Representation Of Numerical Magnitude Research Articles

Endogenous attention modulates automaticity of number processing.

Numerals (i.e., symbolic representations of numerical magnitude) are widespread in our environment and are fundamental to many decisions we make. It has been suggested that the processing of numerical magnitude is automatic. Various robust psychological effects, such as the distance effect (Moyer & Landauer, Nature, 215 (5109), 1519-1520, 1967) and the physical size-congruity effect (SiCE; Henik & Tzelgov, Memory & Cognition, 10 (4), 389-395, 1982), support this claim. Importantly, these effects demonstrate that the processing of numerical magnitude occurs unintentionally and while irrelevant to the task. These aspects often serve as criteria to assess the automatic nature of mental processes. However, evidence for the involvement of attention in the processing of magnitude of numerals somewhat subverts the automaticity account that was originally put forward. To reconcile between evidence in support of the automaticity account and evidence that compromises this account, we drew on another account of automaticity. This account distinguishes between strongly automatic and partly automatic mental processes based on their susceptibility to attentional influences. In the current study, we manipulated endogenous attention while participants completed numerical and physical comparisons of numerals, separately. We observed modulations of the SiCE for physical comparisons but not for numerical comparisons of numerals. That is, the processing of numerical magnitude when irrelevant was subjected to attentional influences, but the processing of their physical magnitude (i.e., size) was not. Therefore, we concluded that processing the numerical magnitude is partly automatic, whereas processing their physical magnitude is strongly automatic.

The numerical stroop effect on consumer preference to order healthy food

This article explores an alternative strategy to subtly nudge decisions in preference for healthy menu items – managing calorie presentation to create a Numerical Stroop Effect drawing upon the numerical magnitude representation theory. Across two experiments, the research finds that incongruent (vs. congruent) numerical magnitude representation of food calories (e.g., small 7 and large 5) increases consumers' preference to order healthy items. The study reveals the effect’s underlying mechanism through perceived healthiness along with the boundary factors of consumers’ health consciousness and time pressure. Specifically, low health-conscious consumers show a preference to order healthy items when exposed to incongruent magnitude representation of food calories, particularly when they experience high time pressure. The research contributes to the hospitality literature with an in-depth understanding of the less explored effect of numerical magnitude representation. This research also provides an easily implementable menu strategy to influence consumers’ dining preferences for healthier alternatives.

Developmental trajectories of numerical magnitude representations of fractions and decimals.

We examined the development of numerical magnitude representations of fractions and decimals from fourth to 12th grade. In Experiment 1, we assessed the rational number magnitude knowledge of 200 Chinese fourth, fifth, sixth, eighth, and 12th graders (92 girls and 108 boys) by presenting fraction and decimal magnitude comparison tasks as well as fraction and decimal 0-1 and 0-5 number line estimation tasks. Magnitude representations of decimals became accurate earlier, improved more rapidly, and reached a higher asymptotic accuracy than magnitude representations of fractions. Analyses of individual differences revealed positive relations between the accuracy of decimal and fraction magnitude representations at all ages. In Experiment 2, we presented an additional set of 24 fourth graders (14 girls and 10 boys) with the same tasks but with the decimals that were being compared varying in the number of decimal digits. The decimal advantage continued to be present for both magnitude comparison and estimation tasks, indicating that the greater accuracy with decimals was not limited to decimals with equal numbers of decimal digits, though unequal numbers of decimal digits did impact performance with decimals on both magnitude comparison and number line estimation tasks. Implications for understanding numerical development and education are discussed. (PsycInfo Database Record (c) 2023 APA, all rights reserved).

Supplemental Material for Developmental Trajectories of Numerical Magnitude Representations of Fractions and Decimals

Supplemental Material for Developmental Trajectories of Numerical Magnitude Representations of Fractions and Decimals

Structural and functional connectivity of the inferior temporal numeral area.

A growing body of evidence suggests that in adults, there is a spatially consistent "inferior temporal numeral area" (ITNA) in the occipitotemporal cortex that appears to preferentially process Arabic digits relative to non-numerical symbols and objects. However, very little is known about why the ITNA is spatially segregated from regions that process other orthographic stimuli such as letters, and why it is spatially consistent across individuals. In the present study, we used diffusion-weighted imaging and functional magnetic resonance imaging to contrast structural and functional connectivity between left and right hemisphere ITNAs and a left hemisphere letter-preferring region. We found that the left ITNA had stronger structural and functional connectivity than the letter region to inferior parietal regions involved in numerical magnitude representation and arithmetic. Between hemispheres, the left ITNA showed stronger structural connectivity with the left inferior frontal gyrus (Broca's area), while the right ITNA showed stronger structural connectivity to the ipsilateral inferior parietal cortex and stronger functional coupling with the bilateral IPS. Based on their relative connectivity, our results suggest that the left ITNA may be more readily involved in mapping digits to verbal number representations, while the right ITNA may support the mapping of digits to quantity representations.

