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.

Brain-mechanistic responses to varying difficulty levels of approximate solutions to arithmetic problems

Approximate strategies are crucial in daily human life. The studies on the “difficulty effect” seen in approximate complex arithmetic have long been neglected. Here, we aimed to explore the brain mechanisms related to this difficulty effect in the case of complex addition, using event-related potential-based methods. Following previous path-finding studies, we used the inequality paradigm and different split sizes to induce the use of two approximate strategies for different difficulty levels. By comparing dependent variables from the medium- and large-split conditions, we anticipated being able to dissociate the effects of task difficulty based on approximate strategy in electrical components. In the fronto−central region, early P2 (150–250 ms) and an N400-like wave (250–700 ms) were significantly different between different difficulty levels. Differences in P2 correlated with the difficulty of separation of the approximate strategy from the early physical stimulus discrimination process, which is dominant before 200 ms, and differences in the putative N400 correlated with different difficulties of approximate strategy execution. Moreover, this difference may be linked to speech processing. In addition, differences were found in the fronto-central region, which may reflect the regulatory role of this part of the cortex in approximate strategy execution when solving complex arithmetic problems.

Current Perspectives on Cognitive Diversity

To what extent is cognition influenced by a person’s cultural background? This question has remained controversial in large fields of the cognitive sciences, including cognitive psychology, and is also underexplored in anthropology. In this perspective article, findings from a recent wave of cross-cultural studies will be outlined with respect to three aspects of cognition: perception and categorization, number representation and counting, and explanatory frameworks and beliefs. Identifying similarities and differences between these domains allows for general conclusions regarding cognitive diversity and helps to highlight the importance of culturally shaped content for a comprehensive understanding of cognition.

Origins of the brain networks for advanced mathematics in expert mathematicians

The origins of human abilities for mathematics are debated: Some theories suggest that they are founded upon evolutionarily ancient brain circuits for number and space and others that they are grounded in language competence. To evaluate what brain systems underlie higher mathematics, we scanned professional mathematicians and mathematically naive subjects of equal academic standing as they evaluated the truth of advanced mathematical and nonmathematical statements. In professional mathematicians only, mathematical statements, whether in algebra, analysis, topology or geometry, activated a reproducible set of bilateral frontal, Intraparietal, and ventrolateral temporal regions. Crucially, these activations spared areas related to language and to general-knowledge semantics. Rather, mathematical judgments were related to an amplification of brain activity at sites that are activated by numbers and formulas in nonmathematicians, with a corresponding reduction in nearby face responses. The evidence suggests that high-level mathematical expertise and basic number sense share common roots in a nonlinguistic brain circuit.

Changing the precision of preschoolers' approximate number system representations changes their symbolic math performance

From early in life, humans have access to an approximate number system (ANS) that supports an intuitive sense of numerical quantity. Previous work in both children and adults suggests that individual differences in the precision of ANS representations correlate with symbolic math performance. However, this work has been almost entirely correlational in nature. Here we tested for a causal link between ANS precision and symbolic math performance by asking whether a temporary modulation of ANS precision changes symbolic math performance. First, we replicated a recent finding that 5-year-old children make more precise ANS discriminations when starting with easier trials and gradually progressing to harder ones, compared with the reverse. Next, we show that this brief modulation of ANS precision influenced children's performance on a subsequent symbolic math task but not a vocabulary task. In a supplemental experiment, we present evidence that children who performed ANS discriminations in a random trial order showed intermediate performance on both the ANS task and the symbolic math task, compared with children who made ordered discriminations. Thus, our results point to a specific causal link from the ANS to symbolic math performance.

