisely than is possible using traditional assessment tools. We provide two sets of recommendations for improving cognitive assessment tools, using the important case of mathematics as an example. (1) neurocognitive tests would benefit substantially from incorporating assessments (based on findings from cognitive neuroscience) that entail systematic manipulation of fundamental aspects of number processing. Tests that focus on evaluating networks of core neurocognitive deficits have considerable potential to lead to more precise diagnosis and to provide the basis for designing specific intervention programs tailored to the deficits exhibited by the individual child. (2) implicit knowledge, derived from inspection of variables that are irrelevant to the task at hand, can also provide a useful assessment tool. Implicit knowledge is powerful and plays an important role in human development, especially in cases of psychiatric or neurological deficiencies (such as math learning disabilities or math anxiety).

Representation of the Numerosity 'zero' in the Parietal Cortex of the Monkey

Zero is a fundamental concept in mathematics and modern science. Empty sets are considered a precursor of the concept of numerosity zero and a part of numerical continuum. How is numerosity zero (the absence of visual items) represented in the primate cortex? To address this question, we trained monkeys to perform numerical operations including numerosity zero. Here we show a group of neurons in the posterior parietal cortex of the monkey activated in response to numerosity ‘zero’. ‘Zero’ neurons are classified into exclusive and continuous types; the exclusive type discretely encodes numerical absence and the continuous type encodes numerical absence as a part of a numerical continuum. “Numerosity-zero” neurons enhance behavioral discrimination of not only zero numerosity but also non-zero numerosities. Representation of numerosity zero in the parietal cortex may be a precursor of non-verbal concept of zero in primates.

Representations of numerical sequences and the concept of middle in preschoolers

The present study concerns preschoolers' understanding of the middle concept as it applies to numerical sequences. Previous research using implicit psychophysical assessment suggests that the numerical midpoint is embedded within numerical representations by 4 years of age. Here, we examined 3- to 5-year-olds' ability to identify the midpoint value in triplets of non-symbolic numbers when explicitly probed to do so. We found that whereas 4- and 5-year-olds were capable of explicit access to numerical midpoint values and showed ratio-dependent performance, a signature of the approximate number system (ANS), 3-year-olds performed at chance. Children's difficulty in identifying numerical midpoint values was not due to comparing multiple arrays, nor was it entirely due to a spatial association with the word "middle" used in the task. We speculate that explicit access to numerical midpoint values may be jointly supported by endogenous control of attentional mechanisms and the development of a mental number line.

The origins of counting algorithms

Humans' ability to count by verbally labeling discrete quantities is unique in animal cognition. The evolutionary origins of counting algorithms are not understood. We report that nonhuman primates exhibit a cognitive ability that is algorithmically and logically similar to human counting. Monkeys were given the task of choosing between two food caches. First, they saw one cache baited with some number of food items, one item at a time. Then, a second cache was baited with food items, one at a time. At the point when the second set was approximately equal to the first set, the monkeys spontaneously moved to choose the second set even before that cache was completely baited. Using a novel Bayesian analysis, we show that the monkeys used an approximate counting algorithm for comparing quantities in sequence that is incremental, iterative, and condition controlled. This proto-counting algorithm is structurally similar to formal counting in humans and thus may have been an important evolutionary precursor to human counting.

Oscillatory brain activity reveals linguistic prints in the quantity code

Number representations change through education, although it is currently unclear whether and how language could impact the magnitude representation that we share with other species. The most prominent view is that language does not play any role in modulating the core numeric representation involved in the contrast of quantities. Nevertheless, possible cultural hints on the numerical magnitude representation are currently on discussion focus. In fact, the acquisition of number words provides linguistic input that the quantity system may not ignore. Bilingualism offers a window to the study of this question, especially in bilinguals where the two number wording systems imply also two different numerical systems, such as in Basque-Spanish bilinguals. The present study evidences linguistic prints in the core number representational system through the analysis of EEG oscillatory activity during a simple number comparison task. Gamma band synchronization appears when Basque-Spanish bilinguals compare pairs of Arabic numbers linked through the Basque base-20 wording system, but it does not if the pairs are related through the base-10 system. Crucially, this gamma activity, originated in a left fronto-parietal network, only appears in bilinguals who learned math in Basque and not in equivalent proficiency bilinguals who learned math in Spanish. Thus, this neural index reflected in gamma band synchrony appears to be triggered by early learning experience with the base-20 numerical associations in Basque number words.

