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According to the dominant view in the literature, several numerical cognition phenomena are explained coherently and parsimoniously by the Approximate Number System (ANS) model, which supposes the existence of an evolutionarily old, simple representation behind many numerical tasks. We offer an alternative account that proposes that only nonsymbolic numbers are processed by the ANS, while symbolic numbers, which are more essential to human mathematical capabilities, are processed by the Discrete Semantic System (DSS). In the DSS, symbolic numbers are stored in a network of nodes, similar to conceptual or linguistic networks. The benefit of the DSS model and the benefit of the more general hybrid ANS–DSS framework are demonstrated using the crucial example of the distance and size effects of comparison tasks.

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Mathematics skills relate to lifelong career, health and financial outcomes. Individuals’ cognitive abilities predict mathematics performance and there is growing recognition that environmental influences, including differences in culture and variability in mathematics engagement, also affect mathematics performance. In this Review, we summarize evidence indicating that differences between languages, exposure to maths-focused language, socioeconomic status, attitudes and beliefs about mathematics, and engagement with mathematics activities influence young children’s mathematics performance. These influences play out at the community and individual levels. However, research on the role of these environmental influences for foundational number skills, including understanding of number words, is limited. Future research is needed to understand individual differences in the development of early emerging mathematics skills such as number word skills, examining to what extent different types of environmental input are necessary and how children’s cognitive abilities shape the impact of environmental input. Children’s individual abilities and environment influence their mathematics skills. In this Review, Silver and Libertus examine how language, socioeconomic status and other environmental factors influence mathematics skills across childhood, with a focus on number word acquisition.

In elementary symbolic number processing, the comparison distance effect (in a comparison task, the task is more difficult with smaller numerical distance between the values) and the priming distance effect (in a number processing task, actual number is easier to process with a numerically close previous number) are two essential phenomena. While a dominant model, the approximate number system model, assumes that the two effects rely on the same mechanism, some other models, such as the discrete semantic system model, assume that the two effects are rooted in different generators. In a correlational study, here we investigate the relation of the two effects. Critically, the reliability of the effects is considered; therefore, a possible null result cannot be attributed to the attenuation of low reliability. The results showed no strong correlation between the two effects, even though appropriate reliabilities were provided. These results confirm the models of elementary number processing that assume distinct mechanisms behind number comparison and number priming.

In the numerical Stroop task, participants are presented with two digits that differ in their numerical and physical size and are requested to respond to which digit is numerically larger. Commonly, slower responses are observed when the numerical distance between the digits is small (the distance effect) and when the numerical and physical size are incongruent (the size-congruency effect). The current study will use proportion manipulation, which consists of two experimental lists with high versus low frequency of trials belonging to different conditions, as a tool to reduce these effects. Specifically, it will be used to examine how these two interference effects depend on each other, and how a reduction of one effect will affect the other. In Experiment 1, the size-congruency proportions were manipulated; in Experiment 2, the distance proportions were manipulated. The results show that manipulating size-congruency proportions modulates the size-congruency effect but not the distance effect, while manipulating the distance proportions modulates the distance effect but not the size-congruency effect. These results demonstrate for the first time that the distance effect can be modulated by the distance proportions. Furthermore, these results indicate that proportion manipulation is specific and only modulates the variable being manipulated. Together, these results shed new light on the specificity of proportion manipulation in the context of numerical information processing. These results are further discussed in the context of various numerical models that suggest a different relationship between these effects and demonstrate how proportion manipulation can aid to investigate numerical processes.

The approximate number system (ANS) is believed to be an essential component of numerical understanding. The sensitivity of the ANS has been found to be correlating with various mathematical abilities. Recently, Chesney (2018, Attention, Perception, & Psychophysics, 80[5], 1057-1063) demonstrated that if the ANS sensitivity is measured with the ratio effect slope, the slope may measure the sensitivity imprecisely. The present work extends her findings by demonstrating that mathematically the usability of the ratio effect slope depends on the Weber fraction range of the sample and the ratios of the numbers in the used test. Various indexes presented here can specify whether the use of the ratio effect slope as a replacement for the sigmoid fit is recommended or not. Detailed recommendations and a publicly available script help the researchers to plan or evaluate the use of the ratio effect slope as an ANS sensitivity index.

