Approximate quantities and exact number words: dissociable systems

Origins of numbers: a shared language-of-thought for arithmetic and geometry?

Concepts of exact number are often thought to originate from counting and the successor function, or from a refinement of the approximate number system (ANS). We argue here for a third origin: a shared language-of-thought (LoT) for geometry and arithmetic that involves primitives of repetition, concatenation, and recursive embedding. Applied to sets, those primitives engender concepts of exact integers through recursive applications of additions and multiplications. Links between geometry and arithmetic also explain the emergence of higher-level notions (squares, primes, etc.). Under our hypothesis, understanding a number means having one or several mental expressions for it, and their minimal description length (MDL) determines how easily they can be mentally manipulated. Several historical, developmental, linguistic, and brain imaging phenomena provide preliminary support for our proposal.

The role of the intraparietal sulcus in numeracy: A review of parietal lesion cases

Prominent theories of numeracy link the intraparietal sulcus (IPS) to approximate representations of quantity that undergird basic math abilities. The goal of this review is to better understand the neural basis of mathematical cognition through the lens of acalculia, by identifying numeracy-focused single case studies of patients with parietal lesions and testing for causal relationships between numeracy impairments and the locus of parietal damage. A systematic literature review identified 27 single case studies with left parietal lesions and categorized administered tasks across four numeracy domains: Approximation, Calculation, Ordinality/Cardinality, and Transcoding. We compared published lesion images by drawing a sphere at the inferred center-of-mass and assigning each case to an anatomical group (IPS or Other Parietal damage) based on overlap with left IPS and original anatomical description. We performed Fisher's Exact Test to compare behavioral performance on each numeracy domain between the two groups. As an exploratory follow-up, we used Activation Likelihood Estimation (ALE) to identify sites of damage within parietal cortex preferentially associated with impairments in each domain. We found that Approximation impairments were significantly more frequent in the IPS group (p = .003). The exploratory ALE analysis revealed that only Approximation impairment cases significantly overlapped with the IPS, while impairments in other domains were localized to different regions of the parietal lobe. Based on the pattern of impairments shown across these cases, we conclude that damage to the left IPS is linked to impairments in approximation ability specifically. Our findings support theoretical claims linking IPS to magnitude representation, but do not provide evidence that IPS critically underpins performance across all numeracy tasks. Instead, our findings are more compatible with models of dissociable circuits of numerical processing within the parietal lobe.

The relationship of spatial visualization ability and number representation: evidence from multiple tasks

It is well established in the literature that the relation of spatial ability and the number representation, but the intrinsic relation of spatial visualization ability and number representation are not well understood. In the Current study, Chinese Preschool children (N = 200; 107 girls; Mage = 5.47years, SD = 0.67) completed two kinds of spatial visualization tasks and six kinds of number representation tasks. The results showed that spatial visualization ability was positively correlated with number representation for 5 years-old Children. Furthermore, spatial visualization ability can positively predicted number representation. These findings are consistent with the view that spatial visualization is recognized for its potential to enhance numerical skills, encompassing competencies associated with fundamental number. Therefore, developing children's spatial visualization ability may be an effective way to enhance their number representation skills. This has important implications for early education and intervention strategies.

Hierarchical representations of relative numerical magnitudes in the human frontoparietal cortex

The ability to estimate numerical magnitude is essential for decision-making and is thought to underlie arithmetic skills. In humans, neural populations in the frontoparietal regions are tuned to represent numerosity. However, it remains unclear whether their response properties are fixed to a specific numerosity (i.e., absolute code) or dynamically scaled according to the range of numerosities relevant to the context (i.e., relative code). Here, using functional magnetic resonance imaging combined with multivariate pattern analysis, we uncover evidence that representations of relative numerosity coding emerge gradually as visual information processing advances in the frontoparietal regions. In contrast, the early sensory areas predominantly exhibit absolute coding. These findings indicate a hierarchical organization of relative numerosity representations that adapt their response properties according to the context. Our results highlight the existence of a context-dependent optimization mechanism in numerosity representation, enabling the efficient processing of infinite magnitude information with finite neural resources.

The calculating brain

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

EEG alpha and theta time-frequency structure during a written mathematical task

Since the first electroencephalogram (EEG) was obtained, there have been many possibilities to use it as a tool to access brain cognitive dynamics. Mathematical (Math) problem solving is one of the most important cortical processes, but it is still far from being well understood. EEG is an inexpensive and simple indirect measure of brain operation, but only recently has low-cost equipment (mobile EEG) allowed sophisticated analyses in non-clinical settings. The main purpose of this work is to study EEG activation during a Math task in a realistic environment, using mobile EEG. A matching pursuit (MP)-based signal analysis technique was employed, since MP properties render it a priori suitable to study induced EEG activity over long time sequences, when it is not tightly locked to a given stimulus. The study sample comprised sixty healthy volunteers. Unlike the majority of previous studies, subjects were studied in a sitting position with their eyes open. They completed a written Math task outside the EEG lab, wearing a mobile EEG device (EPOC+). Theta [4 Hz-7.5 Hz], alpha (7.5 Hz-13 Hz] and 0.5 Hz micro-bands in the [0.5 Hz-20 Hz] range were studied with a low-density stochastic MP dictionary. Over 1-min windows, ongoing EEG alpha and theta activity was decomposed into numerous MP atoms with median duration around 3 s, similar to the duration of induced, time-locked activity obtained with event-related (des)synchronization (ERS/ERD) studies. Relative to Rest, there was lower right-side and posterior MP alpha atom/min during Math, whereas MP theta atom/min was significantly higher on anteriorly located electrodes, especially on the left side. MP alpha findings were particularly significant on a narrow range around 10 Hz-10.5 Hz, consistent with FFT alpha peak findings from ERS/ERD studies. With a streamlined protocol, these results replicate previous findings of EEG alpha and theta activation obtained during Math tasks with different signal analysis techniques and in different time frames. The efficient application to real-world, noisy EEG data with a low-resolution stochastic MP dictionary shows that this technique is very encouraging. These results provide support for studies of mathematical cognition with mobile EEG and matching pursuit.