Cognitive predictors of children’s arithmetic principle understanding

The understanding of arithmetic principles (APs) is an important component of our conceptual mathematical knowledge, but we have limited knowledge about how children acquire APs. The current study examined this issue through a longitudinal study with 273 Chinese fourth graders. These children were assessed on various cognitive capacities (e.g., verbal and visuospatial working memory, processing speed, inhibition skill, numerical magnitude representation) in Grade 4 as well as on their AP understanding using multifaceted assessment and their arithmetic competence in Grade 5. Results from structural equation modeling suggested that verbal memory and inhibition skill directly predicted AP understanding, which in turn predicted arithmetic competence. Visuospatial working memory predicted AP understanding through numerical magnitude representation. The findings allow researchers to better account for the individual differences in AP understanding.

Preschoolers prior formal mathematics education engage numerical magnitude representation rather than counting principles in symbolic ±1 arithmetic: Evidence from the operational momentum effect.

In numerical cognition research, the operational momentum (OM) phenomenon (tendency to overestimate the results of addition and/or binding addition to the right side and underestimating subtraction and/or binding it to the left side) can help illuminate the most basic representations and processes of mental arithmetic and their development. This study is the first to demonstrate OM in symbolic arithmetic in preschoolers. It was modeled on Haman and Lipowska's (2021) non-symbolic arithmetic task, using Arabic numerals instead of visual sets. Seventy-seven children (4-7 years old) who know Arabic numerals and counting principles (CP), but without prior school math education, solved addition and subtraction problems presented as videos with one as the second operand. In principle, such problems may be difficult when involving a non-symbolic approximate number processing system, whereas in symbolic format they can be solved based solely on the successor/predecessor functions and knowledge of numerical orders, without reference to representation of numerical magnitudes. Nevertheless, participants made systematic errors, in particular, overestimating results of addition in line with the typical OM tendency. Moreover, subtraction and addition induced longer response times when primed with left- and right-directed movement, respectively, which corresponds to the reversed spatial form of OM. These results largely replicate those of non-symbolic task and show that children at early stages of mastering symbolic arithmetic may rely on numerical magnitude processing and spatial-numerical associations rather than newly-mastered CP and the concept of an exact number.

A shared numerical magnitude representation evidenced by the distance effect in frequency-tagging EEG

Humans can effortlessly abstract numerical information from various codes and contexts. However, whether the access to the underlying magnitude information relies on common or distinct brain representations remains highly debated. Here, we recorded electrophysiological responses to periodic variation of numerosity (every five items) occurring in rapid streams of numbers presented at 6 Hz in randomly varying codes—Arabic digits, number words, canonical dot patterns and finger configurations. Results demonstrated that numerical information was abstracted and generalized over the different representation codes by revealing clear discrimination responses (at 1.2 Hz) of the deviant numerosity from the base numerosity, recorded over parieto-occipital electrodes. Crucially, and supporting the claim that discrimination responses reflected magnitude processing, the presentation of a deviant numerosity distant from the base (e.g., base “2” and deviant “8”) elicited larger right-hemispheric responses than the presentation of a close deviant numerosity (e.g., base “2” and deviant “3”). This finding nicely represents the neural signature of the distance effect, an interpretation further reinforced by the clear correlation with individuals’ behavioral performance in an independent numerical comparison task. Our results therefore provide for the first time unambiguously a reliable and specific neural marker of a magnitude representation that is shared among several numerical codes.

Influences of hand action on the processing of symbolic numbers: A special role of pointing?