In the number comparison task distance effect (better performance with larger distance between the two numbers) and size effect (better performance with smaller numbers) are used extensively to find the representation underlying numerical cognition. According to the dominant analog number system (ANS) explanation, both effects depend on the extent of the overlap between the noisy representations of the two values. An alternative discrete semantic system (DSS) account supposes that the distance effect is rooted in the association between the numbers and the “small–large” properties with better performance for numbers with relatively high differences in their strength of association, and that the size effect depends on the everyday frequency of the numbers with smaller numbers being more frequent and thus easier to process. A recent study demonstrated that in a new, artificial digit notation—where both association and frequency can be arbitrarily manipulated—the distance and size effects change according to the DSS account. Here, we investigate whether the same manipulations modify the distance and size effects in Indo-Arabic notation, for which associations and frequency are already well established. We found that the distance effect depends on the association between the numbers and the “small–large” responses. It was also found that while the distance effect is flexible, the size effect seems to be unaltered, revealing a dissociation between the two effects. This result challenges the ANS view, which supposes a single mechanism behind the distance and size effects, and supports the DSS account, supposing two independent, statistics-based mechanisms behind the two effects.

HIGHLIGHTSWe test whether symbolic number comparison is handled by an analog noisy system.
Analog system model has systematic biases in describing symbolic number comparison.
This suggests that symbolic and non-symbolic numbers are processed by different systems.
Dominant numerical cognition models suppose that both symbolic and non-symbolic numbers are processed by the Analog Number System (ANS) working according to Weber's law. It was proposed that in a number comparison task the numerical distance and size effects reflect a ratio-based performance which is the sign of the ANS activation. However, increasing number of findings and alternative models propose that symbolic and non-symbolic numbers might be processed by different representations. Importantly, alternative explanations may offer similar predictions to the ANS prediction, therefore, former evidence usually utilizing only the goodness of fit of the ANS prediction is not sufficient to support the ANS account. To test the ANS model more rigorously, a more extensive test is offered here. Several properties of the ANS predictions for the error rates, reaction times, and diffusion model drift rates were systematically analyzed in both non-symbolic dot comparison and symbolic Indo-Arabic comparison tasks. It was consistently found that while the ANS model's prediction is relatively good for the non-symbolic dot comparison, its prediction is poorer and systematically biased for the symbolic Indo-Arabic comparison. We conclude that only non-symbolic comparison is supported by the ANS, and symbolic number comparisons are processed by other representation.

In a comparison task, the larger the distance between the two numbers to be compared, the better the performance—a phenomenon termed as the numerical distance effect. According to the dominant explanation, the distance effect is rooted in a noisy representation, and performance is proportional to the size of the overlap between the noisy representations of the two values. According to alternative explanations, the distance effect may be rooted in the association between the numbers and the small-large categories, and performance is better when the numbers show relatively high differences in their strength of association with the small-large properties. In everyday number use, the value of the numbers and the association between the numbers and the small-large categories strongly correlate; thus, the two explanations have the same predictions for the distance effect. To dissociate the two potential sources of the distance effect, in the present study, participants learned new artificial number digits only for the values between 1 and 3, and between 7 and 9, thus, leaving out the numbers between 4 and 6. It was found that the omitted number range (the distance between 3 and 7) was considered in the distance effect as 1, and not as 4, suggesting that the distance effect does not follow the values of the numbers predicted by the dominant explanation, but it follows the small-large property association predicted by the alternative explanations.

Human number understanding is thought to rely on the analog number system (ANS), working according to Weber’s law. We propose an alternative account, suggesting that symbolic mathematical knowledge is based on a discrete semantic system (DSS), a representation that stores values in a semantic network, similar to the mental lexicon or to a conceptual network. Here, focusing on the phenomena of numerical distance and size effects in comparison tasks, first we discuss how a DSS model could explain these numerical effects. Second, we demonstrate that the DSS model can give quantitatively as appropriate a description of the effects as the ANS model. Finally, we show that symbolic numerical size effect is mainly influenced by the frequency of the symbols, and not by the ratios of their values. This last result suggests that numerical distance and size effects cannot be caused by the ANS, while the DSS model might be the alternative approach that can explain the frequency-based size effect.

