The present study examined whether a dissociation among formats for rational numbers can be obtained in tasks that require comparing a number to a non-symbolic quantity. In Experiment 1, college students saw a discrete or else continuous image followed by a rational number, and had to decide which was numerically larger. In Experiment 2, participants saw the same displays but had to make a judgment about the type of ratio represented by the number. The magnitude task was performed more (...) quickly using decimals, whereas the relation task was performed more accurately with fractions. The pattern observed for percentages was very similar to that for decimals. A dissociation between magnitude comparison and relational processing with rational numbers can be obtained when a symbolic number must be compared to a non-symbolic display. (shrink)

Legal Judgment Prediction (LJP) is an essential component of legal assistant systems, which aims to automatically predict judgment results from a given criminal fact description. As a vital subtask of LJP, researchers have paid little attention to the numerical LJP, i.e., the prediction of imprisonment and penalty. Existing methods ignore numerical information in the criminal facts, making their performances far from satisfactory. For instance, the amount of theft varies, as do the prison terms and penalties. The major challenge (...) is how the model can obtain the ability of numerical comparison and magnitude perception, e.g., 400 < 500 < 800, 500 is closer to 400 than to 800. To this end, we propose a judicial knowledge-enhanced magnitude-aware reasoning architecture, called NumLJP, for the numerical LJP task. Specifically, we first implement a contrastive learning-based judicial knowledge selector to distinguish confusing criminal cases efficiently. Unlike previous approaches that employ the law article as external knowledge, judicial knowledge is a quantitative guideline in real scenarios. It contains many numerals (called anchors) that can construct a reference frame. Then we design a masked numeral prediction task to help the model remember these anchors to acquire legal numerical commonsense from the selected judicial knowledge. We construct a scale-based numerical graph using the anchors and numerals in facts to perform magnitude-aware numerical reasoning. Finally, the representations of fact description, judicial knowledge, and numerals are fused to make decisions. We conduct extensive experiments on three real-world datasets and select several competitive baselines. The results demonstrate that the macro-F1 of NumLJP improves by at least 9.53% and 11.57% on the prediction of penalty and imprisonment, respectively. (shrink)

Quantifier expressions like “many” and “at least” are part of a rich repository of words in language representing magnitude information. The role of numerical processing in comprehending quantifiers was studied in a semantic truth value judgment task, asking adults to quickly verify sentences about visual displays using numerical or proportional quantifiers. The visual displays were composed of systematically varied proportions of yellow and blue circles. The results demonstrated that numerical estimation and numerical reference information are (...) fundamental in encoding the meaning of quantifiers in terms of response times and acceptability judgments. However, a difference emerges in the comparison strategies when a fixed external reference numerosity is used for numerical quantifiers, whereas an internal numerical criterion is invoked for proportional quantifiers. Moreover, for both quantifier types, quantifier semantics and its polarity biased the response direction. Overall, our results indicate that quantifier comprehension involves core numerical and lexical semantic properties, demonstrating integrated processing of language and numbers. (shrink)

Developmental dyscalculia (DD) is a pervasive difficulty affecting number processing and arithmetic. It is encountered in around 6% of school-aged children. While previous studies have mainly focused on general cognitive functions, the present paper aims to further investigate the hypothesis of a specific numerical deficit in dyscalculia. The performance of 10- and 11-year-old children with DD characterised by a weakness in arithmetic facts retrieval and age-matched control children was compared on various number comparison tasks. Participants were asked to (...) compare a quantity presented in either a symbolic (Arabic numerals, number words, canonical dots patterns) or a nonsymbolic format (noncanonical dots patterns, and random sticks patterns) to the reference quantity 5. DD children showed a greater numerical distance effect than control children, irrespective of the number format. This favours a deficit in the specialised cognitive system underlying the processing of number magnitude in children with DD. Results are discussed in terms of access and representation deficit hypotheses. (shrink)

