Magnitude Representation Research Articles

Magnitude processing and complex calculation is negatively impacted by mathematics anxiety while retrieval-based simple calculation is not.

Mathematics anxiety (MA) refers to the experience of negative affect when engaging in mathematical activity. According to Ashcraft and Kirk (2001), MA selectively affects calculation with high working memory (WM) demand. On the other hand, Maloney, Ansari, and Fugelsang (2011) claim that MA affects all mathematical activities, including even the most basic ones such as magnitude comparison. The two theories make opposing predictions on the negative effect of MA on magnitude processing and simple calculation that make minimal demands on WM. We propose that MA has a selective impact on mathematical problem solving that likely involves processing of magnitude representations. Based on our hypothesis, MA will impinge upon magnitude processing even though it makes minimal demand on WM, but will spare retrieval-based, simple calculation, because it does not require magnitude processing. Our hypothesis can reconcile opposing predictions on the negative effect of MA on magnitude processing and simple calculation. In the present study, we observed a negative relationship between MA and performance on magnitude comparison and calculation with high but not low WM demand. These results demonstrate that MA has an impact on a wide range of mathematical performance, which depends on one's sense of magnitude, but spares over-practiced, retrieval-based calculation.

Securing highly penetrated wind energy systems using linearized transmission switching mechanism

The increasing penetration of Wind Energy Sources (WES) in an ac-network requires more Flexibility Resources (FRs), such as thermal units and transmission topology control by transmission switching (TS) action. The FRs, especially the TS action, can help to accommodate the intermittent and volatiles from WESs. Nevertheless, obstacles still remain and must be dealt before the TS action can be implemented by the real power systems. Indeed, the challenges comprise AC feasibility, the ability to handle large-scale real power systems and computational complexity. This paper investigates these challenges by developing the TS based on a linearized AC network (LAC-based TS) model that includes a linear representation of reactive power and bus voltage magnitudes. The proposed LAC-based TS model can handle high penetration of WES uncertainty in the stochastic security constrained unit commitment (SCUC). Also, the WES is the only source of uncertainty considered in this paper, which is modeled through an appropriate set of scenarios. Accordingly, the proposed model is formulated as a two-stage stochastic programming problem, wherein, the first-stage relates to the day-ahead scheduling, and the second-stage refers to the real-time operating conditions. An iterative algorithm based on the Benders decomposition method is used to solve the problem. The performance of the proposed model is investigated in details using a modified 6-bus and IEEE 118/662-bus test systems.

Controlling for continuous variables is not futile: What we can learn about number representation despite imperfect control.

Leibovich et al. argue that because it is impossible to isolate numerosity in a stimulus set, attempts to show that number is processed independently of continuous magnitudes are necessarily in vain. I propose that through clever design and manipulation of confounding variables, we can gain deep insight into number representation and its relationship to the representation of other magnitudes.

Two-Digit Number Processing during Childhood: A Mental Rotation Task?

The association between numerical and spatial thinking manifests itself throughout various cognitive observations, concerning arithmetic problem solving as well as simple number recognition tasks. The spatial component in numerical thinking is assumed to represent the semantic processing pathway, for example, the mental representation of numerical magnitude. There is an interconnection of semantic, linguistic, and visual/symbolic pathways, which is thought to be stronger in adults than in children. During childhood, the semantic pathway is supposed to play a major part in numerical processing. In this study, we conducted a functional magnetic resonance imaging paradigm combining a mental rotation task with a number processing task. The behavioral data showed the highest error rate for two-digit numbers (not for three-digit numbers). In regard to two-digit numbers, we also found the strongest neuronal activation overlap for number processing and mental rotation in the left intraparietal sulcus. This pattern suggests that the mental inversion of digits in native German speakers might be a spatial function related to mental rotation. In terms of overcoming opposed informational input for symbolic and linguistic representation and therefore impeded informational extraction the spatial processing pathway might play a major role. To conclude, spatial training could be an effective educational approach, supporting the interconnection of numerical processing pathways.

A critical analysis of design, facts, bias and inference in the approximate number system training literature: A systematic review

Abstract A popular suggestion states that an evolutionarily grounded analogue magnitude representation, also called an approximate number system (ANS) or ‘number sense’ underlies human mathematical knowledge. During recent years many studies aimed to train the ANS with the intention of transferring improvements to symbolic arithmetic. Methods: We critically evaluated all published studies that aimed to train the ANS. We also carried out a citation analysis of the training studies. Results: There is no conclusive evidence that specific ANS training improves symbolic arithmetic. The citation analysis demonstrates that highly controversial results often get cited in support of specific claims without discussion of controversies. Conclusions: We suggest ways to run future training studies so that clear evidence can be collected and also suggest that data should be discussed in considering both supporting and contrary evidence and arguments.