Language and thought are not the same thing: evidence from neuroimaging and neurological patients

Is thought possible without language? Individuals with global aphasia, who have almost no ability to understand or produce language, provide a powerful opportunity to find out. Surprisingly, despite their near-total loss of language, these individuals are nonetheless able to add and subtract, solve logic problems, think about another person's thoughts, appreciate music, and successfully navigate their environments. Further, neuroimaging studies show that healthy adults strongly engage the brain's language areas when they understand a sentence, but not when they perform other nonlinguistic tasks such as arithmetic, storing information in working memory, inhibiting prepotent responses, or listening to music. Together, these two complementary lines of evidence provide a clear answer: many aspects of thought engage distinct brain regions from, and do not depend on, language.

Symbolic, Nonsymbolic and Conceptual: An Across-Notation Study on the Space Mapping of Numerals

Previous studies suggested that there are interconnections between two numeral modalities of symbolic notation and nonsymbolic notation (array of dots), differences and similarities of the processing, and representation of the two modalities have both been found in previous research. However, whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation is still uninvestigated. The present study aims to examine whether there are differences between the spatial representation and numeral-space mapping of the two numeral modalities of symbolic notation and nonsymbolic notation; especially how zero, as both a symbolic magnitude numeral and a nonsymbolic conceptual numeral, mapping onto space; and if the mapping happens automatically at an early stage of the numeral information processing. Results of the two experiments demonstrate that the low-level processing of symbolic numerals including zero and nonsymbolic numerals except zero can mapping onto space, whereas the low-level processing of nonsymbolic zero as a semantic conceptual numeral cannot mapping onto space, which indicating the specialty of zero in the numeral domain. The present study indicates that the processing of non-semantic numerals can mapping onto space, whereas semantic conceptual numerals cannot mapping onto space.

Spatial Ability Explains the Male Advantage in Approximate Arithmetic

Previous research has shown that females consistently outperform males in exact arithmetic, perhaps due to the former's advantage in language processing. Much less is known about gender difference in approximate arithmetic. Given that approximate arithmetic is closely associated with visuospatial processing, which shows a male advantage we hypothesized that males would perform better than females in approximate arithmetic. In two experiments (496 children in Experiment 1 and 554 college students in Experiment 2), we found that males showed better performance in approximate arithmetic, which was accounted for by gender differences in spatial ability.

Can rodents conceive hyperbolic spaces?

The grid cells discovered in the rodent medial entorhinal cortex have been proposed to provide a metric for Euclidean space, possibly even hardwired in the embryo. Yet, one class of models describing the formation of grid unit selectivity is entirely based on developmental self-organization, and as such it predicts that the metric it expresses should reflect the environment to which the animal has adapted. We show that, according to self-organizing models, if raised in a non-Euclidean hyperbolic cage rats should be able to form hyperbolic grids. For a given range of grid spacing relative to the radius of negative curvature of the hyperbolic surface, such grids are predicted to appear as multi-peaked firing maps, in which each peak has seven neighbours instead of the Euclidean six, a prediction that can be tested in experiments. We thus demonstrate that a useful universal neuronal metric, in the sense of a multi-scale ruler and compass that remain unaltered when changing environments, can be extended to other than the standard Euclidean plane.

Adaptation to number operates on perceived rather than physical numerosity

Humans share with many animals a number sense, the ability to estimate rapidly the approximate number of items in a scene. Recent work has shown that like many other perceptual attributes, numerosity is susceptible to adaptation. It is not clear, however, whether adaptation works directly on mechanisms selective to numerosity, or via related mechanisms, such as those tuned to texture density. To disentangle this issue we measured adaptation of numerosity of 10 pairs of connected dots, as connecting dots makes them appear to be less numerous than unconnected dots. Adaptation to a 20-dot pattern (same number of dots as the test) caused robust reduction in apparent numerosity of the connected-dot pattern, but not of the unconnected dot-pattern. This suggests that adaptation to numerosity, at least for relatively sparse dot-pattern, occurs at neural levels encoding perceived numerosity, rather than at lower levels responding to the number of elements in the scene.