The philosophy of mathematics has been accused of paying insufficient attention to mathematical practice: one way to cope with the problem, the one we will follow in this paper on extensive magnitudes, is to combine the `history of ideas' and the `philosophy of models' in a logical and epistemological perspective. The history of ideas allows the reconstruction of the theory of extensive magnitudes as a theory of ordered algebraic structures; the philosophy of models allows an investigation into the way epistemology (...) might affect relevant mathematical notions. The article takes two historical examples as a starting point for the investigation of the role of numerical models in the construction of a system of non-Archimedean magnitudes. A brief exposition of the theories developed by Giuseppe Veronese and by Rodolfo Bettazzi at the end of the 19th century will throw new light on the role played by magnitudes and numbers in the development of the concept of a non-Archimedean order. Different ways of introducing non-Archimedean models will be compared and the influence of epistemological models will be evaluated. Particular attention will be devoted to the comparison between the models that oriented Veronese's and Bettazzi's works and the mathematical theories they developed, but also to the analysis of the way epistemological beliefs affected the concepts of continuity and measurement. (shrink)

Recently, an associative learning account of cognitive control has been suggested (Verguts & Notebaert, 2009). In this so-called adaptation by binding theory, Hebbian learning of stimulus–stimulus and stimulus–response associations is assumed to drive the adaptation of human behavior. In this study, we evaluated the validity of the adaptation-by-binding account for the case of implicit learning of regularities within a stimulus set (i.e., the frequency of specific unit digit combinations in a two-digit number magnitude comparison task) and their association (...) with a particular response. Our data indicated that participants indeed learned these regularities and adapted their behavior accordingly. In particular, influences of cognitive control were even able to override the numerical distance effect—one of the most robust effects in numerical cognition research. Thus, the general cognitive processes involved in two-digit number magnitude comparison seem much more complex than previously assumed. Multi-digit number magnitude comparison may not be automatic and inflexible but influenced by processes of cognitive control being highly adaptive to stimulus set properties and task demands on multiple levels. (shrink)

Although some cognitive studies provided reasons that children with low socioeconomic status (SES) showed poor mathematical achievements, there was no explicit evidence to directly explain the root of lagged performance in children with low SES. Therefore, the present study explored the differences in neural correlates in the process of symbolic magnitude comparison between children with different SES by the event-related potentials (ERP). A total of 16 second graders from low SES families and 16 from middle/high SES families participated (...) in this study. According to the results of anterior N1 (early attention) and P2 (extraction of numerical meaning) over frontal region, the differences among children with different SES were manifested as differences in general neural activities in terms of attention and top-down cognitive control. In the late stage of cognitive processing, there was no significant difference in the average amplitude of the LPC between the children with different SES, indicating that low SES did not influence the information encoding and memory updating of numerical representation responsible by the parietal lobe. The educational implications of this study were mentioned in the discussion. (shrink)

A Theory of Magnitude suggests that space, time, and quantities are processed through a generalized magnitude system. ATOM posits that task-irrelevant magnitudes interfere with the processing of task-relevant magnitudes as all the magnitudes are processed by a common system. Many behavioral and neuroimaging studies have found support in favor of a common magnitude processing system. However, it is largely unknown whether such cross-domain monotonic mapping arises from a change in the accuracy of the magnitude judgments or (...) results from changes in precision of the processing of magnitude. Therefore, in the present study, we examined whether large numerical magnitude affects temporal accuracy or temporal precision, or both. In other words, whether numerical magnitudes change our temporal experience or simply bias duration judgments. The temporal discrimination paradigm was used to present numerical magnitudes across varied durations. We estimated temporal accuracy and precision for each numerical magnitude. The results revealed that temporal accuracy for large numerical magnitude was significantly lower than that of small and identical magnitudes. This implies that the temporal duration was overestimated for large numerical magnitude compared to small and identical numerical magnitude, in line with ATOM’s prediction. However, no influence of numerical magnitude was observed on temporal precision. The findings of the present study suggest that task-irrelevant numerical magnitude selectively affects the accuracy of processing of duration but not duration discrimination itself. Further, we argue that numerical magnitude may not directly affect temporal processing but could influence via attentional mechanisms. (shrink)