From vision to cognition: potential contributions of cerebral visual impairment to neurodevelopmental disorders

Vision has a crucial role to play in human development and functioning. It is, therefore, not surprising that vision plays a fundamental role in the development of the child. As a consequence, an alteration in visual function is, therefore, likely to hinder the child's development. Although ocular disorders are well known, diagnosed and taken into account, cerebral visual impairments (CVI) resulting from post-chiasmatic damage are largely underdiagnosed. However, among the disorders resulting from an episode of perinatal asphyxia and/or associated with prematurity, or neonatal hypoglycaemia, CVIs are prominent. In this article, we focus on the role of the possible effects of CVI on a child's learning abilities, leading to major difficulty in disentangling the consequences of CVI from other neurodevelopmental disorders (NDD) such as dyslexia, dyscalculia, dysgraphia, attention-deficit/hyperactivity disorder (ADHD), developmental coordination disorder (DCD) and autism spectrum disorders (ASD). Although we focus here on the possible overlap between children with CVI and children with other NDD, De Witt et al. (Wit et al. Ear Hear 39:1-19, 2018) have raised exactly the same question regarding children with auditory processing disorders (the equivalent of CVI in the auditory modality). We underline how motor, social and cognitive development as well as academic success can be impaired by CVI and raise the question of the need for systematic evaluation for disorders of vision, visual perception and cognition in all children presenting with a NDD and/or previously born under adverse neurological conditions.

Omit needless words: Sentence length perception

Short sentences improve readability. Short sentences also promote social justice through accessibility and inclusiveness. Despite this, much remains unknown about sentence length perception-an important factor in producing readable writing. Accordingly, we conducted a psychophysical study using procedures from Signal Detection Theory to examine sentence length perception in naive adults. Participants viewed real-world full-page text samples and judged whether a bolded target sentence contained more or fewer than 17 words. The experiment yielded four findings. First, naïve adults perceived sentence length in real-world text samples quickly (median = 300-400 ms) and precisely (median = ~90% correct). Second, flipping real-world text samples upside-down generated no reaction-time cost and nearly no loss in the precision of sentence length perception. This differs from the large inversion effects that characterize other highly practiced, real-world perceptual tasks involving canonically oriented stimuli, most notably face perception and reading. Third, participants significantly underestimated the length of mirror-reversed sentences-but not upside-down, nor standard sentences. This finding parallels participants' familiarity with commonly occurring left-justified right-ragged text, and suggests a novel demonstration of left-lateralized anchoring in scene syntax. Fourth, error patterns demonstrated that participants achieved their high speed, high precision sentence-length judgments by heuristically counting text lines, not by explicitly counting words. This suggests practical advice for writing instructors to offer students. When copy editing, students can quickly and precisely identify their long sentences via a line-counting heuristic, e.g., "a 17-word sentence spans about 1.5 text lines". Students can subsequently improve a long sentence's readability and inclusiveness by omitting needless words.

Estimation Strategy Utilization Is Modulated by Implicit Emotion Regulation: Evidence from Behavioral and Event-Related Potentials Studies

A large number of studies have studied the influence of emotional experience on an individual's estimation performance, but the influence of implicit emotion regulation is still unknown. Participants were asked to complete the following tasks in order: idiom matching task, multiplication computational estimation task (MCE task), gender judgment task (GJ task), and emotional experience intensity assessment task. The words matching task was adopted to achieve the purpose of implicit emotion regulation (implicit reappraisal and implicit suppression). Behavioral results showed that implicit reappraisal and implicit suppression equally contributed to improving an individual's estimation speed (but not ACC (accuracy)). The MCE task related ERP (event-related potential) results showed that the influence of implicit emotion regulation on estimation consisted of two phases. In the first phase (encoding phase), implicit reappraisal both enhanced (larger P1 amplitudes) and weakened (smaller N170 amplitudes) an individual's encoding sensitivity, while implicit suppression enhanced an individual's encoding sensitivity (larger P1 amplitudes). In the second phase (estimation strategies retrieval phase), implicit reappraisal (but not implicit suppression) cost more attention resources (larger LPC2 and LPC3 amplitudes). The present study suggested that both implicit reappraisal and implicit suppression contributed to improving an individual's estimation performance, and the regulation effect of implicit suppression (vs. implicit reappraisal) was better.

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

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

Neural Correlates of Numerical Estimation: The Role of Strategy Use

Introduction: Computation estimation is the ability to provide an approximate answer to a complex arithmetic problem without calculating it exactly. Despite its importance in daily life, the neuronal network underlying computation estimation is largely unknown. Methods: We looked at the neuronal correlates of two computational estimation strategies: approximated calculation and sense of magnitude (SOM)–intuitive representation of magnitude, without calculation. During an fMRI scan, thirty-one college students judged whether the result of a two-digit multiplication problem was larger or smaller than a given reference number. In two different blocks, they were asked to use a specific strategy (AC or SOM). Results: The two strategies activated brain regions related to calculation, numerical cognition, decision-making, and working memory. AC more than SOM elicited activations in multiple, domain-specific brain regions in the parietal lobule, including the left SMG (BA 40), the bilateral superior parietal lobule (BA 7), and the right inferior parietal lobule (BA 7). The activation level of the IFG was positively correlated to individual accuracy, indicating that the IFG has an essential role in both strategies. Conclusions: These finding suggest that the analogic code of magnitude is more involved in the AC than the SOM strategy.