Methodological aspects to be considered when measuring the approximate number system (ANS) - a research review

According to a dominant view, the approximate number system (ANS) is the foundation of symbolic math abilities. Due to the importance of math abilities for education and career, a lot of research focuses on the investigation of the ANS and its relationship with math performance. However, the results are inconsistent. This might be caused by studies differing greatly regarding the operationalization of the ANS (i.e., tasks, dependent variables). Moreover, many methodological aspects vary from one study to the next. In the present review, we discuss commonly used ANS tasks and dependent variables regarding their theoretical foundation and psychometric features. We argue that the inconsistent findings concerning the relationship between ANS acuity and math performance may be partially explained by differences in reliability. Furthermore, this review summarizes methodological aspects of ANS tasks having important impacts on the results, including stimulus range, visual controls, presentation duration of the stimuli and feedback. Based on this review, we give methodological recommendations on how to assess the ANS most reliably and most validly. All important methodological aspects to be considered when designing an ANS task or comparing results of different studies are summarized in two practical checklists.

Children's mappings between number words and the approximate number system

Humans can represent number either exactly--using their knowledge of exact numbers as supported by language, or approximately--using their approximate number system (ANS). Adults can map between these two systems--they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate (i.e., ANS-to-Word), and can generate an approximation, for example by rapidly tapping, when provided with an exact verbal number (i.e., Word-to-ANS). Here we ask how these mappings are initially formed and whether one mapping direction may become functional before the other during development. In two experiments, we gave 2-5 year old children both an ANS-to-Word task, where they had to give a verbal number response to an approximate presentation (i.e., after seeing rapidly flashed dots, or watching rapid hand taps), and a Word-to-ANS task, where they had to generate an approximate response to a verbal number request (i.e., rapidly tapping after hearing a number word). Replicating previous results, children did not successfully generate numerically appropriate verbal responses in the ANS-to-Word task until after 4 years of age--well after they had acquired the Cardinality Principle of verbal counting. In contrast, children successfully generated numerically appropriate tapping sequences in the Word-to-ANS task before 4 years of age--well before many understood the Cardinality Principle. We further found that the accuracy of the mapping between the ANS and number words, as captured by error rates, continues to develop after this initial formation of the interface. These results suggest that the mapping between the ANS and verbal number representations is not functionally bidirectional in early development, and that the mapping direction from number representations to the ANS is established before the reverse.

Language influences number processing--a quadrilingual study

Reading/writing direction or number word formation influence performance even in basic numerical tasks such as magnitude comparison. However, so far the interaction of these language properties has not been evaluated systematically. In this study we tested English, German, Hebrew, and Arab participants realizing a natural 2 × 2 design of reading/writing direction (left-to-right vs. right-to-left) and number word formation (non-inverted vs. inverted, i.e., forty-seven vs. seven-and-forty). Symbolic number magnitude comparison was specifically influenced by the interaction of reading/writing direction and number word formation: participants from cultures where reading direction and the order of tens and units in number words are incongruent (i.e., German and Hebrew) exhibited more pronounced unit interference in place-value integration. A within-group comparison indicated that this effect was not due to differences in education. Thus, basic cultural differences in numerical cognition were driven by natural language variables and their specific combination.

In how many ways is the approximate number system associated with exact calculation?