Embodied and grounded cognition theories state that cognitive processing is built upon sensorimotor systems. In the context of numerical cognition, support to this framework comes from the interactions between numerical processing and the hand actions of reaching and grasping documented in skilled adults. Accordingly, mechanisms for the processing of object size and location during reach and grasp actions might scaffold the development of mental representations of numerical magnitude. The present study exploited motor adaptation to test the hypothesis of a functional overlap between neurocognitive mechanisms of hand action and numerical processing. Participants performed repetitive grasping of an object, repetitive pointing, repetitive tapping, or passive viewing. Subsequently, they performed a symbolic number comparison task. Importantly, hand action and number comparison were functionally and temporally dissociated, thereby minimizing context-based effects. Results showed that executing the action of pointing slowed down the responses in number comparison. Moreover, the typical distance effect (faster responses for numbers far from the reference as compared to close ones) was not observed for small numbers after pointing, while it was enhanced by grasping. These findings confirm the functional link between hand action and numerical processing, and suggest new hypotheses on the role of pointing as a meaningful gesture in the development and embodiment of numerical skills.

Numerical estimation strategies are correlated with math ability in school-aged children

Various laboratory-based numerical cognition measures are often found to be linked with standardized math scores, yet the exact nature of these links over development remains in question. In this study, we investigated how 0–100 number line estimation performance and dot numerosity discrimination explained variance in 6- to 8-year-olds’ standardized math ability scores, and we assessed links between tasks. Our primary goal was to assess the link between estimation performance and math ability from a theoretical perspective that models number line estimation as an instance of proportion estimation. As age increased, children’s proportion-based strategies for number line estimation became more sophisticated and estimation bias decreased. Children who spontaneously inferred and used a central reference point to guide their proportion estimates scored higher on the math ability measure (TEMA-3), demonstrating a relation between more advanced proportion estimation strategies and higher math abilities. Individual differences in estimation bias also made a lesser contribution to explaining variance in children’s math ability. Dot discrimination accuracy uniquely contributed to children’s math ability. Number line estimation performance and dot discrimination accuracy were not associated, consistent with prior research suggesting that number line estimation does not directly map onto mental numerical magnitude representations. Educational implications and future directions are discussed.

Operational momentum during children’s approximate arithmetic relates to symbolic math skills and space–magnitude association

Operational momentum (OM) refers to the behavioral tendency to overestimate or underestimate the results of addition or subtraction, respectively. The cognitive mechanism of the OM effect and how it is related to the development of symbolic math abilities are not well understood. The current study examined whether individual differences in the OM effect are related to symbolic arithmetic abilities, number line estimation performance, and the space–magnitude association effect in young children. In this study, first-grade elementary school children manifested the OM effect during approximate addition and subtraction. Individual differences in the OM effect were not correlated with number line estimation error. Interestingly, children who showed a greater degree of the OM effect performed not worse, but better on the symbolic arithmetic task. In addition, the OM effect was correlated with the space–magnitude association (size congruity) effect measured with the Numerical Stroop task. More specifically, the OM bias was correlated with the ability to inhibit interference from competing information on the incongruent trials of the Numerical Stroop task. Our results suggest that the inaccuracy of numerical magnitude representations is not the source of the OM effect. Given that children with better math ability showed a greater OM bias, a stronger OM effect may reflect better intuition in arithmetic operations. Altogether, we carefully interpret these findings as suggesting that a greater OM effect reflects superior intuition or fundamental knowledge of arithmetic operations and a more adult-like maturation of the reorienting component of the attentional system.

Numerical magnitude understanding in kindergartners: A specific and sensitive predictor of later mathematical difficulties?

Difficulties in mathematics are argued to stem from impairments of a specialized system of numerical magnitude representation. This study investigates whether different measures of numerical magnitude understanding in kindergarten uniquely predict mathematical achievement concurrently and 6 months later, and also examines the relative explanatory of each aspect of magnitude understanding. It also examines the utility of such tasks as sensitive and specific predictors of children at risk of mathematical learning difficulties. Kindergartners (N = 215, M age = 74 months) were administered a number-line estimation task, a nonsymbolic numerical discrimination task, and a nonsymbolic to word mapping task. Multiple measures of math achievement were administered concurrently and upon entering formal education (M age = 80 months). Path analysis revealed that all 3 aspects of magnitude understanding had similar explanatory power in predicting concurrent and prospective math achievement; the only exception was that nonsymbolic numerical discrimination was not a significant predictor of concurrent mathematical reasoning. Of the 3 measures of magnitude understanding, number-line estimation was found to be the most accurate predictor of persistent mathematical difficulties. Approximately half of the children misclassified as having mathematical difficulties showed inconsistent mathematical achievement across the multiple measures. The implications for screening for mathematical difficulties and for designing interventions are discussed. (PsycInfo Database Record (c) 2021 APA, all rights reserved)