Constructing rationals through conjoint measurement of numerator and denominator as approximate integer magnitudes in tradeoff relations

To investigate mechanisms of rational representation, I consider (1) construction of an ordered continuum of psychophysical scale of magnitude of sensation; (2) counting mechanism leading to an approximate numerosity scale for integers; and (3) conjoint measurement structure pitting the denominator against the numerator in tradeoff positions. Number sense of resulting rationals is neither intuitive nor expedient in their manipulation.

Children With Dyscalculia Show Hippocampal Hyperactivity During Symbolic Number Perception

Dyscalculia is a learning disability affecting the acquisition of arithmetical skills in children with normal intelligence and age-appropriate education. Two hypotheses attempt to explain the main cause of dyscalculia. The first hypothesis suggests that a problem with the core mechanisms of perceiving (non-symbolic) quantities is the cause of dyscalculia (core deficit hypothesis), while the alternative hypothesis suggests that dyscalculics have problems only with the processing of numerical symbols (access deficit hypothesis). In the present study, the symbolic and non-symbolic numerosity processing of typically developing children and children with dyscalculia were examined with functional magnetic resonance imaging (fMRI). Control (n = 15, mean age: 11.26) and dyscalculia (n = 12, mean age: 11.25) groups were determined using a wide-scale screening process. Participants performed a quantity comparison paradigm in the fMRI with two number conditions (dot and symbol comparison) and two difficulty levels (0.5 and 0.7 ratio). The results showed that the bilateral intraparietal sulcus (IPS), left dorsolateral prefrontal cortex (DLPFC) and left fusiform gyrus (so-called “number form area”) were activated for number perception as well as bilateral occipital and supplementary motor areas. The task difficulty engaged bilateral insular cortex, anterior cingulate cortex, IPS, and DLPFC activation. The dyscalculia group showed more activation in the left orbitofrontal cortex, left medial prefrontal cortex, and right anterior cingulate cortex than the control group. The dyscalculia group showed left hippocampus activation specifically for the symbolic condition. Increased left hippocampal and left-lateralized frontal network activation suggest increased executive and memory-based compensation mechanisms during symbolic processing for dyscalculics. Overall, our findings support the access deficit hypothesis as a neural basis for dyscalculia.

The Evolutionary History of Brains for Numbers

Humans and other animals share a number sense', an intuitive understanding of countable quantities. Having evolved independent from one another for hundreds of millions of years, the brains of these diverse species, including monkeys, crows, zebrafishes, bees, and squids, differ radically. However, in all vertebrates investigated, the pallium of the telencephalon has been implicated in number processing. This suggests that properties of the telencephalon make it ideally suited to host number representations that evolved by convergent evolution as a result of common selection pressures. In addition, promising candidate regions in the brains of invertebrates, such as insects, spiders, and cephalopods, can be identified, opening the possibility of even deeper commonalities for number sense.

The Number Sense Represents (Rational) Numbers

On a now orthodox view, humans and many other animals possess a "number sense," or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique-the arguments from congruency, confounds, and imprecision-and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical confounds or exotic substitutes for number, such as "numerosities" or "quanticals," as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind(s) of number being represented. In response, we propose that the ANS represents not only natural numbers (e.g. 7), but also non-natural rational numbers (e.g. 3.5). It does not represent irrational numbers (e.g. √2), however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research.

Neuroethology of number sense across the animal kingdom

Many species from diverse and often distantly related animal groups (e.g. monkeys, crows, fish and bees) have a sense of number. This means that they can assess the number of items in a set - its 'numerosity'. The brains of these phylogenetically distant species are markedly diverse. This Review examines the fundamentally different types of brains and neural mechanisms that give rise to numerical competence across the animal tree of life. Neural correlates of the number sense so far exist only for specific vertebrate species: the richest data concerning explicit and abstract number representations have been collected from the cerebral cortex of mammals, most notably human and nonhuman primates, but also from the pallium of corvid songbirds, which evolved independently of the mammalian cortex. In contrast, the neural data relating to implicit and reflexive numerical representations in amphibians and fish is limited. The neural basis of a number sense has not been explored in any protostome so far. However, promising candidate regions in the brains of insects, spiders and cephalopods - all of which are known to have number skills - are identified in this Review. A comparative neuroscientific approach will be indispensable for identifying evolutionarily stable neuronal circuits and deciphering codes that give rise to a sense of number across phylogeny.

Comprehension of computer code relies primarily on domain-general executive brain regions

Computer programming is a novel cognitive tool that has transformed modern society. What cognitive and neural mechanisms support this skill? Here, we used functional magnetic resonance imaging to investigate two candidate brain systems: the multiple demand (MD) system, typically recruited during math, logic, problem solving, and executive tasks, and the language system, typically recruited during linguistic processing. We examined MD and language system responses to code written in Python, a text-based programming language (Experiment 1) and in ScratchJr, a graphical programming language (Experiment 2); for both, we contrasted responses to code problems with responses to content-matched sentence problems. We found that the MD system exhibited strong bilateral responses to code in both experiments, whereas the language system responded strongly to sentence problems, but weakly or not at all to code problems. Thus, the MD system supports the use of novel cognitive tools even when the input is structurally similar to natural language.