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 adults.

Numerical and non-numerical order processing share empirical characteristics (distance effect and semantic congruity), but there are also important differences (in size effect and end effect). At the same time, models and theories of numerical and non-numerical order processing developed largely separately. Currently, we combine insights from 2 earlier models to integrate them in a common framework. We argue that the same learning principle underlies numerical and non-numerical orders, but that environmental features determine the empirical differences. Implications for current theories on order processing are pointed out. (PsycINFO Database Record (c) 2014 APA, all rights reserved).

The representation of 0 in healthy adults was studied with the physical comparison task. Automatic processing of numbers, as indicated by the size congruity effect, was used for detecting the basic numerical representations stored in long-term memory. The size congruity effect usually increases with numerical distance between the physically compared numbers. This increase was attenuated for comparisons to 0 or 1 (but not to 2) when they were perceived as the smallest number in the set. Furthermore, the size congruity effect was enlarged in these cases. These results indicate an end effect in automatic processing of numbers and suggest that 0, or 1 in the absence of 0, is perceived as the smallest entity on the mental number line. The implications of these findings are discussed with regard to models of number representation.

Nine experiments of timed odd-even judgments examined how parity and number magnitude are accessed from Arabic and verbal numerals. With Arabic numerals, Ss used the rightmost digit to access a store of semantic number knowledge. Verbal numerals went through an additional stage of transcoding to base 10. Magnitude information was automatically accessed from Arabic numerals. Large numbers preferentially elicited a rightward response, and small numbers a leftward response. The Spatial-Numerical Association of Response Codes (SNARC) effect depended only on relative number magnitude and was weaker or absent with letters or verbal numerals. Direction did not vary with handedness or hemispheric dominance but was linked to the direction of writing, as it faded or even reversed in right-to-left writing Iranian Ss. The results supported a modular architecture for number processing, with distinct but interconnected Arabic, verbal, and magnitude representations.

Studies on developmental dyscalculia (DD) have tried to identify a basic numerical deficit that could account for this specific learning disability. The first proposition was that the number magnitude representation of these children was impaired. However, Rousselle and Noël (2007) brought data showing that this was not the case but rather that these children were impaired when processing the magnitude of symbolic numbers only. Since then, incongruent results have been published. In this paper, we will propose a developmental perspective on this issue. We will argue that the first deficit shown in DD regards the building of an exact representation of numerical value, thanks to the learning of symbolic numbers, and that the reduced acuity of the approximate number magnitude system appears only later and is secondary to the first deficit.

Although infants and animals respond to the approximate number of elements in visual, auditory, and tactile arrays, only human children and adults have been shown to possess abstract numerical representations that apply to entities of all kinds (e.g., 7 samurai, seas, or sins). Do abstract numerical concepts depend on language or culture, or do they form a part of humans' innate, core knowledge? Here we show that newborn infants spontaneously associate stationary, visual-spatial arrays of 4-18 objects with auditory sequences of events on the basis of number. Their performance provides evidence for abstract numerical representations at the start of postnatal experience.

Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

To account for the size effect in numerical comparison, three assumptions about the internal structure of the mental number line (e.g., Dehaene, 1992) have been proposed. These are magnitude coding (e.g., Zorzi & Butterworth, 1999), compressed scaling (e.g., Dehaene, 1992), and increasing variability (e.g., Gallistel & Gelman, 1992). However, there are other tasks besides numerical comparison for which there is clear evidence that the mental number line is accessed, and no size effect has been observed in these tasks. This is contrary to the predictions of these three assumptions. Moreover, all three assumptions have difficulties explaining certain symmetries in priming studies of number naming and parity judgment. We propose a neural network model that avoids these three assumptions but, instead, uses place coding, linear scaling, and constant variability on the mental number line. We train the model on naming, parity judgment, and comparison and show that the size effect appears in comparison, but not in naming or parity judgment. Moreover, no asymmetries appear in primed naming or primed parity judgment with this model, in line with empirical data. Implications of our findings are discussed.