The approximate number system (ANS) has been consistently found to be associated with math achievement. However, little is known about the interactions between the different instantiations of the ANS and in how many ways they are related to exact calculation. In a cross-sectional design, we investigated the relationship between three measures of ANS acuity (non-symbolic comparison, non-symbolic estimation and non-symbolic addition), their cross-sectional trajectories and specific contributions to exact calculation. Children with mathematical difficulties (MD) and typically achieving (TA) controls attending the first six years of formal schooling participated in the study. The MD group exhibited impairments in multiple instantiations of the ANS compared to their TA peers. The ANS acuity measured by all three tasks positively correlated with age in TA children, while no correlation was found between non-symbolic comparison and age in the MD group. The measures of ANS acuity significantly correlated with each other, reflecting at least in part a common numerosity code. Crucially, we found that non-symbolic estimation partially and non-symbolic addition fully mediated the effects of non-symbolic comparison in exact calculation.

The role of ANS acuity and numeracy for the calibration and the coherence of subjective probability judgments

The purpose of the study was to investigate how numeracy and acuity of the approximate number system (ANS) relate to the calibration and coherence of probability judgments. Based on the literature on number cognition, a first hypothesis was that those with lower numeracy would maintain a less linear use of the probability scale, contributing to overconfidence and nonlinear calibration curves. A second hypothesis was that also poorer acuity of the ANS would be associated with overconfidence and non-linearity. A third hypothesis, in line with dual-systems theory (e.g., Kahneman and Frederick, 2002) was that people higher in numeracy should have better access to the normative probability rules, allowing them to decrease the rate of conjunction fallacies. Data from 213 participants sampled from the Swedish population showed that: (i) in line with the first hypothesis, overconfidence and the linearity of the calibration curves were related to numeracy, where people higher in numeracy were well calibrated with zero overconfidence. (ii) ANS was not associated with overconfidence and non-linearity, disconfirming the second hypothesis. (iii) The rate of conjunction fallacies was slightly, but to a statistically significant degree decreased by numeracy, but still high at all numeracy levels. An unexpected finding was that participants with better ANS acuity gave more realistic estimates of their performance relative to others.

The association between higher education and approximate number system acuity

Humans are equipped with an approximate number system (ANS) supporting non-symbolic numerosity representation. Studies indicate a relationship between ANS-precision (acuity) and math achievement. Whether the ANS is a prerequisite for learning mathematics or if mathematics education enhances the ANS remains an open question. We investigated the association between higher education and ANS acuity with university students majoring in subjects with varying amounts of mathematics (mathematics, business, and humanities), measured either early (First year) or late (Third year) in their studies. The results suggested a non-significant trend where students taking more mathematics had better ANS acuity and a significant improvement in ANS acuity as a function of study length that was mainly confined to the business students. The results provide partial support for the hypothesis that education in mathematics can enhance the ANS acuity.

Single-digit arithmetic processing-anatomical evidence from statistical voxel-based lesion analysis

Different specific mechanisms have been suggested for solving single-digit arithmetic operations. However, the neural correlates underlying basic arithmetic (multiplication, addition, subtraction) are still under debate. In the present study, we systematically assessed single-digit arithmetic in a group of acute stroke patients (n = 45) with circumscribed left- or right-hemispheric brain lesions. Lesion sites significantly related to impaired performance were found only in the left-hemisphere damaged (LHD) group. Deficits in multiplication and addition were related to subcortical/white matter brain regions differing from those for subtraction tasks, corroborating the notion of distinct processing pathways for different arithmetic tasks. Additionally, our results further point to the importance of investigating fiber pathways in numerical cognition.

Predicting first-grade mathematics achievement: the contributions of domain-general cognitive abilities, nonverbal number sense, and early number competence

Early number competence, grounded in number-specific and domain-general cognitive abilities, is theorized to lay the foundation for later math achievement. Few longitudinal studies have tested a comprehensive model for early math development. Using structural equation modeling and mediation analyses, the present work examined the influence of kindergarteners' nonverbal number sense and domain-general abilities (i.e., working memory, fluid intelligence, and receptive vocabulary) and their early number competence (i.e., symbolic number skills) on first grade math achievement (i.e., arithmetic, shape and space skills, and number line estimation) assessed 1 year later. Latent regression models revealed that nonverbal number sense and working memory are central building blocks for developing early number competence in kindergarten and that early number competence is key for first grade math achievement. After controlling for early number competence, fluid intelligence significantly predicted arithmetic and number line estimation while receptive vocabulary significantly predicted shape and space skills. In sum we suggest that early math achievement draws on different constellations of number-specific and domain-general mechanisms.