Number line tasks and their relation to arithmetics in second to fourth graders

Considering the importance of mathematical knowledge for STEM careers, we aimed to better understand the cognitive mechanisms underlying the commonly observed relation between number line estimations (NLEs) and arithmetics. We used a within-subject design to model NLEs in an unbounded and bounded task and to assess their relations to arithmetics in second to fourth grades. Our results mostly agree with previous findings, indicating that unbounded and bounded NLEs likely index different cognitive constructs at this age. Bounded NLEs were best described by cyclic power models including the subtraction bias model, likely indicating proportional reasoning. Conversely, mixed log-linear and single scalloped power models provided better fits for unbounded NLEs, suggesting direct estimation. Moreover, only bounded but not unbounded NLEs related to addition and subtraction skills. This thus suggests that proportional reasoning probably accounts for the relation between NLEs and arithmetics, at least in second to fourth graders. This was further confirmed by moderation analysis, showing that relations between bounded NLEs and subtraction skills were only observed in children whose estimates were best described by the cyclic power models. Depending on the aim of future studies, our results suggest measuring estimations on unbounded number lines if one is interested in directly assessing numerical magnitude representations. Conversely, if one aims to predict arithmetic skills, one should assess bounded NLEs, probably indexing proportional reasoning, at least in second to fourth graders. The present outcomes also further highlight the potential usefulness of training the positioning of target numbers on bounded number lines for arithmetic development.

Neurodevelopmental differences in child and adult number processing: An fMRI-based validation of the triple code model

The triple code model of numerical cognition (TCM) details the neurocognitive mechanisms associated with perceiving and manipulating numerical information in exact symbolic (Arabic digits and number words) and approximate nonsymbolic numerical magnitude (e.g., dot arrays) representation codes. The current study provides a first empirical fMRI-based investigation into neurodevelopmental differences in 30 healthy children’s and 44 healthy adults’ recruitment of neural correlates associated with the Arabic digit, number word, and nonsymbolic magnitude codes. Differences between the two groups were found in cingulate regions commonly associated with domain-general aspects of cognitive control, as opposed to neural correlates of number processing per se. A primary developmental difference was identified in verbal number discrimination, where only adults recruited left-lateralized perisylvian language areas in accordance with the TCM. We therefore call for a revision of the verbal code and a formulation of separate child and adult-specific neurocognitive mechanisms associated with the discrimination of number words. Although further research is necessary, results indicate that numerical discrimination abilities in middle-school-aged children operate close to adult-level maturity. Neurodevelopmental differences may be more apparent in younger children, or on the level of functional network dynamics as opposed to a shift in recruited neural substrates.

The Force of Numbers: Investigating Manual Signatures of Embodied Number Processing.

The study has two objectives: (1) to introduce grip force recording as a new technique for studying embodied numerical processing; and (2) to demonstrate how three competing accounts of numerical magnitude representation can be tested by using this new technique: the Mental Number Line (MNL), A Theory of Magnitude (ATOM) and Embodied Cognition (finger counting-based) account. While 26 healthy adults processed visually presented single digits in a go/no-go n-back paradigm, their passive holding forces for two small sensors were recorded in both hands. Spontaneous and unconscious grip force changes related to number magnitude occurred in the left hand already 100–140 ms after stimulus presentation and continued systematically. Our results support a two-step model of number processing where an initial stage is related to the automatic activation of all stimulus properties whereas a later stage consists of deeper conscious processing of the stimulus. This interpretation generalizes previous work with linguistic stimuli and elaborates the timeline of embodied cognition. We hope that the use of grip force recording will advance the field of numerical cognition research.

Nonsymbolic and Symbolic Numerical Magnitude Processing in the Brazilian Children with Mathematics Difficulties.

ABSTRACTIt is still debated if the main deficit in mathematical difficulties (MD) is nonsymbolic or symbolic numerical magnitude processing.Objectives:In the present study, our main goal was to investigate nonsymbolic and symbolic numerical magnitude processing in MD and the relationship between these abilities and arithmetic.Methods:The Brazilian school-age children with MD completed a nonsymbolic and a symbolic numerical magnitude comparison task and an arithmetic task. We compared their performance with a group of children with typical achievement (TA) and investigated the association between numerical magnitude processing and arithmetic with a series of regression analyses.Results:Results indicated that children with MD had low performance in the nonsymbolic numerical magnitude comparison task. Performance in both nonsymbolic and symbolic numerical magnitude comparison tasks predicted arithmetic abilities in children with TA, but not in children with MD.Conclusions:These results indicate that children with MD have difficulties in nonsymbolic numerical magnitude processing, and do not engage basic numerical magnitude representations to solve arithmetic.