The goals of the present study were to test whether (and which) basic numerical abilities can be improved with training and whether training effects transfer to improvement in children’s math achievement. The literature is mixed with evidence that does or does not substantiate the efficacy of training basic numerical ability. In the present study, we developed a child-friendly software named ‘123 Bakery’ which includes four training modules; non-symbolic numerosity comparison, non-symbolic numerosity estimation, approximate arithmetic and symbol-to-numerosity mapping. (...) Fifty-six first graders were randomly assigned to either the training or control group. The training group participated in 6 weeks of training (5 times a week, 30 minutes per day). All participants underwent pre- and post- training assessment of their basic numerical processing ability (including numerosity discrimination acuity, symbolic/non-symbolic magnitude estimation, approximate arithmetic, symbol-to-numerosity mapping), overall math achievement and intelligence, 6 weeks apart. The acuity for numerosity discrimination (approximate number sense acuity; ANS acuity) significantly improved after training, but this training effect did not transfer to improvement in symbolic, exact calculation, or any other math ability. We conclude that basic numerical cognition training leads to improvement in ANS acuity, but whether this effect transfers to symbolic math ability remains to be further tested. (shrink)

Although most deaf individuals could use sign language or sign/spoken language mix, hearing loss would still affect their language acquisition. Compensatory plasticity holds that the lack of auditory stimulation experienced by deaf individuals, such as congenital deafness, can be met by enhancements in visual cognition. And the studies of hearing individuals have showed that visual form perception is the cognitive mechanism that could explain the association between numerical magnitude processing and arithmetic computation. Therefore, we examined numerical (...) class='Hi'>magnitude processing and its contribution to arithmetical ability in deaf adolescents, and explored the differences between the congenital and acquired deafness. 112 deaf adolescents and 58 hearing adolescents performed a series of cognitive and mathematical tests, and it was found there was no significant differences between the congenital group and the hearing group, but congenital group outperformed acquired group in numerical magnitude processing and arithmetic computation. It was also found there was a close association between numerical magnitude processing and arithmetic computation in all deaf adolescents, and after controlling for the demographic variables and general cognitive abilities, numerical magnitude processing could predict arithmetic computation in all deaf adolescents but not in congenital group. The role of numerical magnitude processing in deaf adolescents' mathematical performance should be paid attention in the training of arithmetical ability. (shrink)

Spatial, physical, and semantic magnitude dimensions can influence action decisions in human cognitive processing and interact with each other. For example, in the SNARC effect, semantic numerical magnitude facilitates left-hand or right-hand responding dependent on the small or large magnitude of number symbols. SNARC-like interactions of numerical magnitudes with the radial spatial dimension (depth) were postulated from early on. Usually, the SNARC effect in any direction is investigated using fronto-parallel computer monitors for presentation of stimuli. (...) In such 2D setups, however, the metaphorical and literal interpretation of the radial depth axis with seemingly close / far stimuli or responses are not distinct. Hence, it is difficult to draw clear conclusions with respect to the contribution of different spatial mappings to the SNARC effect. In order to disentangle the different mappings in a natural way, we studied parametrical interactions between semantic numerical magnitude, horizontal directional responses, and perceptual distance by means of stereoscopic depth in an immersive virtual reality (VR). Two VR experiments show horizontal SNARC effects across all spatial displacements in traditional latency measures and kinematic response parameters. No indications of a SNARC effect along the depth axis, as it would be predicted by a direct mapping account, were observed, but the results show a non-linear relationship between horizontal SNARC slopes and physical distance. Steepest SNARC slopes were observed for digits presented close to the hands. We conclude that spatial-numerical processing is susceptible to effector-based processes but relatively resilient to task-irrelevant variations of radial-spatial magnitudes. (shrink)