Meaning to multiply: Electrophysiological evidence that children and adults treat multiplication facts differently

Multiplication tables are typically memorized verbally, with fluent retrieval leading to better performance in advanced math. Arithmetic development is characterized by strategy shifts from procedural operations to direct fact retrieval, which would not necessitate access to the facts' conceptual meaning. This study tested this hypothesis using a combination of event related brain potentials (ERP) and behavioral measures with 3rd-5th grade children and young adults. Participants verified the solutions to simple multiplication problems (2 × 3 = 6 or = 7) and the semantic fit of word-picture pairs, separately. Children showed an N400 effect to multiplication solutions with larger (more negative) amplitude for incorrect than correct solutions, reflecting meaning-level processing. A similar ERP response was observed in the word-picture verification task, with larger negative amplitude for word-picture pairs that were semantically mismatched compared to matched. In contrast, adults showed a P300 response for correct solutions, suggesting that they treated these solutions as potential targets in over-rehearsed mathematical expressions. This P300 response was specific to math fact processing, as the word-picture verification task elicited a classic N400 in adults. These ERP findings reveal an overlooked developmental transition that occurs after fifth grade, and speak to theories of arithmetic that have been based primarily on adult data.

EEG correlation during the solving of simple and complex logical-mathematical problems

Solving logical-mathematical word problems is a complex task that requires numerous cognitive operations, including comprehension, reasoning, and calculation. These abilities have been associated with activation of the parietal, temporal, and prefrontal cortices. It has been suggested that the reasoning involved in solving logical-mathematical problems requires the coordinated functionality of all these cortical areas. In this study was evaluated the activation and electroencephalographic (EEG) correlation of the prefrontal, temporal, and parietal regions in young men while solving logical-mathematical word problems with two degrees of difficulty: simple and complex. During the solving of complex problems, higher absolute power and EEG correlation of the alpha and fast bands between the left frontal and parietal cortices were observed. A temporal deactivation and functional decoupling of the right parietal-temporal cortices also were obtained. Solving complex problems probably require activation of a left prefrontal-parietal circuit to maintain and manipulate multiple pieces of information. The temporal deactivation and decreased parietal-temporal correlation could be associated to text processing and suppression of the content-dependent reasoning to focus cognitive resources on the mathematical reasoning. Together, these findings support a pivotal role for the left prefrontal and parietal cortices in mathematical reasoning and of the temporal regions in text processing required to understand and solve written mathematical problems.

Damage to the Intraparietal Sulcus Impairs Magnitude Representations of Results of Complex Arithmetic Problems

Past research investigating the role of the intraparietal sulcus (IPS) in numerical processes focused mainly on quantity and numerical comparisons as well on single digit arithmetic. The present study investigates the involvement of the IPS in estimating the results of multi-digit multiplication problems. For this purpose, the performance a 24-year-old female (JD) with brain damage in the left IPS was compared to an age-matched control group in the computation estimation task. When required to estimate whether the results of multi-digit multiplication problems are smaller or larger than given reference numbers, JD, in contrast to controls, did not show the common patterns of distance and size effects. Her strategy use was also atypical. Most control participants used both the approximated calculation strategy that involves rounding and calculation procedures and the sense of magnitude strategy that relies on an intuitive approximated magnitude representation of the results. In contrast, JD used only the former but not the latter strategy. Together, these findings suggest that the damage to the IPS impaired JD's representations of magnitude that play an important role in this computation estimation task.

Language-specific numerical estimation in bilingual children

We tested 5- to 7-year-old bilingual learners of French and English (N = 91) to investigate how language-specific knowledge of verbal numerals affects numerical estimation. Participants made verbal estimates for rapidly presented random dot arrays in each of their two languages. Estimation accuracy differed across children's two languages, an effect that remained when controlling for children's familiarity with number words across their two languages. In addition, children's estimates were equivalently well ordered in their two languages, suggesting that differences in accuracy were due to how children represented the relative distance between number words in each language. Overall, these results suggest that bilingual children have different mappings between their verbal and nonverbal counting systems across their two languages and that those differences in mappings are likely driven by an asymmetry in their knowledge of the structure of the count list across their languages. Implications for bilingual math education are discussed.

Detection of arithmetic violations during sleep

Can the sleeping brain develop predictions of future auditory stimuli? Past research demonstrated disrupted prediction capabilities during sleep in the context of novel, arbitrary auditory sequences, but the availability of overlearned knowledge already stored in long-term memory could still be preserved. We tested the sleeping brain capabilities to detect violations of simple arithmetic facts. Sleeping participants were presented with spoken arithmetic facts such as "two plus two is nine" and brain responses to correct or incorrect results were recorded in electro and magneto-encephalography. Sleep responses were compared to both attentive and inattentive wakefulness. During attentive wakefulness, arithmetic violations elicited a succession of N400 and P600 effects, whereas no such activations could be recorded in sleep or in inattentive wakefulness. Still, small but significant effects remained in sleep, advocating for a preserved but partial accessibility to arithmetic facts stored in long-term memory and preserved predictions of low-level and already learned knowledge. Those effects were very different from residual activities seen in inattention, highlighting the differences of information processing between the sleeping and the inattentive brain.