Are the acuities of magnitude representations of different types and ranges of numbers related? Testing the core assumption of the integrated theory of numerical development

The current study tested whether the magnitude representation acuities of different types and ranges of numbers, as proposed in the integrated theory of numerical development (Siegler, 2016), are significantly related to each other. A sample of 123 kindergarteners was assessed four times over the period from kindergarten to fourth grade on magnitude representation acuities of different types and ranges of numbers (nonsymbolic numerical magnitude in kindergarten, small whole-number magnitude in grade 1, large whole-number magnitude in grade 2, rational number magnitude in grade 4). The children were also evaluated for their mathematics achievement, intelligence, working memory capacity, reading skills, attention level, and multiplication skills. The results showed that the magnitude representation acuities of different types and ranges of numbers were significantly related to each other, and these numerical magnitude representation acuities were either directly or indirectly related to children’s mathematics achievement in grade 4. The findings from this work provide empirical support to the core assumption of the integrated theory of numerical development and highlight the significance of numerical magnitude representations at an early developmental stage to the acquisition of more advanced numerical magnitude representations in later elementary school years.

Judging the order of numbers relies on familiarity rather than activating the mental number line

A series of effects characterises the processing of symbolic numbers (i.e., distance effect, size effect, SNARC effect, size congruency effect). The combination of these effects supports the view that numbers are represented on a compressed and spatially oriented mental number line (MNL) as well as the presence of an interaction between numerical and other magnitude representations. However, when individuals process the order of digits, response times are faster when the distance between digits is small (e.g., 1-2-3) compared to large (e.g., 1-3-5; i.e., reversed distance effect), suggesting that the processing of magnitude and order may be distinct. Here, we investigated whether the effects related to the MNL also emerge in the processing of symbolic number ordering. In Experiment 1, participants judged whether three digits were presented in order while spatial distance, numerical distance, numerical size, and the side of presentation were manipulated. Participants were faster in determining the ascending order of small triplets compared to large ones (i.e., size effect) and faster when the numerical distance between digits was small (i.e., reversed distance effect). In Experiment 2, we explored the size effect across all possible consecutive triplets between 1 and 9 and the effect that physical size has on order processing. Participants showed faster reactions times only for the triplet 1-2-3 compared to the other triplets, and the effect of physical magnitude was negligible. Symbolic order processing lacks the signatures of the MNL and suggests the presence of a familiarity effect related to well-known consecutive triplets in the long-term memory.

Serial dependence generalizes across different stimulus formats, but not different sensory modalities

Visual perception is thought to be supported by a stabilization mechanism integrating information over time, resulting in a systematic attractive bias in experimental contexts. Previous studies show that this effect, whereby a current stimulus appears more similar to the one previous to it, depends on attention, suggesting an active high-level mechanism that modulates perception. Here, we test the hypothesis that such a mechanism generalizes across different stimulus formats or sensory modalities, effectively abstracting from the low-level properties of the stimuli. Participants performed a numerosity discrimination task, with task-relevant dot-array stimuli preceded by a sequence of visual (flashes) or auditory (tones) stimuli encompassing different numerosities. Our results show a clear attractive bias induced by visual sequential numerosity affecting an array of simultaneously presented dots, thus operating across different stimulus formats. Conversely, auditory sequences did not affect the judgment on visual numerosities. Overall, our results demonstrate that serial dependence in numerosity perception operates according to the abstract representation of numerical magnitude of visual stimuli irrespective of their format. These results thus support the idea that a high-level mechanism mediates visual stability and continuity, which integrates relevant information over time irrespective of the low-level sensory properties of the stimuli.

Visuospatial pathways to mathematical achievement

Children having better visuospatial working memory, or the capacity to store and manipulate visuospatial information, perform better in mathematics. Yet, the underlying mechanisms are not well understood. In this study, we proposed a pathway model which linked up visuospatial working memory and mathematical achievement through two routes: numerical magnitude representation and problem representation. The data were drawn from 541 children who were assessed in both Grades 1 and 2. Using path analysis, we found that the association between visuospatial working memory and mathematical achievement was mediated by the deployment of visuospatial processes for magnitude representation (magnitude representation pathway) as well as for problem representation in solving mathematical problems (problem representation pathway). Such findings offer an important framework for developing intervention strategies to help children with poor visuospatial working memory in learning mathematics.