Improving Preschoolers' Arithmetic through Number Magnitude Training: The Impact of Non-Symbolic and Symbolic Training.

The numerical cognition literature offers two views to explain numerical and arithmetical development. The unique-representation view considers the approximate number system (ANS) to represent the magnitude of both symbolic and non-symbolic numbers and to be the basis of numerical learning. In contrast, the dual-representation view suggests that symbolic and non-symbolic skills rely on different magnitude representations and that it is the ability to build an exact representation of symbolic numbers that underlies math learning. Support for these hypotheses has come mainly from correlative studies with inconsistent results. In this study, we developed two training programs aiming at enhancing the magnitude processing of either non-symbolic numbers or symbolic numbers and compared their effects on arithmetic skills. Fifty-six preschoolers were randomly assigned to one of three 10-session-training conditions: (1) non-symbolic training (2) symbolic training and (3) control training working on story understanding. Both numerical training conditions were significantly more efficient than the control condition in improving magnitude processing. Moreover, symbolic training led to a significantly larger improvement in arithmetic than did non-symbolic training and the control condition. These results support the dual-representation view.

Spatial reference frames and SNARC

ABSTRACTIn this work, the role that spatial reference frames play in determining the spatial-numerical association of response codes (SNARC) effect is examined. Participants were instructed to generate an image of the numbers 1–9 oriented spatially in either a horizontal, vertical, or proximo-distal manner. Responses to a magnitude comparison task were then provided manually using keys located either to the left and right or at the bottom and top of a computer keyboard. For conditions in which the orientation of the generated image and the response locations were orthogonally misaligned, SNARC effects were not evident. It is argued that such results imply that alignment of the spatial reference frames associated with the responses and with the representation of numerical magnitude is a necessary condition for the elicitation of the SNARC effect.

Arithmetic Training Does Not Improve Approximate Number System Acuity.

The approximate number system (ANS) is thought to support non-symbolic representations of numerical magnitudes in humans. Recently much debate has focused on the causal direction for an observed relation between ANS acuity and arithmetic fluency. Here we investigate if arithmetic training can improve ANS acuity. We show with an experimental training study consisting of six 45-min training sessions that although feedback during arithmetic training improves arithmetic performance substantially, it does not influence ANS acuity. Hence, we find no support for a causal link where symbolic arithmetic training influences ANS acuity. Further, although short-term number memory is likely involved in arithmetic tasks we did not find that short-term memory capacity for numbers, measured by a digit-span test, was effected by arithmetic training. This suggests that the improvement in arithmetic fluency may have occurred independent of short-term memory efficiency, but rather due to long-term memory processes and/or mental calculation strategy development. The theoretical implications of these findings are discussed.

Independent coding of absolute duration and distance magnitudes in the prefrontal cortex.

Human behavioral studies have shown that spatial and duration judgments can interfere with each other. We investigated the neural representation of such magnitudes in the prefrontal cortex. We found that the two magnitudes are independently coded by prefrontal neurons. We suggest that the interference among magnitude judgments might depend on the goal rather than the perceptual resource sharing.

Symbolic and non-symbolic number processing in children with developmental dyslexia

This study examined number processing in 10-year-olds with developmental dyslexia (DD). The phonological deficit and double deficit hypotheses imply that children with DD might have a connection deficit that affects their ability to establish links between number symbols and magnitude representations. The double deficit hypothesis also posits that symbolic number difficulties may emerge due to difficulties with processes underlying rapid automatic naming (RAN). The DD group displayed difficulties with symbolic number processing but not with non-symbolic number processing. However, the underlying processes of this access or connection deficit appeared not to be related to phonological awareness or RAN. The DD group displayed impaired arithmetic fluency and calculation that were accounted for by defective processes underlying RAN. In view of the triple-code model, children with DD have impaired verbal number codes or defective access to verbal number codes but an intact core magnitude representation.