Different brains process numbers differently: structural bases of individual differences in spatial and nonspatial number representations

A dominant hypothesis on how the brain processes numerical size proposes a spatial representation of numbers as positions on a "mental number line." An alternative hypothesis considers numbers as elements of a generalized representation of sensorimotor-related magnitude, which is not obligatorily spatial. Here we show that individuals' relative use of spatial and nonspatial representations has a cerebral counterpart in the structural organization of the posterior parietal cortex. Interindividual variability in the linkage between numbers and spatial responses (faster left responses to small numbers and right responses to large numbers; spatial-numerical association of response codes effect) correlated with variations in gray matter volume around the right precuneus. Conversely, differences in the disposition to link numbers to force production (faster soft responses to small numbers and hard responses to large numbers) were related to gray matter volume in the left angular gyrus. This finding suggests that numerical cognition relies on multiple mental representations of analogue magnitude using different neural implementations that are linked to individual traits.

Differences in the acuity of the Approximate Number System in adults: the effect of mathematical ability

It is largely admitted that processing numerosity relies on an innate Approximate Number System (ANS), and recent research consistently observed a relationship between ANS acuity and mathematical ability in childhood. However, studies assessing this relationship in adults led to contradictory results. In this study, adults with different levels of mathematical expertise performed two tasks on the same pairs of dot collections, based either on numerosity comparison or on cumulative area comparison. Number of dots and cumulative area were congruent in half of the stimuli, and incongruent in the other half. The results showed that adults with higher mathematical ability obtained lower Weber fractions in the numerical condition than participants with lower mathematical ability. Further, adults with lower mathematical ability were more affected by the interference of the continuous dimension in the numerical comparison task, whereas conversely higher-expertise adults showed stronger interference of the numerical dimension in the continuous comparison task. Finally, ANS acuity correlated with arithmetic performance. Taken together, the data suggest that individual differences in ANS acuity subsist in adulthood, and that they are related to mathematical ability.

Bilateral bi-cephalic tDCS with two active electrodes of the same polarity modulates bilateral cognitive processes differentially [corrected]

Transcranial direct current stimulation (tDCS) is an innovative method to explore the causal structure-function relationship of brain areas. We investigated the specificity of bilateral bi-cephalic tDCS with two active electrodes of the same polarity (e.g., cathodal on both hemispheres) applied to intraparietal cortices bilaterally using a combined between- and within-task approach. Regarding between-task specificity, we observed that bilateral bi-cephalic tDCS affected a numerical (mental addition) but not a control task (colour word Stroop), indicating a specific influence of tDCS on numerical but not on domain general cognitive processes associated with the bilateral IPS. In particular, the numerical effect of distractor distance was more pronounced under cathodal than under anodal stimulation. Moreover, with respect to within-task specificity we only found the numerical distractor distance effect in mental addition to be modulated by direct current stimulation, whereas the effect of target identity was not affected. This implies a differential influence of bilateral bi-cephalic tDCS on the recruitment of different processing components within the same task (number magnitude processing vs. recognition of familiarity). In sum, this first successful application of bilateral bi-cephalic tDCS with two active electrodes of the same polarity in numerical cognition research corroborates the specific proposition of the Triple Code Model that number magnitude information is represented bilaterally in the intraparietal cortices.

Measuring acuity of the approximate number system reliably and validly: the evaluation of an adaptive test procedure

Two studies investigated the reliability and predictive validity of commonly used measures and models of Approximate Number System acuity (ANS). Study 1 investigated reliability by both an empirical approach and a simulation of maximum obtainable reliability under ideal conditions. Results showed that common measures of the Weber fraction (w) are reliable only when using a substantial number of trials, even under ideal conditions. Study 2 compared different purported measures of ANS acuity as for convergent and predictive validity in a within-subjects design and evaluated an adaptive test using the ZEST algorithm. Results showed that the adaptive measure can reduce the number of trials needed to reach acceptable reliability. Only direct tests with non-symbolic numerosity discriminations of stimuli presented simultaneously were related to arithmetic fluency. This correlation remained when controlling for general cognitive ability and perceptual speed. Further, the purported indirect measure of ANS acuity in terms of the Numeric Distance Effect (NDE) was not reliable and showed no sign of predictive validity. The non-symbolic NDE for reaction time was significantly related to direct w estimates in a direction contrary to the expected. Easier stimuli were found to be more reliable, but only harder (7:8 ratio) stimuli contributed to predictive validity.