Differences in performance with various stimulus-response mappings are among the most prevalent findings for binary choice reaction tasks. The authors show that perceptual or conceptual similarity is not necessary to obtain mapping effects; a type of structural similarity is sufficient. Specifically, stimulus and response alternatives are coded as positive and negative polarity along several dimensions, and polarity correspondence is sufficient to produce mapping effects. The authors make the case for this polarity correspondence principle using the literature on word-picture verification and then provide evidence that polarity correspondence is a determinant of mapping effects in orthogonal stimulus-response compatibility, numerical judgment, and implicit association tasks. The authors conclude by discussing implications of this principle for interpretation of results from binary choice tasks and future model development.

The representation of negative numbers was explored during intentional processing (i.e., when participants performed a numerical comparison task) and during automatic processing (i.e., when participants performed a physical comparison task). Performance in both cases suggested that negative numbers were not represented as a whole but rather their polarity and numerical magnitudes were represented separately. To explore whether this was due to the fact that polarity and magnitude are marked by two spatially separated symbols, participants were trained to mark polarity by colour. In this case there was still evidence for a separate representation of polarity and magnitude. However, when a different set of stimuli was used to refer to positive and negative numbers, and polarity was not marked separately, participants were able to represent polarity and magnitude together when numerical processing was performed intentionally but not when it was conducted automatically. These results suggest that notation is only partly responsible for the components representation of negative numbers and that the concept of negative numbers can be grasped only through that of positive numbers.

In the symbolic number comparison task, the size effect (better performance for small than for large numbers) is usually interpreted as the result of the more general ratio effect, in line with Weber's law. In alternative models, the size effect might be a result of stimulus frequency: smaller numbers are more frequent, and more frequent stimuli are easier to process. It has been demonstrated earlier, that in artificial new number digits, the size effect reflects the frequencies of those digits. In the present work we investigate whether frequency also directs the size effect in Indo-Arabic numbers, in which notation, unlike in new symbols, the frequencies are already firmly established for the participants. We found that frequency has an effect on the size effect in Indo-Arabic notation, but this influence is limited. However, this limited size effect change is acquired fast at the beginning of the session. We argue that these results are more in line with the frequency-based accounts of the size effect.

Individual differences in the ability to compare and evaluate nonsymbolic numerical magnitudes—approximate number system (ANS) acuity—are emerging as an important predictor in many research areas. Unfortunately, recent empirical studies have called into question whether a historically common ANS-acuity metric—the size of the numerical distance effect (NDE size)—is an effective measure of ANS acuity. NDE size has been shown to frequently yield divergent results from other ANS-acuity metrics. Given these concerns and the measure’s past popularity, it behooves us to question whether the use of NDE size as an ANS-acuity metric is theoretically supported. This study seeks to address this gap in the literature by using modeling to test the basic assumption underpinning use of NDE size as an ANS-acuity metric: that larger NDE size indicates poorer ANS acuity. This assumption did not hold up under test. Results demonstrate that the theoretically ideal relationship between NDE size and ANS acuity is not linear, but rather resembles an inverted J-shaped distribution, with the inflection points varying based on precise NDE task methodology. Thus, depending on specific methodology and the distribution of ANS acuity in the tested population, positive, negative, or null correlations between NDE size and ANS acuity could be predicted. Moreover, peak NDE sizes would be found for near-average ANS acuities on common NDE tasks. This indicates that NDE size has limited and inconsistent utility as an ANS-acuity metric. Past results should be interpreted on a case-by-case basis, considering both specifics of the NDE task and expected ANS acuity of the sampled population.