Common and distinct brain regions in both parietal and frontal cortex support symbolic and nonsymbolic number processing in humans: A functional neuroimaging meta-analysis

In recent years, there has been substantial growth in neuroimaging studies investigating neural correlates of symbolic (e.g. Arabic numerals) and non-symbolic (e.g. dot arrays) number processing. At present it remains contested whether number is represented abstractly, or if number representations in the brain are format-dependent. In order to quantitatively evaluate the available neuroimaging evidence, we used activation likelihood estimation (ALE) to conduct quantitative meta-analyses of the results reported in 57 neuroimaging papers. Consistent with the existence of an abstract representation of number in the brain, conjunction analyses revealed overlapping activation for symbolic and nonsymbolic numbers in frontal and parietal lobes. Consistent with the notion of format-dependent activation, contrast analyses demonstrated anatomically distinct fronto-parietal activation for symbolic and non-symbolic processing. Therefore, symbolic and non-symbolic numbers are subserved by format-dependent and abstract neural systems. Moreover, the present results suggest that regions across the parietal cortex, not just the intraparietal sulcus, are engaged in both symbolic and non-symbolic number processing, challenging the notion that the intraparietal sulcus is the key region for number processing. Additionally, our analyses indicate that regions in the frontal cortex subserve magnitude representations rather than non-numerical cognitive processes associated with number tasks, thereby highlighting the importance of considering both frontal and parietal regions as important for number processing.

Relativistic locality and the action reaction principle predict de Broglie fields

The action reaction principle is violated in the standard formulation of Quantum Mechanics, so that its phase (Hilbert) space is incomplete. Moreover, projection of state of a quantum system under indirect measurement, when there are alternative virtual paths and one of them is discarded by negative detection implies, according to the action reaction principle, a reaction on the detector, although its macroscopic state does not change. If all interactions are local, mediated by fields with relativistically causal evolution, some system different from the particle (which follows another path) must locally interact with the detector. Relativistic locality and the action reaction principle predict the existence of de Broglie fields. It is presented a formulation of Quantum Mechanics in extended Hilbert spaces, where kinematic and dynamical representations of physical magnitudes are distinguished.

ViSimpl: Multi-View Visual Analysis of Brain Simulation Data.

After decades of independent morphological and functional brain research, a key point in neuroscience nowadays is to understand the combined relationships between the structure of the brain and its components and their dynamics on multiple scales, ranging from circuits of neurons at micro or mesoscale to brain regions at macroscale. With such a goal in mind, there is a vast amount of research focusing on modeling and simulating activity within neuronal structures, and these simulations generate large and complex datasets which have to be analyzed in order to gain the desired insight. In such context, this paper presents ViSimpl, which integrates a set of visualization and interaction tools that provide a semantic view of brain data with the aim of improving its analysis procedures. ViSimpl provides 3D particle-based rendering that allows visualizing simulation data with their associated spatial and temporal information, enhancing the knowledge extraction process. It also provides abstract representations of the time-varying magnitudes supporting different data aggregation and disaggregation operations and giving also focus and context clues. In addition, ViSimpl tools provide synchronized playback control of the simulation being analyzed. Finally, ViSimpl allows performing selection and filtering operations relying on an application called NeuroScheme. All these views are loosely coupled and can be used independently, but they can also work together as linked views, both in centralized and distributed computing environments, enhancing the data exploration and analysis procedures.

Cognitive factors affecting children’s nonsymbolic and symbolic magnitude judgment abilities: A latent profile analysis

Early math abilities are claimed to be linked to magnitude representation ability. Some claim that nonsymbolic magnitude abilities scaffold the acquisition of symbolic (Arabic number) magnitude abilities and influence math ability. Others claim that symbolic magnitude abilities, and ipso facto math abilities, are independent of nonsymbolic abilities and instead depend on the ability to process number symbols (e.g., 2, 7). Currently, the issue of whether symbolic abilities are or are not related to nonsymbolic abilities, and the cognitive factors associated with nonsymbolic–symbolic relationships, remains unresolved. We suggest that different nonsymbolic–symbolic relationships reside within the general magnitude ability distribution and that different cognitive abilities are likely associated with these different relationships. We further suggest that the different nonsymbolic–symbolic relationships and cognitive abilities in combination differentially predict math abilities. To test these claims, we used latent profile analysis to identify nonsymbolic–symbolic judgment patterns of 124, 5- to 7-year-olds. We also assessed four cognitive factors (visuospatial working memory [VSWM], naming numbers, nonverbal IQ, and basic reaction time [RT]) and two math abilities (number transcoding and single-digit addition abilities). Four nonsymbolic–symbolic ability profiles were identified. Naming numbers, VSWM, and basic RT abilities were differentially associated with the different ability profiles and in combination differentially predicted math abilities. Findings show that different patterns of nonsymbolic–symbolic magnitude abilities can be identified and suggest that an adequate account of math development should specify the inter-relationship between cognitive factors and nonsymbolic–symbolic ability patterns.

The Spatial Representation of Angles.