Objects, numbers, fingers, space: clustering of ventral and dorsal functions in young children and adults

In the primate brain, sensory information is processed along two partially segregated cortical streams: the ventral stream, mainly coding for objects' shape and identity, and the dorsal stream, mainly coding for objects' quantitative information (including size, number, and spatial position). Neurophysiological measures indicate that such functional segregation is present early on in infancy, and that the two streams follow independent maturational trajectories during childhood. Here we collected, in a large sample of young children and adults, behavioural measures on an extensive set of functions typically associated with either the dorsal or the ventral stream. We then used a correlational approach to investigate the presence of inter-individual variability resulting in clustering of functions. Results show that dorsal- and ventral-related functions follow two uncorrelated developmental trajectories. Moreover, within each stream, some functions show age-independent correlations: finger gnosis, non-symbolic numerical abilities and spatial abilities within the dorsal stream, and object and face recognition abilities within the ventral stream. This pattern of clear within-stream cross-task correlation seems to be lost in adults, with two notable exceptions: performance in face and object recognition on one side, and in symbolic and non-symbolic comparison on the other, remain correlated, pointing to distinct shape recognition and quantity comparison systems.

Processing pathways in mental arithmetic--evidence from probabilistic fiber tracking

Numerical cognition is a case of multi-modular and distributed cerebral processing. So far neither the anatomo-functional connections between the cortex areas involved nor their integration into established frameworks such as the differentiation between dorsal and ventral processing streams have been specified. The current study addressed this issue combining a re-analysis of previously published fMRI data with probabilistic fiber tracking data from an independent sample. We aimed at differentiating neural correlates and connectivity for relatively easy and more difficult addition problems in healthy adults and their association with either rather verbally mediated fact retrieval or magnitude manipulations, respectively. The present data suggest that magnitude- and fact retrieval-related processing seem to be subserved by two largely separate networks, both of them comprising dorsal and ventral connections. Importantly, these networks not only differ in localization of activation but also in the connections between the cortical areas involved. However, it has to be noted that even though seemingly distinct anatomically, these networks operate as a functionally integrated circuit for mental calculation as revealed by a parametric analysis of brain activation.

Neural activity during natural viewing of Sesame Street statistically predicts test scores in early childhood

Neural activity that is evoked naturalistically in children during educational television viewing can be used to predict math and verbal knowledge.

It is not currently possible to measure the real-world thought process that a child has while observing an actual school lesson. However, if it could be done, children's neural processes would presumably be predictive of what they know. Such neural measures would shed new light on children's real-world thought. Toward that goal, this study examines neural processes that are evoked naturalistically, during educational television viewing. Children and adults all watched the same Sesame Street video during functional magnetic resonance imaging (fMRI). Whole-brain intersubject correlations between the neural timeseries from each child and a group of adults were used to derive maps of “neural maturity” for children. Neural maturity in the intraparietal sulcus (IPS), a region with a known role in basic numerical cognition, predicted children's formal mathematics abilities. In contrast, neural maturity in Broca's area correlated with children's verbal abilities, consistent with prior language research. Our data show that children's neural responses while watching complex real-world stimuli predict their cognitive abilities in a content-specific manner. This more ecologically natural paradigm, combined with the novel measure of “neural maturity,” provides a new method for studying real-world mathematics development in the brain.