Developmental alterations of the numerical processing networks in the brain

Neuroimaging studies revealed that number perception is mainly located in parietal cortex. Although controversial, it was suggested that number is processed in the frontal lobe in childhood and in the parietal cortex in adulthood. The purpose of this study is to investigate developmental differences in the neural correlates of number representation with fMRI. Sixteen healthy young adults (age:21.69 ± 0.79) and 15 healthy children (age:11.87 ± 0.52) performed a numerosity comparison paradigm which consists of two numerical conditions with two difficulty levels. Adults showed broad parietal cortex activation, as well as activation in the inferior parietal lobes, dorsolateral and medial prefrontal cortex, anterior and posterior cingulate cortex, and peristriate cortex (PC) during number processing. Children showed activations in the intraparietal sulcus and PC. Group differences were observed in the posterior insula, fusiform gyrus, and PC whose coordinates correspond to the number form area (NFA). Region of interest analysis was performed for these clusters to get the time series of hemodynamic responses which were estimated with a finite impulse response function. In contrast to the prominent frontoparietal shift theory, no age-related differences were observed in the frontoparietal regions. Overall, the presented study suggests developmental changes in the brain's number processing revolving around the NFA.

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

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

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

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Our findings generally agree with TCM predictions.

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

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

Neural constraints on human number concepts

True counting and arithmetic abilities are unique to humans and are inextricably linked to symbolic competence. However, our unprecedented numerical skills are deeply rooted in our neuronal heritage as primates and vertebrates. In this article, I argue that numerical competence in humans is the result of three neural constraints. First, I propose that the neuronal mechanisms of quantity estimation are part of our evolutionary heritage and can be witnessed across primate and vertebrate phylogeny. Second, I suggest that a basic understanding of number, what numerical quantity means, is innately wired into the brain and gives rise to an intuitive number sense, or number instinct. Third and finally, I argue that symbolic counting and arithmetic in humans is rooted in an evolutionarily and ontogenetically primeval neural system for non-symbolic number representations. These three neural constraints jointly determine the basic processing of number concepts in the human mind.

Mental Images and School Learning: A Longitudinal Study on Children

Recent literature have underlined the connections between children's reading skills and capacity to create and use mental representations or mental images; furthermore data highlighted the involvement of visuospatial abilities both during math learning and during subsequent developmental phases in performing math tasks. The present research adopted a longitudinal design to assess whether the processes of mental imagery in preschoolers (ages 4-5 years) are predictive of mathematics skills, writing and reading, in the early years of primary school (ages 6-7 years). The research lasted for two school years; in the first phase, the general group of participants consisted of 100 children, and although all participants agreed to be part of the research, in the second phase, there was a mortality rate of 30%. In order to measure school learning and mental imagery processes four batteries of tests were used. The mental imagery battery evaluated mental generation, inspection and transformation processes. Data underlined that the different aspects in which mental imagery processes are articulated are differently implied in some skills that constitute school learning. These findings emphasize the potential usefulness of a screening for mental imagery ability for schoolchildren to adopt effective measures to increase their mental imagery abilities.

Language and Math: What If We Have Two Separate Naming Systems?

The role of language in numerical processing has traditionally been restricted to counting and exact arithmetic. Nevertheless, the impact that each of a bilinguals’ languages may have in core numerical representations has not been questioned until recently. What if the language in which math has been first acquired (LLmath) had a bigger impact in our math processing? Based on previous studies on language switching we hypothesize that balanced bilinguals would behave like unbalanced bilinguals when switching between the two codes for math. In order to address this question, we measured the brain activity with magneto encephalography (MEG) and source estimation analyses of 12 balanced Basque-Spanish speakers performing a task in which participants were unconscious of the switches between the two codes. The results show an asymmetric switch cost between the two codes for math, and that the brain areas responsible for these switches are similar to those thought to belong to a general task switching mechanism. This implies that the dominances for math and language could run separately from the general language dominance.

Numerical processing profiles in children with varying degrees of arithmetical achievement

Recent studies show basic cognitive abilities such as the rapid and precise apprehension of small numerosities in object sets ("subitizing"), verbal counting and numerical magnitude comparison significantly influence the acquisition of arithmetic and continues to modulate more advanced stages of mathematical cognition. Additionally, children with low arithmetic achievement (LAA) and Developmental Dyscalculia (DD) exhibit significant deficits in these cognitive processes. Nevertheless, the different cognitive profiles of children with varying degrees of numerical and arithmetic processing deficits have not been sufficiently characterized, despite its potential relevance to the stimulation of numerical cognition and the design of appropriate intervention strategies. Here, the cognitive profiles of groups of typically developing children, children with low arithmetical achievement and DD, exhibiting typical and atypical subitizing ability were contrasted. The results suggest that relatively independent neurocognitive mechanisms may produce distinct profiles at the behavioral level and suggest children with low arithmetic performance exhibiting atypical subitizing abilities are not only significantly slower, but rely on compensatory mechanisms and strategies compared to typical subitizers. The role of subitizing as a correlate of arithmetic fluency is revised in the light of the present findings.

