Mental Representation Of Numbers Research Articles

Practice-induced SNARC: evidence from a null-SNARC sample.

The mental representation of numbers inherently involves a spatial organization, often positioning smaller numbers to the left and larger numbers to the right. The SNARC effect, characterized by faster responses to small numbers using the left hand and vice versa for large numbers, is typically attributed to this left-to-right oriented mental number line (MNL). However, the direction of the SNARC effect seems to rely on reading direction, with most research exploring these mechanisms conducted within left-to-right reading cultures where the SNARC effect is prevalent. This study takes advantage of a sample from a left-to-right reading culture that does not exhibit the SNARC effect, allowing us to isolate and elucidate the stand-alone effects of recent experiences on SNARC. Therefore, the current study aims to investigate how MNL-compatible and MNL-incompatible practices induce an effect within a sample lacking the SNARC effect. To accomplish this, we reinvited the individuals from the sample which had previously shown no SNARC, and retested those who agreed to take part in the current study after an MNL-compatible or MNL-incompatible practice manipulation. The findings revealed an absence of the SNARC effect with MNL-compatible practices. Conversely, MNL-incompatible practices yielded a reverse SNARC effect. These results prompt a discussion on SNARC mechanisms within the framework of practice effects.

Distance in depth: A comparison of explicit and implicit numerical distances in the horizontal and radial dimensions.

Numbers are a constant presence in our daily lives: A brain devoid of the ability to process numbers would not be functional in its external environment. Comparing numerical magnitudes is a fundamental ability that requires the processing of numerical distances. From magnitude comparison tasks, a comparison distance effect (DE) emerges: It describes better performance when comparing numerically distant rather than close numbers. Unlike other signatures of number processing, the comparison DE has been assessed only implicitly, with numerical distance as nonsalient task property. Different assessments permit identification of different cognitive processes underlying a specific effect. To investigate whether explicit and implicit assessment of the comparison DE influences numerical cognition differently, we introduced the distance classification task, involving explicit classification of numbers as close or far from a reference. N = 93 healthy adults classified numbers either by magnitude or by numerical distance. To investigate associations between numerical and physical distance, response buttons were positioned horizontally (Experiment 1) or radially (Experiment 2). In both experiments, there was an advantage for both the closest and farthest numbers with respect to the reference during distance classification, but not during magnitude classification. In Experiment 2, numerically close/far numbers were classified faster with the close/far response button, respectively, suggesting radial correspondence between physical and representational distances. These findings provide new theoretical and methodological insights into the mental representation of numbers. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Spatial biases in approximate arithmetic are subject to sequential dependency effects and dissociate from attentional biases

The notion that mental arithmetic is associated with shifts of spatial attention along a spatially organised mental number representation has received empirical support from three lines of research. First, participants tend to overestimate results of addition and underestimate those of subtraction problems in both exact and approximate formats. This has been termed the operational momentum (OM) effect. Second, participants are faster in detecting right-sided targets presented in the course of addition problems and left-sided targets in subtraction problems (attentional bias). Third, participants are biased toward choosing right-sided response alternatives to indicate the results of addition problems and left-sided response alternatives for subtraction problems (Spatial Association Of Responses [SOAR] effect). These effects potentially have their origin in operation-specific shifts of attention along a spatially organised mental number representation: rightward for addition and leftward for subtraction. Using a lateralised target detection task during the calculation phase of non-symbolic additions and subtractions, the current study measured the attentional focus, the OM and SOAR effects. In two experiments, we replicated the OM and SOAR effects but did not observe operation-specific biases in the lateralised target-detection task. We describe two new characteristics of the OM effect: First, a time-resolved, block-wise analysis of both experiments revealed sequential dependency effects in that the OM effect builds up over the course of the experiment, driven by the increasing underestimation of subtraction over time. Second, the OM effect was enhanced after arithmetic operation repetition compared to trials where arithmetic operation switched from one trial to the next. These results call into question the operation-specific attentional biases as the sole generator of the observed effects and point to the involvement of additional, potentially decisional processes that operate across trials.

Early visual deprivation disrupts the mental representation of numbers in visually impaired children

Early visual deprivation disrupts the mental representation of numbers in visually impaired children

Dyslexia and dyscalculia: which neuropsychological processes distinguish the two developmental disorders?