In the real world, children learn new information by participating in classrooms, interacting with their family and friends, and watching educational videos. While previous neuroimaging research has typically used simple tasks and short-lasting stimuli, in this study we examined brain development using a more complex and naturalistic educational stimulus. Children and adults all watched the same Sesame Street video as we measured their neural activity using functional magnetic resonance imaging (fMRI). We examined the timecourses of neural activity over the length of the video for children and adults. We found that the degree to which children showed adult-like brain responses was correlated with their math and verbal knowledge levels. In the intraparietal sulcus, children's neural correlation with adults depended on their mathematics knowledge whereas in Broca's area, it depended on their verbal knowledge. Additional experiments showed that children's neural responses in the intraparietal sulcus are selectively driven by numerical content both when children are watching Sesame Street and when they engage in a number matching task. These convergent results highlight the broad role of the intraparietal sulcus in processing numerical information. In addition, our study validates the use of naturalistic stimuli and child-to-adult neural timecourse correlations for studying brain development. We suggest that this new approach can enrich our understanding of how children's brains process information in the real world.

Neuro-anatomical correlates of a number bisection bias: A neuropsychological voxel-based morphometry study

The number bisection tasks, whereby participants estimate the midpoint of a given number interval, is frequently used to explore the idea that numbers are spatially represented within the brain across a 'mental number line'. Some neuropsychological research supports the argument that number bisection is a spatial task, recruiting parietal brain regions, whereas other data suggest that number bisection is dissociable from spatial processing and is instead dependent on working memory in the prefrontal cortices. This study explored the anatomical correlates of deficits in the number bisection task, using voxel-based morphometry in a sample of 25 neuropsychological patients with both left and right hemisphere damage. Interestingly, impairments in number bisection were strongly associated with grey matter lesions in the left hemisphere including both frontal and prefrontal cortices, extending to inferior parietal cortex. Similar prefrontal and frontal grey matter areas were found to be associated with increased leftward deviations (underestimations of the midpoint), whereas no suprathreshold clusters were observed for rightward deviations from the midpoint. Analysis of white matter integrity revealed that lesions in the tracts connecting the parietal and frontal cortices (i.e. the superior longitudinal fasciculus) were highly associated with leftward deviation impairments in number bisection. The data suggest that there is a common parieto-frontal number processing network underlying performances on number bisection, with larger numbers represented on the left side.

The approximate number system and its relation to early math achievement: evidence from the preschool years

Humans rely on two main systems of quantification; one is nonsymbolic and involves approximate number representations (known as the approximate number system or ANS), and the other is symbolic and allows for exact calculations of number. Despite the pervasiveness of the ANS across development, recent studies with adolescents and school-aged children point to individual differences in the precision of these representations that, importantly, have been shown to relate to symbolic math competence even after controlling for general aspects of intelligence. Such findings suggest that the ANS, which humans share with nonhuman animals, interfaces specifically with a uniquely human system of formal mathematics. Other findings, however, point to a less straightforward picture, leaving open questions about the nature and ontogenetic origins of the relation between these two systems. Testing children across the preschool period, we found that ANS precision correlated with early math achievement but, critically, that this relation was nonlinear. More specifically, the correlation between ANS precision and math competence was stronger for children with lower math scores than for children with higher math scores. Taken together, our findings suggest that early-developing connections between the ANS and mathematics may be fundamentally discontinuous. Possible mechanisms underlying such nonlinearity are discussed.

An electrophysiological investigation of non-symbolic magnitude processing: numerical distance effects in children with and without mathematical learning disabilities

INTRODUCTION

The aim of the present study was to probe electrophysiological effects of non-symbolic numerical processing in 20 children with mathematical learning disabilities (mean age = 99.2 months) compared to a group of 20 typically developing matched controls (mean age = 98.4 months).

METHODS

EEG data were obtained while children were tested with a standard non-symbolic numerical comparison paradigm that allowed us to investigate the effects of numerical distance manipulations for different set sizes, i.e., the classical subitizing, counting and estimation ranges. Effects of numerical distance manipulations on event-related potential (ERP) amplitudes as well as activation patterns of underlying current sources were analyzed.