Where words meet numbers: Comprehension of measurement unit terms in posterior cortical atrophy

Units of measurement (e.g., metre, week, gram) are critically important concepts in everyday life. Little is known about how knowledge of units is represented in the brain or how this relates to other forms of semantic knowledge. As unit terms are intimately connected with numerical quantity, we might expect knowledge for these concepts to be supported by parietally-mediated representations of space, time and magnitude. We investigated knowledge for measurement units in patients with posterior cortical atrophy (PCA), who display profound impairments of spatial and numerical cognition associated with occipital and parietal lobe atrophy. Relative to healthy controls, PCA patients displayed impairments for a range of unit-based knowledge, including the ability to specify the dimension which a unit refers to (e.g., grams measure mass), to select the appropriate units to measure everyday quantities (e.g., grams for sugar) and to determine the relative magnitudes of different unit terms (e.g., gram is smaller than kilogram). In most cases, their performance was also significantly poorer than a patient control group diagnosed with typical Alzheimer's disease. Our results suggest that impairment to systems that code numerical and spatial magnitudes has an effect on non-numerical verbal knowledge for measurement units. Units of measurement appear to lie at the intersection of the brain's verbal and numerical semantic systems, making them a critical class of concepts in which to investigate how magnitude-based codes contribute to verbal semantic representation.

Tactile enumeration: A case study of acalculia

Enumeration is one of the building blocks of arithmetic and fingers are used as a counting tool in early steps. Subitizing-fast and accurate enumeration of small quantities-has been vastly studied in the visual modality, but less in the tactile modality. We explored tactile enumeration using fingers, and gray matter (GM) changes using voxel-based morphometry (VBM), in acalculia. We examined JD, a 22-year-old female with acalculia following a stroke to the left inferior parietal cortex. JD and a neurologically healthy normal comparison (NC) group reported how many fingers were stimulated. JD was tested at several time points, including at acute and chronic phases. Using the sensory intact hand for tactile enumeration, JD showed deficit in the acute phase, compared to the NC group, and improvement in the chronic phase of (1) the RT slope of enumerating up to four stimuli, (2) enumerating neighboring fingers, and (3) arithmetic fluency performance. Moreover, VBM analysis showed a larger GM volume for JD relative to the NC group in the right middle occipital cortex, most profoundly in the chronic phase. JD's performance serves as a first glance of tactile enumeration in acalculia. Pattern-recognition-based results support the suggestion of subitizing being the enumeration process when using one hand. Moreover, the increase in GM in the occipital cortex lays the groundwork for studying the innate and primitive ability to perceive and evaluate sizes or amounts-"sense of magnitude"- as a multisensory magnitude area and as part of a recovery path for deficits in basic numerical abilities.

Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks

Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Reassessing lateralization in calculation

The role of the left hemisphere in calculation has been unequivocally demonstrated in numerous studies in the last decades. The right hemisphere, on the other hand, had been traditionally considered subsidiary to the left hemisphere functions, although its role was less clearly defined. Recent clinical studies as well as investigations conducted with other methodologies (e.g. neuroimaging, transcranial magnetic stimulation and direct cortical electro-stimulation) leave several unanswered questions about the contribution of the right hemisphere in calculation. In particular, novel clinical studies show that right hemisphere acalculia encompasses a wide variety of symptoms, affecting even simple calculation, which cannot be easily attributed to spatial disorders or to a generic difficulty effect as previously believed. The studies reported here also show how the right hemisphere has its own specific role and that only a bilateral orchestration between the respective functions of each hemisphere guarantees, in fact, precise calculation. Vis-à-vis these data, the traditional wisdom that attributes to the right hemisphere a role mostly confined to spatial aspects of calculation needs to be significantly reshaped. The question for the future is whether it is possible to precisely define the specific contribution of the right hemisphere in several aspects of calculation while highlighting the nature of the cross-talk between the two hemispheres.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

Spatial Arrangement and Set Size Influence the Coding of Non-symbolic Quantities in the Intraparietal Sulcus

Performance in visual quantification tasks shows two characteristic patterns as a function of set size. A precise subitizing process for small sets (up to four) was contrasted with an approximate estimation process for larger sets. The spatial arrangement of elements in a set also influences visual quantification performance, with frequently perceived arrangements (e.g., dice patterns) being faster enumerated than random arrangements. Neuropsychological and imaging studies identified the intraparietal sulcus (IPS), as key brain area for quantification, both within and above the subitizing range. However, it is not yet clear if and how set size and spatial arrangement of elements in a set modulate IPS activity during quantification. In an fMRI study, participants enumerated briefly presented dot patterns with random, canonical or dice arrangement within and above the subitizing range. We evaluated how activity amplitude and pattern in the IPS were influenced by size and spatial arrangement of a set. We found a discontinuity in the amplitude of IPS response between subitizing and estimation range, with steep activity increase for sets exceeding four elements. In the estimation range, random dot arrangements elicited stronger IPS response than canonical arrangements which in turn elicited stronger response than dice arrangements. Furthermore, IPS activity patterns differed systematically between arrangements. We found a signature in the IPS response for a transition between subitizing and estimation processes during quantification. Differences in amplitude and pattern of IPS activity for different spatial arrangements indicated a more precise representation of non-symbolic numerical magnitude for dice and canonical than for random arrangements. These findings challenge the idea of an abstract coding of numerosity in the IPS even within a single notation.

Evaluation of the Triple Code Model of numerical processing-Reviewing past neuroimaging and clinical findings

This review reconciles past findings on numerical processing with key assumptions of the most predominant model of arithmetic in the literature, the Triple Code Model (TCM). This is implemented by reporting diverse findings in the literature ranging from behavioral studies on basic arithmetic operations over neuroimaging studies on numerical processing to developmental studies concerned with arithmetic acquisition, with a special focus on developmental dyscalculia (DD). We evaluate whether these studies corroborate the model and discuss possible reasons for contradictory findings. A separate section is dedicated to the transfer of TCM to arithmetic development and to alternative accounts focusing on developmental questions of numerical processing. We conclude with recommendations for future directions of arithmetic research, raising questions that require answers in models of healthy as well as abnormal mathematical development.