Numerical magnitude information is assumed to be spatially represented in the form of a mental number line defined with respect to a body-centred, egocentric frame of reference. In this context, spatial language skills such as mastery of verbal descriptions of spatial position have been proposed to be relevant for grasping spatial relations between numerical magnitudes on the mental number line. We examined 4- to 5-year-old’s spatial language skills in tasks that allow responses in egocentric and allocentric frames (...) of reference, as well as their relative understanding of numerical magnitude. In addition, we evaluated influences of children’s absolute understanding of numerical magnitude assessed by their number word comprehension and of their knowledge on numerical sequences. Results indicated that when considering responses that corresponded to the egocentric perspective, children’s spatial language was associated significantly with their relative numerical magnitude understanding, even after controlling for covariates, such as children’s SES, mental rotation skills, and also absolute magnitude understanding or knowledge on numerical sequences. This suggests that the use of egocentric reference frames in spatial language may facilitate spatial representation of numbers along a mental number line and thus seem important for preschoolers’ relative understanding of numerical magnitude. (shrink)

Recent findings indicate that the constituting digits of multi-digit numbers are processed, decomposed into units, tens, and so on, rather than integrated into one entity. This is suggested by interfering effects of unit digit processing on two-digit number comparison. In the present study, we extended the computational model for two-digit number magnitude comparison of Moeller, Huber, Nuerk, and Willmes (2011a) to the case of three-digit number comparison (e.g., 371_826). In a second step, we evaluated how hundred-decade (...) and hundred-unit compatibility effects were moderated by varying the percentage of within-hundred (e.g., 539_582) and within-hundred-and-decade filler items (e.g., 483_489). From the results we predict that numerical distance as well as compatibility effects should indeed be modulated by the relevance of tens and units in three-digit number magnitude comparison: While in particular the hundred distance effect should decrease, we predict hundred-decade and hundred-unit compatibility effects to increase with the relevance of tens and units. (shrink)

In this study, we directly contrast two approaches that have been proposed to explain the SNARC effect. The traditional direct mapping account suggests that a direct association exists between the position of a number on the mental number line and the location of the response. On the other hand, accounts are considered that propose an intermediate step in which numbers are categorized as either small or large between the number magnitude and the response representations. In a magnitude (...) class='Hi'>comparison task, we departed from the usual bimanual left/right response dimension and instead introduced the unimanual close/far dimension. A spatial-numerical association was observed: small numbers were associated with a close response, while large numbers were associated with a far response, regardless of the movement direction (left/right). We discuss why these results cannot be explained by assuming a direct mapping from the representation of numbers on a mental number line to response locations and discuss how the results can be explained by the alternative accounts. (shrink)

A theory of magnitudes involves criteria for their comparison, equivalence and addition. We examine these aspects from an abstract viewpoint, stressing independence and definability. These considerations are triggered by the so-called De Zolt’s principle in the theory of equivalence of plane polygons.

Educational interventions are necessary to develop mathematical competence at early ages and prevent widespread mathematics learning failure in the education system as indicated by the results of European reports. Numerous studies agree that domain-specific predictors related to mathematics are symbolic and non-symbolic magnitude comparison, as well as, number line estimation. The goal of this study was to design 4 digital learning app games to train specific cognitive bases of mathematical learning in order to create resources and promote the (...) use of these technologies in the educational community and to promote effective scientific transfer and increase the research visibility. This study involved 193 preschoolers aged 57–79 months. A quasi-experimental design was carried out with 3 groups created after scores were obtained in a standardised mathematical competence assessment test, i.e., low-performance group, high-performance group, and control group. The results show that training with the 4 digital learning app games focusing on magnitude, subitizing, number facts, and estimation tasks improved the numerical skills of the experimental groups, compared to the control group. The implications of the study were, on the one hand, provided verified technological tools for teaching early mathematical competence. On the other hand, this study supports other studies on the importance of cognitive precursors in mathematics performance. (shrink)

We concur with Cohen Kadosh & Walsh (CK&W) that representation of numbers in the parietal cortex is format dependent. In addition, we suggest that all formats do not automatically, and equally, access analog magnitude representation in the intraparietal sulcus (IPS). Understanding how development, learning, and context lead to differential access of analog magnitude representation is a key question for future research.