Interference between number magnitude and other properties can be explained by either an analogue magnitude system interfering with a continuous representation of the other properties or by discrete, categorical representations in which the corresponding number and property categories interfere. In this study, we investigated whether parity, a discrete property which supposedly cannot be stored on an analogue representation, could interfere with number magnitude. We found that in a parity decision task the magnitude interfered with the parity, highlighting the role of discrete representations in numerical interference. Additionally, some participants associated evenness with large values, while others associated evenness with small values, therefore, a new interference index, the dual index was introduced to detect this heterogeneous interference. The dual index can be used to reveal any heterogeneous interference that were missed in previous studies. Finally, the magnitude-parity interference did not correlate with the magnitude-response side interference (Spatial-Numerical Association of Response Codes [SNARC] effect) or with the parity-response side interference (Markedness Association of Response Codes [MARC] effect), suggesting that at least some of the interference effects are not the result of the stimulus property markedness.

Numerical distance and size effects (easier number comparison with large distance or small size) are mostly supposed to reflect a single effect, the ratio effect, which is the consequence of the analogue number system (ANS) activation, working according to Weber’s law. In an alternative model, symbolic numbers can be processed by a discrete semantic system (DSS), in which the distance and the size effects could originate in two independent factors: the distance effect depends on the semantic distance of the units, and the size effect depends on the frequency of the symbols. While in the classic view both symbolic and nonsymbolic numbers are processed by the ANS, in the alternative view only nonsymbolic numbers are processed by the ANS, but symbolic numbers are handled by the DSS. The current work contrasts the two views, investigating whether the size of the distance and the size effects correlate in nonsymbolic dot comparison and in symbolic Indo-Arabic comparison tasks. If a comparison is backed by the ANS, the distance and the size effects should correlate, because the two effects are merely two ways to measure the same ratio effect, however, if a comparison is supported by other system, for example the DSS, the two effects might dissociate. In the current measurements the distance and the size effects correlated very strongly in the dot comparison task, but they did not correlate in the Indo-Arabic comparison task. Additionally, the effects did not correlate between the Indo-Arabic and the dot comparison tasks. These results suggest that symbolic number comparison is not handled by the ANS, but by an alternative representation, such as the DSS.
Find the postprint at the project's website at https://sites.google.com/site/mathematicalcognition/home/discrete-semantic-system

How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the "symbol-grounding problem," is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture of the symbol-grounding problem. (PsycINFO Database Record

Are symbolic and nonsymbolic numbers coded differently in the brain? Neuronal data indicate that overlap in numerical tuning curves is a hallmark of the approximate, analogue nature of nonsymbolic number representation. Consequently, patterns of fMRI activity should be more correlated when the representational overlap between two numbers is relatively high. In bilateral intraparietal sulci (IPS), for nonsymbolic numbers, the pattern of voxelwise correlations between pairs of numbers mirrored the amount of overlap in their tuning curves under the assumption of approximate, analogue coding. In contrast, symbolic numbers showed a flat field of modest correlations more consistent with discrete, categorical representation (no systematic overlap between numbers). Directly correlating activity patterns for a given number across formats (e.g., the numeral “6” with six dots) showed no evidence of shared symbolic and nonsymbolic number-specific representations. Overall (univariate) activity in bilateral IPS was well fit by the log of the number being processed for both nonsymbolic and symbolic numbers. IPS activity is thus sensitive to numerosity regardless of format; however, the nature in which symbolic and nonsymbolic numbers are encoded is fundamentally different. Hum Brain Mapp, 2014. © 2014 Wiley Periodicals, Inc.

The relation between the approximate number system (ANS) and symbolic number processing skills remains unclear. Some theories assume that children acquire the numerical meaning of symbols by mapping them onto the preexisting ANS. Others suggest that in addition to the ANS, children also develop a separate, exact representational system for symbolic number processing. In the current study, we contribute to this debate by investigating whether the nonsymbolic number processing of kindergarteners is predictive for symbolic number processing. Results revealed no association between the accuracy of the kindergarteners on a nonsymbolic number comparison task and their performance on the symbolic comparison task six months later, suggesting that there are two distinct representational systems for the ANS and numerical symbols.