WHAT THIS PAPER ADDS

This review assesses the leading model in the field of arithmetic processing (Triple Code Model) by presenting knowledge from interdisciplinary research. It assesses the observed contradictory findings and integrates the resulting opposing viewpoints. The focus is on the development of arithmetic expertise as well as abnormal mathematical development. The original aspect of this article is that it points to a gap in research on these topics and provides possible solutions for future models.

Visual and Imagery Magnitude Comparisons Are Affected Following Left Parietal Lesion

We describe Jane Dow (JD), a young right-handed female with acalculia following a cerebral infarction in the left intraparietal sulcus. We investigated automatic processing of different types of magnitudes that were presented visually or through imagery. We employed the size congruity task and the mental clock task that differ in stimuli presentation and in working memory load. In the size congruity task, for physical comparisons, JD presented a lack of facilitation effect, suggesting a deficit in the automatic processing of numerical values. In the mental clock task, JD performed as accurate as controls did but much slower. In both tasks, JD presented a steeper distance effect compared to controls, suggesting a deficit in a domain-general comparison process. Our findings present an atypical pattern of magnitude processing following a left parietal lesion that appears not only for visually presented stimuli but also for imagery-based magnitudes. These finding support recent theories suggesting different types of magnitudes are interconnected with each other.

Mental arithmetic in the bilingual brain: Language matters

How do bilinguals solve arithmetic problems in each of their languages? We investigated this question by exploring the neural substrates of mental arithmetic in bilinguals. Critically, our population was composed of a homogeneous group of adults who were fluent in both of their instruction languages (i.e., German as first instruction language and French as second instruction language). Twenty bilinguals were scanned with fMRI (3T) while performing mental arithmetic. Both simple and complex problems were presented to disentangle memory retrieval occuring in very simple problems from arithmetic computation occuring in more complex problems. In simple additions, the left temporal regions were more activated in German than in French, whereas no brain regions showed additional activity in the reverse constrast. Complex additions revealed the reverse pattern, since the activations of regions for French surpassed the same computations in German and the extra regions were located predominantly in occipital regions. Our results thus highlight that highly proficient bilinguals rely on differential activation patterns to solve simple and complex additions in each of their languages, suggesting different solving procedures. The present study confirms the critical role of language in arithmetic problem solving and provides novel insights into how highly proficient bilinguals solve arithmetic problems.

Simultanagnosia and object individuation

Simultanagnosic patients have difficulty in perceiving multiple objects when presented simultaneously. In this review article, I discuss how neuropsychological research on simultanagnosia has been inspirational for two interconnected lines of research related to the core mechanisms by which the visual system processes cluttered scenes. First, I review previous studies on enumeration tasks indicating that, despite their inability to identify multiple objects, simultanagnosic patients can enumerate up to 2-3 elements as efficiently as healthy individuals (the so-called "subitizing" phenomenon). This intriguing observation is one of the first results to support the existence of an "object individuation" mechanism that can spatially tag a limited set of objects simultaneously, and resonates with recent research on the brain dynamics of enumeration in healthy individuals. Second, I further develop the implications of the dissociation between object identification and object enumeration in simultanagnosia specifically for the distinction between object identification and individuation. The latter distinction has been the subject of recent neuroimaging research that has provided fine-grained information on the spatial as well as temporal aspects of object individuation and recognition. The lessons learned from neuropsychological research on exact enumeration in simultanagnosia can be generalized to the normal functioning of the human mind, and have provided insightful clues for cognitive neuroscience.

Possible Associations between Subitizing, Estimation and Visuospatial Working Memory (VSWM) in Children

Researchers have focused on identifying the mechanisms involved in subitizing and its differences with estimation. Some suggest that subitizing relies on a visual indexing system in charge of the simultaneous individuation of objects that is also used by visuospatial working memory (VSWM). In adults, studies found associations between subitizing and VSWM, in the absence of correlation between VSWM and estimation. The present study analyzed the performance of 120 4 and 6-year-old children in three tasks: dot enumeration to measure subitizing capacity, quantity discrimination for estimation, and Corsi Block-tapping task for VSWM. In the enumeration task RTs (F(9, 1062)=720.59, MSE=734394, p<.001, η2=.86) and errors (F(9, 1062)=42.15, MSE=.194, p<.001, η2=.26.) increased with the array, but this growth was statistically significant only from 4 dots onward. Each subject's subitizing range was estimated by fitting RTs with a sigmoid function of number of dots and obtaining the bend point of the curve. Data fit (age 4: R 2 = .88; SD = .08; age 6: R 2 = .91, SD = .08) showed a mean subitizing range of 2.79 (SD = .66) for 4 year-olds and of 3.11 (SD = .64) for 6 year-olds. Subitizing ranges and average RTs showed low association with storage (r = .274; p < .05; r = -.398; p < .001) and average RTs with concurrent processing (r = -.412; p < .001) in VSWM. Subitizing range and speed showed no association with estimation speed and a poor association with accuracy (r = .234, p < .01; r = -.398, p < .001), which suggests independent systems for small and large quantities. Subitizing and estimation measures correlated with VSWM (p < .01), which suggests that both processes may require VSWM resources.