Multicultural environments require learning multiple number notations wherein some are encountered more frequently than others. This leads to differences in exposure and consequently differences in usage between notations. We find that differential notational usage imposes a significant neurocognitive load on number processing. Despite simultaneous acquisition, forty-two adult binumerate populations, familiar with two positional writing systems namely Hindu Nagari digits and Hindu Arabic digits, reported significantly lower preference and usage for Nagari as compared to Arabic. Twenty-four participants showed significantly increased reaction (...) times and reduced accuracy while performing magnitude comparison tasks in Nagari with respect to Arabic. Functional magnetic resonance imaging revealed that processing Nagari elicited significantly greater activity in number processing and attention networks. A direct subtraction of networks for Nagari and Arabic notations revealed a neural circuit comprising of bilateral intra-parietal sulcus, inferior and mid frontal gyri, fusiform gyrus and the anterior cingulate cortex (FDR p<0.005). Additionally, whole brain correlation analysis showed that activity in the left inferior parietal region was modulated by task performance in Nagari. We attribute the increased activation in Ng to increased task difficulty due to infrequent exposure and usage. Our results reiterate the role of the left intra-parietal sulcus in modulating performance in numeric tasks and highlight that of the attention network for monitoring symbolic notation mode in binumerates. (shrink)

The number of elements in a small set of items is appraised in a fast and exact manner, a phenomenon called subitizing. In contrast, humans provide imprecise responses when comparing larger numerosities, with decreasing precision as the number of elements increases. Estimation is thought to rely on a dedicated system for the approximate representation of numerosity. While previous behavioral and neuroimaging studies associate subitizing to a domain-general system related to object tracking and identification, the nature of small numerosity processing is (...) still debated. We investigated the neural processing of numerosity across subitizing and estimation ranges by examining electrophysiological activity during the memory retention period in a delayed numerical match-to-sample task. We also assessed potential differences in the neural signature of numerical magnitude in a fully non-symbolic or cross-format comparison. In line with behavioral performance, we observed modulation of parietal-occipital neural activity as a function of numerosity that differed in two ranges, with distinctive neural signatures of small numerosities showing clear similarities with those observed in visuospatial working memory tasks. We also found differences in neural activity related to numerical information in anticipation of single vs. cross-format comparison, suggesting a top-down modulation of numerical processing. Finally, behavioral results revealed enhanced performance in the mixed-format conditions and a significant correlation between task performance and symbolic mathematical skills. Overall, we provide evidence for distinct mechanisms related to small and large numerosity and differences in numerical encoding based on task demands. (shrink)

In this paper, we present an intensive investigation of the finite volume method compared to the finite difference methods. In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov’s scheme to simulate the (...) results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods. (shrink)

A dual-code model of number processing needs to take into account the difference between a number symbol and its meaning. The transition of automatic non-abstract number representations into intentional abstract representations could be conceptualized as a translation of perceptual asemantic representations of numerals into semantic representations of the associated magnitude information. The controversy about the nature of number representations should be thus related to theories on embodied grounding of symbols.

The so-called Ricardian trade model of contemporary economic textbooks is not a rational reconstruction of Ricardo's famous numerical example in chapter seven of the Principles. It differs from the latter in terms of the definition of the four numbers, relevant cost comparison, rule for specialisation, assumptions and theoretical implications. Thus, the widespread critique regarding the unrealistic assumptions of the textbook trade model does not apply to Ricardo's original proof of comparative advantage.

The anchoring heuristic refers to phenomena when an arbitrary number affects subsequent numerical estimations. Oppenheimer, LeBoeuf and Brewer showed that it is not necessary for the anchor to be a numerical value, yet current models describing the anchoring heuristic do not fully account for the mechanism of non-numerical anchoring. However, this effect shows similarity to the basic anchoring effect - obtained without the comparative question and based on the availability of the given number in working memory. In (...) this study, we attempt to verify whether those two effect share the same psychological mechanism. In Experiment 1, we show that non-numerical anchoring based on magnitude priming cannot be obtained when the lines are just observed. The examined mechanism proves to be dependent on the act of drawing, displaying limitations similar to the basic anchoring effect, previously pointed out by Brewer and Chapman. By using the same numerical anchors in different size formats, in Experiment 2 we showed that anchoring based on magnitude priming occurs even when the numerical values do not affect the estimations. The results are discussed in the light of a possible mechanism that underlies the investigated effect. (shrink)