Significant Inter-Test Reliability across Approximate Number System Assessments

The approximate number system (ANS) is the hypothesized cognitive mechanism that allows adults, infants, and animals to enumerate large sets of items approximately. Researchers usually assess the ANS by having subjects compare two sets and indicate which is larger. Accuracy or Weber fraction is taken as an index of the acuity of the system. However, as Clayton et al. (2015) have highlighted, the stimulus parameters used when assessing the ANS vary widely. In particular, the numerical ratio between the pairs, and the way in which non-numerical features are varied often differ radically between studies. Recently, Clayton et al. (2015) found that accuracy measures derived from two commonly used stimulus sets are not significantly correlated. They argue that a lack of inter-test reliability threatens the validity of the ANS construct. Here we apply a recently developed modeling technique to the same data set. The model, by explicitly accounting for the effect of numerical ratio and non-numerical features, produces dependent measures that are less perturbed by stimulus protocol. Contrary to their conclusion we find a significant correlation in Weber fraction across the two stimulus sets. Nevertheless, in agreement with Clayton et al. (2015) we find that different protocols do indeed induce differences in numerical acuity and the degree of influence of non-numerical stimulus features. These findings highlight the need for a systematic investigation of how protocol idiosyncrasies affect ANS assessments.

Basic and advanced numerical performances relate to mathematical expertise but are fully mediated by visuospatial skills

Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic number line task, which required mapping positive and negative numbers on a physical horizontal line, and has been shown to correlate with more advanced numerical abilities and mathematical achievement. We found that mathematicians were more accurate compared with nonmathematicians when mapping positive, but not negative numbers, which are considered numerical primitives and cultural artifacts, respectively. Moreover, performance on positive number mapping could predict whether one is a mathematician or not, and was mediated by more advanced mathematical skills. This finding might suggest a link between basic and advanced mathematical skills. However, when we included visuospatial skills, as measured by block design subtest, the mediation analysis revealed that the relation between the performance in the number line task and the group membership was explained by non-numerical visuospatial skills. These results demonstrate that relation between basic, even specific, numerical skills and advanced mathematical achievement can be artifactual and explained by visuospatial processing.

Categories of Large Numbers in Line Estimation

How do people stretch their understanding of magnitude from the experiential range to the very large quantities and ranges important in science, geopolitics, and mathematics? This paper empirically evaluates how and whether people make use of numerical categories when estimating relative magnitudes of numbers across many orders of magnitude. We hypothesize that people use scale words-thousand, million, billion-to carve the large number line into categories, stretching linear responses across items within each category. If so, discontinuities in position and response time are expected near the boundaries between categories. In contrast to previous work (Landy, Silbert, & Goldin, 2013) that suggested only that a minority of college undergraduates employed categorical boundaries, we find that discontinuities near category boundaries occur in most or all participants, but that accurate and inaccurate participants respond in opposite ways to category boundaries. Accurate participants highlight contrasts within a category, whereas inaccurate participants adjust their responses toward category centers.

Numerical Activities and Information Learned at Home Link to the Exact Numeracy Skills in 5-6 Years-Old Children

It is currently accepted that certain activities within the family environment contribute to develop early numerical skills before schooling. However, it is unknown whether this early experience influences both the exact and the approximate representation of numbers, and if so, which is more important for numerical tasks. In the present study the mathematical performance of 110 children (mean age 5 years 11 months) was evaluated using a battery that included tests of approximate and exact numerical abilities, as well as everyday numerical problems. Moreover, children were assessed on their knowledge of number information learned at home. The parents of the participants provided information regarding daily activities of the children and socio-demographic characteristics of the family. The results showed that the amount of numerical information learned at home was a significant predictor of participants' performance on everyday numerical problems and exact number representations, even after taking account of age, memory span and socio-economic and educational status of the family. We also found that particular activities, such as board games, correlate with the children's counting skills, which are foundational for arithmetic. Crucially, tests relying on approximate representations were not predicted by the numerical knowledge acquired at home. The present research supports claims about the importance and nature of home experiences in the child's acquisition of mathematics.