Individual differences in the learning potential of human beings

To the best of our knowledge, the genetic foundations that guide human brain development have not changed fundamentally during the past 50,000 years. However, because of their cognitive potential, humans have changed the world tremendously in the past centuries. They have invented technical devices, institutions that regulate cooperation and competition, and symbol systems, such as script and mathematics, that serve as reasoning tools. The exceptional learning ability of humans allows newborns to adapt to the world they are born into; however, there are tremendous individual differences in learning ability among humans that become obvious in school at the latest. Cognitive psychology has developed models of memory and information processing that attempt to explain how humans learn (general perspective), while the variation among individuals (differential perspective) has been the focus of psychometric intelligence research. Although both lines of research have been proceeding independently, they increasingly converge, as both investigate the concepts of working memory and knowledge construction. This review begins with presenting state-of-the-art research on human information processing and its potential in academic learning. Then, a brief overview of the history of psychometric intelligence research is combined with presenting recent work on the role of intelligence in modern societies and on the nature-nurture debate. Finally, promising approaches to integrating the general and differential perspective will be discussed in the conclusion of this review.

Age-Related Differences of Individuals' Arithmetic Strategy Utilization with Different Level of Math Anxiety

The present study used the choice/no-choice method to investigate the effect of math anxiety on the strategy used in computational estimation and mental arithmetic tasks and to examine age-related differences in this regard. Fifty-seven fourth graders, 56 sixth graders, and 60 adults were randomly selected to participate in the experiment. Results showed the following: (1) High-anxious individuals were more likely to use a rounding-down strategy in the computational estimation task under the best-choice condition. Additionally, sixth-grade students and adults performed faster than fourth-grade students on the strategy execution parameter. Math anxiety affected response times (RTs) and the accuracy with which strategies were executed. (2) The execution of the partial-decomposition strategy was superior to that of the full-decomposition strategy on the mental arithmetic task. Low-math-anxious persons provided more accurate answers than did high-math-anxious participants under the no-choice condition. This difference was significant for sixth graders. With regard to the strategy selection parameter, the RTs for strategy selection varied with age.

Dissociation between exact and approximate addition in developmental dyslexia

Previous research has suggested that number sense and language are involved in number representation and calculation, in which number sense supports approximate arithmetic, and language permits exact enumeration and calculation. Meanwhile, individuals with dyslexia have a core deficit in phonological processing. Based on these findings, we thus hypothesized that children with dyslexia may exhibit exact calculation impairment while doing mental arithmetic. The reaction time and accuracy while doing exact and approximate addition with symbolic Arabic digits and non-symbolic visual arrays of dots were compared between typically developing children and children with dyslexia. Reaction time analyses did not reveal any differences across two groups of children, the accuracies, interestingly, revealed a distinction of approximation and exact addition across two groups of children. Specifically, two groups of children had no differences in approximation. Children with dyslexia, however, had significantly lower accuracy in exact addition in both symbolic and non-symbolic tasks than that of typically developing children. Moreover, linguistic performances were selectively associated with exact calculation across individuals. These results suggested that children with dyslexia have a mental arithmetic deficit specifically in the realm of exact calculation, while their approximation ability is relatively intact.

The role of physical digit representation and numerical magnitude representation in children's multiplication fact retrieval

Arithmetic facts, in particular multiplication tables, are thought to be stored in long-term memory and to be interference prone. At least two representations underpinning these arithmetic facts have been suggested: a physical representation of the digits and a numerical magnitude representation. We hypothesized that both representations are possible sources of interference that could explain individual differences in multiplication fact performance and/or in strategy use. We investigated the specificity of these interferences on arithmetic fact retrieval and explored the relation between interference and performance on the different arithmetic operations and on general mathematics achievement. Participants were 79 fourth-grade children (M_age=9.6 years) who completed a products comparison and a multiplication production task with verbal strategy reports. Performances on a speeded calculation test including the four operations and on a general mathematics achievement test were also collected. Only the interference coming from physical representations was a significant predictor of the performance across multiplications. However, both the magnitude and physical representations were unique predictors of individual differences in multiplication. The frequency of the retrieval strategy across multiplication problems and across individuals was determined only by the physical representation, which therefore is suggested as being responsible for memory storage issues. Interestingly, this impact of physical representation was not observed when predicting performance on subtraction or on general mathematical achievement. In contrast, the impact of the numerical magnitude representation was more general in that it was observed across all arithmetic operations and in general mathematics achievement.

Similarity interference in learning and retrieving arithmetic facts

Storing the solution of simple calculations in long-term memory is an important learning in primary school that is subsequently essential in adult daily living. While most children succeed in storing arithmetic facts to which they have been trained at school, huge individual differences are reported, particularly in children with developmental dyscalculia, who show a severe and persistent deficit in arithmetic facts learning. This chapter reports important advances in the understanding of the development of an arithmetic facts network and focuses on the detrimental effect of similarity interference. First, at the retrieval stage, connectionist models highlighted that the similarity of the neighbor problems in the arithmetic facts network creates interference. More recently, the similarity interference during the learning stage was pointed out in arithmetic facts learning. The interference parameter, that captures the proactive interference that a problem receives from previously learned problems, was shown as a substantial determinant of the performance across multiplication problems. This proactive interference was found both in children and adults and showed that when a problem is highly similar to previously learned ones, it is more difficult to remember it. Furthermore, the sensitivity to this similarity interference determined individual differences in the learning and retrieving of arithmetic facts, giving new insights for interindividual differences. Regarding the atypical development, hypersensitivity-to-interference in memory was related to arithmetic facts deficit in a single case of developmental dyscalculia and in a group of fourth-grade children with low arithmetic facts knowledge. In sum, the impact of similarity interference is shown in the learning stage of arithmetic facts and concerns the typical and atypical development.