RESULTS

In typically developing children, the amplitudes of a late parietal positive-going ERP component showed systematic numerical distance effects that did not depend on set size. For the group of children with mathematical learning disabilities, ERP distance effects were found only for stimuli within the subitizing range. Current source density analysis of distance-related group effects suggested that areas in right inferior parietal regions are involved in the generation of the parietal ERP amplitude differences.

CONCLUSION

Our results suggest that right inferior parietal regions are recruited differentially by controls compared to children with mathematical learning disabilities in response to non-symbolic numerical magnitude processing tasks, but only for stimuli with set sizes that exceed the subitizing range.

Hysteresis affects approximate number discrimination in young children

Perceptual decisions are often affected not only by the evidence gathered during a trial but also by the history of preceding trials. This effect--termed perceptual hysteresis--provides evidence for how perceptual information is represented and how it is used. The present research focuses on how the difficulty of preceding trials affects subsequent ones--we find that how well 5-year-old children perform in a 2-alternative forced-choice numerical discrimination task depends on whether they have had a prior history of easier discriminations or a prior history of harder discriminations. Furthermore, this effect is modulated by the feedback children receive. In 3 experiments, we demonstrate that these effects are not related to practice or loss of interest due to negative feedback, or simply to trial difficulty or discriminability. Instead, children appear to have state-dependent confidence states such that prolonged experience making low-confidence decisions degrades performance, whereas prolonged experience making high-confidence decisions improves it. These results are discussed in the context of dynamical psychophysics, representations of confidence, and work on children's and adults' number perception abilities.

High-resolution EEG analysis of power spectral density maps and coherence networks in a proportional reasoning task

Proportional reasoning is very important logical skill required in mathematics and science problem solving as well as in everyday life decisions. However, there is a lack of studies on neurophysiological correlates of proportional reasoning. To explore the brain activity of healthy adults while performing a balance scale task, we used high-resolution EEG techniques and graph-theory based connectivity analysis. After unskilled subjects learned how to properly solve the task, their cortical power spectral density (PSD) maps revealed an increased parietal activity in the beta band. This indicated that subjects started to perform calculations. In addition, the number of inter-hemispheric connections decreased after learning, implying a rearrangement of the brain activity. Repeated performance of the task led to the PSD decrease in the beta and gamma bands among parietal and frontal regions along with a synchronization of lower frequencies. These findings suggest that repetition led to a more automatic task performance. Subjects were also divided in two groups according to their scores on the test of logical thinking (TOLT). Although no group differences in the accuracy and reaction times were found, EEG data showed higher activity in the beta and gamma bands for the group that scored better on TOLT. Learning and repetition induced changes in the pattern of functional connectivity were evident for all frequency bands. Overall, the results indicated that higher frequency oscillations in frontal and parietal regions are particularly important for proportional reasoning.

Spatiotemporal dynamics of processing nonsymbolic number: an event-related potential source localization study

Coordinated studies with adults, infants, and nonhuman animals provide evidence for two distinct systems of nonverbal number representation. The "parallel individuation" (PI) system selects and retains information about one to three individual entities and the "numerical magnitude" system establishes representations of the approximate cardinal value of a group. Recent event-related potential (ERP) work has demonstrated that these systems reliably evoke functionally and temporally distinct patterns of brain response that correspond to established behavioral signatures. However, relatively little is known about the neural generators of these ERP signatures. To address this question, we targeted known ERP signatures of these systems, by contrasting processing of small versus large nonsymbolic numbers, and used a source localization algorithm (LORETA) to identify their cortical origins. Early processing of small numbers, showing the signature effects of PI on the N1 (∼150 ms), was localized primarily to extrastriate visual regions. In contrast, qualitatively and temporally distinct processing of large numbers, showing the signatures of approximate number representation on the mid-latency P2p (∼200-250 ms), was localized primarily to right intraparietal regions. In comparison, mid-latency small number processing was localized to the right temporal-parietal junction and left-lateralized intraparietal regions. These results add spatial information to the emerging ERP literature documenting the process by which we represent number. Furthermore, these results substantiate recent claims that early attentional processes determine whether a collection of objects will be represented through PI or as an approximate numerical magnitude by providing evidence that downstream processing diverges to distinct cortical regions.