The Mediating Relation between Symbolic and Nonsymbolic Foundations of Math Competence

This study investigated the relation between symbolic and nonsymbolic magnitude processing abilities with 2 standardized measures of math competence (WRAT Arithmetic and KeyMath Numeration) in 150 3^rd- grade children (mean age 9.01 years). Participants compared sets of dots and pairs of Arabic digits with numerosities 1–9 for relative numerical magnitude. In line with previous studies, performance on both symbolic and nonsymbolic magnitude processing was related to math ability. Performance metrics combining reaction and accuracy, as well as weber fractions, were entered into mediation models with standardized math test scores. Results showed that symbolic magnitude processing ability fully mediates the relation between nonsymbolic magnitude processing and math ability, regardless of the performance metric or standardized test.

Effects of Presentation Type and Visual Control in Numerosity Discrimination: Implications for Number Processing?

Performance in a non-symbolic comparison task in which participants are asked to indicate the larger numerosity of two dot arrays, is assumed to be supported by the Approximate Number System (ANS). This system allows participants to judge numerosity independently from other visual cues. Supporting this idea, previous studies indicated that numerosity can be processed when visual cues are controlled for. Consequently, distinct types of visual cue control are assumed to be interchangeable. However, a previous study showed that the type of visual cue control affected performance using a simultaneous presentation of the stimuli in numerosity comparison. In the current study, we explored whether the influence of the type of visual cue control on performance disappeared when sequentially presenting each stimulus in numerosity comparison. While the influence of the applied type of visual cue control was significantly more evident in the simultaneous condition, sequentially presenting the stimuli did not completely exclude the influence of distinct types of visual cue control. Altogether, these results indicate that the implicit assumption that it is possible to compare performances across studies with a differential visual cue control is unwarranted and that the influence of the type of visual cue control partly depends on the presentation format of the stimuli.

Precocious quantitative cognition in monkeys

Basic quantitative abilities are thought to have an innate basis in humans partly because the ability to discriminate quantities emerges early in child development. If humans and nonhuman primates share this developmentally primitive foundation of quantitative reasoning, then this ability should be present early in development across species and should emerge earlier in monkeys than in humans because monkeys mature faster than humans. We report that monkeys spontaneously make accurate quantity choices by 1 year of age in a task that human children begin to perform only at 2.5 to 3 years of age. Additionally, we report that the quantitative sensitivity of infant monkeys is equal to that of the adult animals in their group and that rates of learning do not differ between infant and adult animals. This novel evidence of precocious quantitative reasoning in infant monkeys suggests that human quantitative reasoning shares its early developing foundation with other primates. The data further suggest that early developing components of primate quantitative reasoning are constrained by maturational factors related to genetic development as opposed to learning experience alone.

Non-symbolic division in childhood

The approximate number system (ANS) underlies representations of large numbers of objects as well as the additive, subtractive, and multiplicative relationships between them. In this set of studies, 5- and 6-year-old children were shown a series of video-based events that conveyed a transformation of a large number of objects into one-half or one-quarter of the original number. Children were able to estimate correctly the outcomes to these halving and quartering problems, and they based their responses on scaling by number, not on continuous quantities or guessing strategies. Children's performance exhibited the ratio signature of the ANS. Moreover, children performed above chance on relatively early trials, suggesting that this scaling operation is easily conveyed and readily performed. The results support the existence of a flexible and substantially untrained capacity to scale numerical amounts.

Associations of non-symbolic and symbolic numerical magnitude processing with mathematical competence: a meta-analysis

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non-symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta-analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two-level random-effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non-symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non-symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school-aged children and for adults.