In this chapter, I put together the first elements of a mathematical theory relating neuro- biological observations to psychological laws in the domain of numerical cognition. The starting point is the postulate of a neuronal code whereby numerosity—the cardinal of a set of objects—is represented approximately by the firing of a population of numerosity detectors. Each of these neurons fires to a certain preferred numerosity, with a tuning curve which is a Gaussian function of the logarithm of numerosity. From this log- Gaussian code, decisions are taken using Bayesian mechanisms of log-likelihood compu- tation and accumulation. The resulting equations for response times and errors in classical tasks of number comparison and same-different judgments are shown to tightly fit behavioral and neural data. Two more speculative issues are discussed. First, new chronometric evidence is presented supporting the hypothesis that the acquisition of number symbols changes the mental number line, both by increasing its precision and by changing its coding scheme from logarithmic to linear. Second, I examine how symbolic and nonsymbolic representations of numbers affect performance in arithmetic compu- tations such as addition and subtraction.

Nine experiments of timed odd–even judgments examined how parity and number magnitude are accessed from Arabic and verbal numerals. With Arabic numerals, Ss used the rightmost digit to access a store of semantic number knowledge. Verbal numerals went through an additional stage of transcoding to base 10. Magnitude information was automatically accessed from Arabic numerals. Large numbers preferentially elicited a rightward response, and small numbers a leftward response. The Spatial–Numerical Association of Response Codes effect depended only on relative number magnitude and was weaker or absent with letters or verbal numerals. Direction did not vary with handedness or hemispheric dominance but was linked to the direction of writing, as it faded or even reversed in right-to-left writing Iranian Ss. The results supported a modular architecture for number processing, with distinct but interconnected Arabic, verbal, and magnitude representations. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

We examine the frequency of numerals and ordinals in seven different languages and/or cultures. Many cross-cultural and cross-linguistic patterns are identified. The most striking is a decrease of frequency with numerical magnitude, with local increases for reference numerals such as 10, 12, 15, 20, 50 or 100. Four explanations are considered for this effect: sampling artifacts, notational regularities, environmental biases and psychological limitations on number representations. The psychological explanation, which appeals to a Fechnerian encoding of numerical magnitudes and to the existence of numerical points of reference, accounts for most of the data. Our finding also has practical importance since it reveals the frequent confound of two experimental variables: numerical magnitude and numeral frequency.

Attaching meaning to arbitrary symbols (i.e. words) is a complex and lengthy process. In the case of numbers, it was previously suggested that this process is grounded on two early pre-verbal systems for numerical quantification: the approximate number system (ANS or 'analogue magnitude'), and the object tracking system (OTS or 'parallel individuation'), which children are equipped with before symbolic learning. Each system is based on dedicated neural circuits, characterized by specific computational limits, and each undergoes a separate developmental trajectory. Here, I review the available cognitive and neuroscientific data and argue that the available evidence is more consistent with a crucial role for the ANS, rather than for the OTS, in the acquisition of abstract numerical concepts that are uniquely human.

Developmental dyscalculia is a learning disability that affects the acquisition of knowledge about numbers and arithmetic. It is widely assumed that numeracy is rooted on the "number sense", a core ability to grasp numerical quantities that humans share with other animals and deploy spontaneously at birth. To probe the links between number sense and dyscalculia, we used a psychophysical test to measure the Weber fraction for the numerosity of sets of dots, hereafter called number acuity. We show that number acuity improves with age in typically developing children. In dyscalculics, numerical acuity is severely impaired, with 10-year-old dyscalculics scoring at the level of 5-year-old normally developing children. Moreover, the severity of the number acuity impairment predicts the defective performance on tasks involving the manipulation of symbolic numbers. These results establish for the first time a clear association between dyscalculia and impaired "number sense", and they may open up new horizons for the early diagnosis and rehabilitation of mathematical learning deficits.

This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.

Does the human capacity for mathematical intuition depend on linguistic competence or on visuo-spatial representations? A
series of behavioral and brain-imaging experiments provides evidence for both sources. Exact arithmetic is acquired in a language-specific
format, transfers poorly to a different language or to novel facts, and recruits networks involved in word-association processes.
In contrast, approximate arithmetic shows language independence, relies on a sense of numerical magnitudes, and recruits bilateral
areas of the parietal lobes involved in visuo-spatial processing. Mathematical intuition may emerge from the interplay of
these brain systems.