When time is space: evidence for a mental time line

Time and space are tightly linked in the physical word. Recently, several lines of evidence have suggested that the mental representation of time might be spatial in nature. For instance, time-space interactions have been described as a strong preference to associate the past with the left space and the future with the right space. Here we review the growing evidence of interactions between time and space processing, systematized according to the type of interaction being investigated. We present the empirical findings supporting the possibility that humans represent the subjective time flow on a spatially oriented "mental time line" that is accessed through spatial attention mechanisms. The heterogeneous time-space interactions are then compared with the number-space interactions described in the numerical cognition literature. An alternative hypothesis, which maintains a common system for magnitude processing, including time, space, and number, is also discussed. Finally, we extend the discussion to the more general issue of how the representation of these concepts might be grounded into the cortical circuits that support spatial attention and sensorimotor transformations.

Developmental change in the acuity of approximate number and area representations

From very early in life, humans can approximate the number and surface area of objects in a scene. The ability to discriminate between 2 approximate quantities, whether number or area, critically depends on the ratio between the quantities, with the most difficult ratio that a participant can reliably discriminate known as the Weber fraction. While developmental improvements in the Weber fraction have been demonstrated for number, the developmental trajectory of improvement in area discrimination remains unknown. Here we investigated whether the development of area discrimination parallels that of number discrimination. We tested forty 3- to 6-year-old children and adults in both a number and an area discrimination task in which participants selected the greater of 2 quantities across a range of ratios. We used formal psychophysical models to derive, for each participant and each age group, the Weber fraction for both number and area discrimination. We found that, like number acuity, area acuity steadily improves during childhood. However, we also found area acuity to be consistently higher than number acuity, suggesting a potential difference in the underlying mechanisms that encode and/or represent approximate area and approximate number. We discuss these findings in the context of quantity processing and its development.

Grey parrot number acquisition: the inference of cardinal value from ordinal position on the numeral list

A Grey parrot (Psittacus erithacus) had previously been taught to use English count words ("one" through "sih" [six]) to label sets of one to six individual items (Pepperberg, 1994). He had also been taught to use the same count words to label the Arabic numerals 1 through 6. Without training, he inferred the relationship between the Arabic numerals and the sets of objects (Pepperberg, 2006b). In the present study, he was then trained to label vocally the Arabic numerals 7 and 8 ("sih-none", "eight", respectively) and to order these Arabic numerals with respect to the numeral 6. He subsequently inferred the ordinality of 7 and 8 with respect to the smaller numerals and he inferred use of the appropriate label for the cardinal values of seven and eight items. These data suggest that he constructed the cardinal meanings of "seven" ("sih-none") and "eight" from his knowledge of the cardinal meanings of one through six, together with the place of "seven" ("sih-none") and "eight" in the ordered count list.

Pitch-interval discrimination and musical expertise: is the semitone a perceptual boundary?

The ability to discriminate pitch changes (or intervals) is foundational for speech and music. In an auditory psychophysical experiment, musicians and non-musicians were tested with fixed- and roving-pitch discrimination tasks to investigate the effects of musical expertise on interval discrimination. The tasks were administered parametrically to assess performance across varying pitch distances between intervals. Both groups showed improvements in fixed-pitch interval discrimination as a function of increasing interval difference. Only musicians showed better roving-pitch interval discrimination as interval differences increased, suggesting that this task was too demanding for non-musicians. Musicians had better interval discrimination than non-musicians across most interval differences in both tasks. Interestingly, musicians exhibited improved interval discrimination starting at interval differences of 100 cents (a semitone in Western music), whereas non-musicians showed enhanced discrimination at interval differences exceeding 125 cents. Although exposure to Western music and speech may help establish a basic interval-discrimination threshold between 100 and 200 cents (intervals that occur often in Western languages and music), musical training presumably enhances auditory processing and reduces this threshold to a semitone. As musical expertise does not decrease this threshold beyond 100 cents, the semitone may represent a musical training-induced intervallic limit to acoustic processing.