Asymmetric Processing of Numerical and Nonnumerical Magnitudes in the Brain: An fMRI Study

It is well established that, when comparing nonsymbolic magnitudes (e.g., dot arrays), adults can use both numerical (i.e., the number of items) and nonnumerical (density, total surface areas, etc.) magnitudes. It is less clear which of these magnitudes is more salient or processed more automatically. In this fMRI study, we used a nonsymbolic comparison task to ask if different brain areas are responsible for the automatic processing of numerical and nonnumerical magnitudes, when participants were instructed to attend to either the numerical or the nonnumerical magnitudes of the same stimuli. An interaction of task (numerical vs. nonnumerical) and congruity (congruent vs. incongruent) was found in the right TPJ. Specifically, this brain region was more strongly activated during numerical processing when the nonnumerical magnitudes were negatively correlated with numerosity (incongruent trials). In contrast, such an interference effect was not evident during nonnumerical processing when the task-irrelevant numerical magnitude was incongruent. In view of the role of the right TPJ in the control of stimulus-driven attention, we argue that these data demonstrate that the processing of nonnumerical magnitudes is more automatic than that of numerical magnitudes and that, therefore, the influence of numerical and nonnumerical variables on each other is asymmetrical.

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

The Approximate Number System Acuity Redefined: A Diffusion Model Approach

While all humans are capable of non-verbally representing numerical quantity using so-called the approximate number system (ANS), there exist considerable individual differences in its acuity. For example, in a non-symbolic number comparison task, some people find it easy to discriminate brief presentations of 14 dots from 16 dots while others do not. Quantifying individual ANS acuity from such a task has become an essential practice in the field, as individual differences in such a primitive number sense is thought to provide insights into individual differences in learned symbolic math abilities. However, the dominant method of characterizing ANS acuity-computing the Weber fraction (w)-only utilizes the accuracy data while ignoring response times (RT). Here, we offer a novel approach of quantifying ANS acuity by using the diffusion model, which accounts both accuracy and RT distributions. Specifically, the drift rate in the diffusion model, which indexes the quality of the stimulus information, is used to capture the precision of the internal quantity representation. Analysis of behavioral data shows that w is contaminated by speed-accuracy tradeoff, making it problematic as a measure of ANS acuity, while drift rate provides a measure more independent from speed-accuracy criterion settings. Furthermore, drift rate is a better predictor of symbolic math ability than w, suggesting a practical utility of the measure. These findings demonstrate critical limitations of the use of w and suggest clear advantages of using drift rate as a measure of primitive numerical competence.

Monkeys display classic signatures of human symbolic arithmetic

Non-human primates compare quantities in a crude manner, by approximating their values. Less is known about the mental transformations that non-humans can perform over approximate quantities, such as arithmetic transformations. There is evidence that human symbolic arithmetic has a deep psychological connection with the primitive, approximate forms of quantification of non-human animals. Here, we ask whether the subtle performance signatures that humans exhibit during symbolic arithmetic also bear a connection to primitive arithmetic. Specifically, we examined the problem size effect, the tie effect, and the practice effect-effects which are commonly observed in children's math performance in school. We show that, like humans, monkeys exhibited the problem size and tie effects, indicating commonalities in arithmetic algorithms with humans. Unlike humans, however, monkeys did not exhibit a practice effect. Together, these findings provide new evidence for a cognitive relation between non-symbolic and symbolic arithmetic.

The Multilingual CID-5: A New Tool to Study the Perception of Communicative Interactions in Different Languages

The investigation of the ability to perceive, recognize, and judge upon social intentions, such as communicative intentions, on the basis of body motion is a growing research area. Cross-cultural differences in ability to perceive and interpret biological motion, however, have been poorly investigated so far. Progress in this domain strongly depends on the availability of suitable stimulus material. In the present method paper, we describe the multilingual CID-5, an extension of the CID-5 database, allowing for the investigation of how non-conventional communicative gestures are classified and identified by speakers of different languages. The CID-5 database contains 14 communicative interactions and 7 non-communicative actions performed by couples of agents and presented as point-light displays. For each action, the database provides movie files with the point-light animation, text files with the 3-D spatial coordinates of the point-lights, and five different response alternatives. In the multilingual CID-5 the alternatives were translated into seven languages (Chinese, Dutch, English, French, German, Italian, and Polish). Preliminary data collected to assess the recognizability of the actions in the different languages suggest that, for most of the action stimuli, information presented in point-light displays is sufficient for the distinctive classification of the action as communicative vs. individual, as well as for identification of the specific communicative gesture performed by the actor in all the available languages.