Graded interference effects were tested in a naming task, in parallel for objects and actions. Participants named either object or action pictures presented in the context of other pictures (blocks) that were either semantically very similar, or somewhat semantically similar or semantically dissimilar. We found that naming latencies for both object and action words were modulated by the semantic similarity between the exemplars in each block, providing evidence in both domains of graded semantic effects.

Patterns of neural firing linked to eye movement decisions show that behavioral decisions are predicted by the differential firing rates of cells coding selected and nonselected stimulus alternatives. These results can be interpreted using models developed in mathematical psychology to model behavioral decisions. Current models assume that decisions are made by accumulating noisy stimulus information until sufficient information for a response is obtained. Here, the models, and the techniques used to test them against response-time distribution and accuracy data, are described. Such models provide a quantitative link between the time-course of behavioral decisions and the growth of stimulus information in neural firing data.

This article addresses the representation of numerical information conveyed by nonsymbolic and symbolic stimuli. In a first simulation study, we show how number-selective neurons develop when an initially uncommitted neural network is given nonsymbolic stimuli as input (e.g., collections of dots) under unsupervised learning. The resultant network is able to account for the distance and size effects, two ubiquitous effects in numerical cognition. Furthermore, the properties of the network units conform in detail to the characteristics of recently discovered number-selective neurons. In a second study, we simulate symbol learning by presenting symbolic and nonsymbolic input simultaneously. The same number-selective neurons learn to represent the numerical meaning of symbols. In doing so, they show properties reminiscent of the originally available number-selective neurons, but at the same time, the representational efficiency of the neurons is increased when presented with symbolic input. This finding presents a concrete proposal on the linkage between higher order numerical cognition and more primitive numerical abilities and generates specific predictions on the neural substrate of number processing.

Both the speed and the accuracy of a perceptual judgment depend on the strength of the sensory stimulation. When stimulus strength is high, accuracy is high and response time is fast; when stimulus strength is low, accuracy is low and response time is slow. Although the psychometric function is well established as a tool for analyzing the relationship between accuracy and stimulus strength, the corresponding chronometric function for the relationship between response time and stimulus strength has not received as much consideration. In this article, we describe a theory of perceptual decision making based on a diffusion model. In it, a decision is based on the additive accumulation of sensory evidence over time to a bound. Combined with simple scaling assumptions, the proportional-rate and power-rate diffusion models predict simple analytic expressions for both the chronometric and psychometric functions. In a series of psychophysical experiments, we show that this theory accounts for response time and accuracy as a function of both stimulus strength and speed-accuracy instructions. In particular, the results demonstrate a close coupling between response time and accuracy. The theory is also shown to subsume the predictions of Piéron's Law, a power function dependence of response time on stimulus strength. The theory's analytic chronometric function allows one to extend theories of accuracy to response time.

The diffusion decision model allows detailed explanations of behavior in two-choice discrimination tasks. In this article, the model is reviewed to show how it translates behavioral data-accuracy, mean response times, and response time distributions-into components of cognitive processing. Three experiments are used to illustrate experimental manipulations of three components: stimulus difficulty affects the quality of information on which a decision is based; instructions emphasizing either speed or accuracy affect the criterial amounts of information that a subject requires before initiating a response; and the relative proportions of the two stimuli affect biases in drift rate and starting point. The experiments also illustrate the strong constraints that ensure the model is empirically testable and potentially falsifiable. The broad range of applications of the model is also reviewed, including research in the domains of aging and neurophysiology.

Modelling SNARC by using polarity codes to adjust drift rates

- C Leth-Steensen
- J Lucas
- W M Petrusic

Leth-Steensen, C., Lucas, J., & Petrusic, W. M. (2011). Modelling SNARC by using polarity codes
to adjust drift rates. Proceedings of the 27th Annual Meeting of the International Society for
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Lengyel: Processing symbolic number comparison task

- Preprint Krajcsi
- Kojouharova

PREPRINT Krajcsi, Kojouharova, Lengyel: Processing symbolic number comparison task 14/16