We investigated whether angle magnitude, similarly to numerical quantities (i.e., the spatial-numerical association of response codes effect), is associated to the side of response execution. In addition, we investigated whether this association has the properties of a spatially oriented mental line, since angles are taught in a right-to-left progression. We tested two groups of participants: civil engineering students (high familiarity with angles) and psychology students (low familiarity with angles). In Experiment 1, participants were asked to judge the continuity of the angles' arms (continuous vs. dashed). Magnitude of the angles was task-irrelevant. In Experiment 2, they were asked to judge whether the presented angles were smaller or larger than a right angle (90°). Therefore, the angle magnitude was relevant for performing the task. Overall, engineering students responded faster with their left hand to large angles and with their right hand to small angles. Conversely, psychology students did not show any reliable differences between left- and right-hand responses. In the case of engineering students, the spatial association has a right-to-left (counter clockwise) direction, suggesting the influence of education and practice on the mental representation of angle magnitude.

Delaware Longitudinal Study of Fraction Learning: Implications for Helping Children With Mathematics Difficulties

The goal of the present article is to synthesize findings to date from the Delaware Longitudinal Study of Fraction Learning. The study followed a large cohort of children (N = 536) between Grades 3 and 6. The findings showed that many students, especially those with diagnosed learning disabilities, made minimal growth in fraction knowledge and that some showed only a basic grasp of the meaning of a fraction even after several years of instruction. Children with low growth in fraction knowledge during the intermediate grades were much more likely to fail to meet state standards on a broad mathematics measure at the end of Grade 6. Although a range of general and mathematics-specific competencies predicted fraction outcomes, the ability to estimate numerical magnitudes on a number line was a uniquely important marker of fraction success. Many children with mathematics difficulties have deep-seated problems related to whole number magnitude representations that are complicated by the introduction of fractions into the curriculum. Implications for helping students with mathematics difficulties are discussed.

Relational quantitative reasoning in kindergarten predicts mathematical achievement in third grade

Tremendous variation in elementary school children’s mathematical achievement can partly be traced back to differences in early domain-specific quantitative competencies. While previous research mainly focused on numerical magnitude representation and counting, we tested the long-term effects of relational quantitative reasoning. Before children (N = 51) entered school (i.e. at age 5-6), we assessed this competence with a test that required no knowledge about Arabic numerals. Two and a half years later, when children were in third grade of elementary school, we gauged mathematical achievement, general reasoning ability, and reading skills. A multiple regression analysis with mathematical achievement as outcome variable revealed a small but unique impact of children’s relational quantitative reasoning in kindergarten on their later mathematical achievement after controlling for general reasoning and reading abilities. Thus, a considerable amount of individual differences in mathematics achievement in elementary school results from differences in early relational quantity understanding that emerge before systematic instruction starts.

Defective Number Sense or Impaired Access? Differential Impairments in Different Subgroups of Children With Mathematics Difficulties

Developmental dyscalculia (DD) is a specific learning disability in mathematics that affects around 6% of the population. Currently, the core deficit of DD remains unknown. While the number sense deficit hypothesis suggests that the core deficit of DD lies in the inability to represent nonsymbolic numerosity, the access deficit hypothesis suggests that the origin of this disability lies in the inability to associate numbers with the underlying magnitude representation. The present study compared the performance of DDs with their low-achieving (LA) and normally achieving peers in nonsymbolic numerosity processing and number-magnitude mapping over 1 year (from kindergarten to 1st grade). The results demonstrated differential impairments in different subgroups of children with mathematics difficulties. While DDs showed deficits in both nonsymbolic numerosity processing and number-magnitude mapping, LAs showed deficit only in the number-magnitude mapping. Furthermore, the deficit in number-magnitude mapping among the DD group was partially explained by their number sense deficit. The number sense deficit hypothesis is supported. Theoretical and practical implications are discussed.

Neural representations of magnitude for natural and rational numbers

Humans have developed multiple symbolic representations for numbers, including natural numbers (positive integers) as well as rational numbers (both fractions and decimals). Despite a considerable body of behavioral and neuroimaging research, it is currently unknown whether different notations map onto a single, fully abstract, magnitude code, or whether separate representations exist for specific number types (e.g., natural versus rational) or number representations (e.g., base-10 versus fractions). We address this question by comparing brain metabolic response during a magnitude comparison task involving (on different trials) integers, decimals, and fractions. Univariate and multivariate analyses revealed that the strength and pattern of activation for fractions differed systematically, within the intraparietal sulcus, from that of both decimals and integers, while the latter two number representations appeared virtually indistinguishable. These results demonstrate that the two major notations formats for rational numbers, fractions and decimals, evoke distinct neural representations of magnitude, with decimals representations being more closely linked to those of integers than to those of magnitude-equivalent fractions. Our findings thus suggest that number representation (base-10 versus fractions) is an important organizational principle for the neural substrate underlying mathematical cognition.

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 (Mage=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.