Analog Magnitudes Support Large Number Ordinal Judgments in Infancy

Few studies have explored the source of infants' ordinal knowledge, and those that have are equivocal regarding the underlying representational system. The present study sought clear evidence that the approximate number system, which underlies children's cardinal knowledge, may also support ordinal knowledge in infancy; 10 - to 12-month-old infants' were tested with large sets (>3) in an ordinal choice task in which they were asked to choose between two hidden sets of food items. The difficulty of the comparison varied as a function of the ratio between the sets. Infants reliably chose the greater quantity when the sets differed by a 2:3 ratio (4v6 and 6v9), but not when they differed by a 3:4 ratio (6v8) or a 7:8 ratio (7v8). This discrimination function is consistent with previous studies testing the precision of number and time representations in infants of roughly this same age, thus providing evidence that the approximate number system can support ordinal judgments in infancy. The findings are discussed in light of recent proposals that different mechanisms underlie infants' reasoning about small and large numbers.

Children’s Approximate Number System in Haptic Modality

The approximate number system (ANS) is a primitive system used to estimate quantities. It can process quantities in visual and auditory modalities. The aim of the present study was to examine whether ANS can process quantities presented haptically. Moreover, to assess age-related changes, two groups of children (5- and 7-year-olds) were compared. In a newly designed haptic task, children compared two arrays of dots by touching them simultaneously using both hands, without seeing them, and for limited duration to prevent counting. Using Panamath, a frequently used visual ANS task, we verified that our population exhibited the typical pattern of approximation with visual arrays: Older children outperformed younger children, and an increased ratio between the two quantities to be compared led to more accurate responses. Performance in the haptic task revealed that children, in both age-groups, were able to haptically compare two quantities above chance level, with improved performance in older compared with younger children. Moreover, our results revealed a ratio effect, a well-known signature of the ANS. These findings suggest that haptic numerical discrimination in children is dictated by the ANS, and that ANS acuity measured with a haptic task improves with age, as commonly observed with the visual task.

Core geometry in perspective

Research on animals, infants, children, and adults provides evidence that distinct cognitive systems underlie navigation and object recognition. Here we examine whether and how these systems interact when children interpret 2D edge-based perspectival line drawings of scenes and objects. Such drawings serve as symbols early in development, and they preserve scene and object geometry from canonical points of view. Young children show limits when using geometry both in non-symbolic tasks and in symbolic map tasks that present 3D contexts from unusual, unfamiliar points of view. When presented with the familiar viewpoints in perspectival line drawings, however, do children engage more integrated geometric representations? In three experiments, children successfully interpreted line drawings with respect to their depicted scene or object. Nevertheless, children recruited distinct processes when navigating based on the information in these drawings, and these processes depended on the context in which the drawings were presented. These results suggest that children are flexible but limited in using geometric information to form integrated representations of scenes and objects, even when interpreting spatial symbols that are highly familiar and faithful renditions of the visual world.

Ratio dependence in small number discrimination is affected by the experimental procedure

Adults, infants and some non-human animals share an approximate number system (ANS) to estimate numerical quantities, and are supposed to share a second, 'object-tracking,' system (OTS) that supports the precise representation of a small number of items (up to 3 or 4). In relative numerosity judgments, accuracy depends on the ratio of the two numerosities (Weber's Law) for numerosities >4 (the typical ANS range), while for numerosities ≤4 (OTS range) there is usually no ratio effect. However, recent studies have found evidence for ratio effects for small numerosities, challenging the idea that the OTS might be involved for small number discrimination. Here we tested the hypothesis that the lack of ratio effect in the numbers 1-4 is largely dependent on the type of stimulus presentation. We investigated relative numerosity judgments in college students using three different procedures: a simultaneous presentation of intermingled and separate groups of dots in separate experiments, and a further experiment with sequential presentation. As predicted, in the large number range, ratio dependence was observed in all tasks. By contrast, in the small number range, ratio insensitivity was found in one task (sequential presentation). In a fourth experiment, we showed that the presence of intermingled distractors elicited a ratio effect, while easily distinguishable distractors did not. As the different ratio sensitivity for small and large numbers has been often interpreted in terms of the activation of the OTS and ANS, our results suggest that numbers 1-4 may be represented by both numerical systems and that the experimental context, such as the presence/absence of task-irrelevant items in the visual field, would determine which system is activated.

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.

"Zeroing" in on mathematics in the monkey brain

A new study documented that monkeys showed selective neuronal responding to the concept of zero during a numerical task, and that there were two distinct classes of neurons that coded the absence of stimuli either through a discrete activation pattern (zero or not zero) or a continuous one for which zero was integrated with other numerosities in the relative rate of activity. These data indicate that monkeys, like humans, have a concept of zero that is part of their analog number line but that also may have unique properties compared to other numerosities.

Stable measures of number sense accuracy in math learning disability: Is it time to proceed from basic science to clinical application?

Math learning disability (MLD) or developmental dyscalculia is a highly prevalent and persistent difficulty in learning arithmetic that may be explained by different cognitive mechanisms. The accuracy of the number sense has been implicated by some evidence as a core deficit in MLD. However, research on this topic has been mainly conducted in demographically selected samples, using arbitrary cut-off scores to characterize MLD. The clinical relevance of the association between number sense and MLD remains to be investigated. In this study, we aimed at assessing the stability of a number sense accuracy measure (w) across five experimental sessions, in two clinically defined cases of MLD. Stable measures of number sense accuracy estimate are required to clinically characterize subtypes of MLD and to make theoretical inferences regarding the underlying cognitive mechanisms. G. A. was a 10-year-old boy with MLD in the context of dyslexia and phonological processing impairment and his performance remained steadily in the typical scores range. The performance of H. V., a 9-year-old girl with MLD associated with number sense inaccuracy, remained consistently impaired across measurements, with a nonsignificant tendency to worsen. Qualitatively, H. V.'s performance was also characterized by greater variability across sessions. Concomitant clinical observations suggested that H. V.'s difficulties could be aggravated by developing symptoms of mathematics anxiety. Results in these two cases are in line with the hypotheses that at least two reliable patterns of cognitive impairment may underlie math learning difficulties in MLD, one related to number sense inaccuracy and the other to phonological processing impairment. Additionally, it indicates the need for more translational research in order to examine the usefulness and validity of theoretical advances in numerical cognition to the clinical neuropsychological practice with MLD.

Prefrontal Goal Codes Emerge as Latent States in Probabilistic Value Learning

The prefrontal cortex (PFC) supports goal-directed actions and exerts cognitive control over behavior, but the underlying coding and mechanism are heavily debated. We present evidence for the role of goal coding in PFC from two converging perspectives: computational modeling and neuronal-level analysis of monkey data. We show that neural representations of prospective goals emerge by combining a categorization process that extracts relevant behavioral abstractions from the input data and a reward-driven process that selects candidate categories depending on their adaptive value; both forms of learning have a plausible neural implementation in PFC. Our analyses demonstrate a fundamental principle: goal coding represents an efficient solution to cognitive control problems, analogous to efficient coding principles in other (e.g., visual) brain areas. The novel analytical-computational approach is of general interest because it applies to a variety of neurophysiological studies.