ABSTRACT This study analyses the specific neuropsychological profiles of children with dyslexia and/or dyscalculia, in particular concerning phonological awareness, lexical access, working memory and numerical processing. Four groups were selected, through a screening process that used strict criteria, from 1568 7-10-year-old children: 90 with typical development, 61 with dyslexia, 13 with dyscalculia, and 14 with dyslexia + dyscalculia. Children with dyslexia show a deficit in phonological processing, lexical access, and verbal working memory, especially with alphabetic stimuli. Children with developmental dyscalculia show a deficit of phonological processing, verbal working memory with digits and visual-spatial working memory. They also show an impairment in spatial representation of numbers and in the automatic access to numerical semantics to a greater extent than those with double disturbance. Children with dyslexia + dyscalculia show a profile generally characterized by the summation of the deficits of the two disorders, although they have a lower deficit in access to numerical semantics and mental representation of numbers.

Mental Number Representations Are Spatially Mapped Both by Their Magnitudes and Ordinal Positions

The Spatial-Numerical Association of Response Codes (SNARC) effect – i.e., faster responses to small numbers with the left compared to the right side and to large numbers with the right compared to the left side – suggests that numbers are associated with space. However, it remains unclear whether the SNARC effect evolves from a number’s magnitude or the ordinal position of a number in working memory. One problem is that, in different paradigms, the task demands influence the role of ordinality and magnitude. While single-task setups in which participants judge the parity of a displayed number indicate the importance of magnitude for the SNARC effect, evidence for ordinal influences usually comes from experiments where ordinal sequences have to be memorized or setups in which participants possess pre-existing knowledge of the ordinality of stimuli. Therefore, in this preregistered study, we employed a SNARC task without secondary ordinal sequence memorization. We dissociate ordinal and magnitude accounts by carefully manipulating experimental stimulus sets. The results indicate that even though the magnitude model better accounts for the observed data, the ordinal position seems to matter as well. Hence, numbers are associated with space in both a magnitude- and an order-respective manner, yielding a mixture of both compatibility effects. Moreover, a multiple coding framework may most accurately explain the roots of the SNARC effect.

Early visual deprivation disrupts the mental representation of numbers in visually impaired children

Several shreds of evidence indicate that visual deprivation does not alter numerical competence neither in adults nor in children. However, studies reporting non-impaired numerical abilities in the visually impaired population present some limitations: (a) they mainly assessed the ability to process numbers (e.g. mathematical competence) rather than represent numbers (e.g. mental number line); (b) they principally focused on positive rather than negative number estimates; (c) they investigated numerical abilities in adult individuals except one focusing on children (Crollen et al. in Cognition 210:104586, 2021). Overall, this could limit a comprehensive explanation of the role exerted by vision on numerical processing when vision is compromised. Here we investigated how congenital visual deprivation affects the ability to represent positive and negative numbers in horizontal and sagittal planes in visually impaired children (thirteen children with low vision, eight children with complete blindness, age range 6–15 years old). We adapted the number-to-position paradigm adopted by Crollen et al. (Cognition 210:104586, 2021), asking children to indicate the spatial position of positive and negative numbers on a graduated rule positioned horizontally or sagittally in the frontal plane. Results suggest that long-term visual deprivation alters the ability to identify the spatial position of numbers independently of the spatial plane and the number polarity. Moreover, results indicate that relying on poor visual acuity is detrimental for low vision children when asked to localize both positive and negative numbers in space, suggesting that visual experience might have a differential role in numerical processing depending on number polarity. Such findings add knowledge related to the impact of visual experience on numerical processing. Since both positive and negative numbers are fundamental aspects of learning mathematical principles, the outcomes of the present study inform about the need to implement early rehabilitation strategies to prevent the risk of numerical difficulties in visually impaired children.

Bayesian modelling of the latent structure of individual differences in the numerical size-congruity effect

ABSTRACT When people choose the physically larger of two numerals, they are often slower when relative physical size is incongruent with numerical magnitude. This size-congruity effect not only informs our understanding of mental representations of number, but also serves as an index for numerical ability in individuals. In this paper, we apply methods of Haaf and Rouder (2017) to examine the size-congruity effect through the lens of individual differences. We develop and compare hierarchical Bayesian mixed models with varying levels of constraint on the individual size-congruity effects. In three data sets, the winning model was one in which the size-congruity effect was constrained to be positive. This indicates that, at least in the context of a physical comparison task with Arabic numerals, everyone exhibits a positive size-congruity effect. We discuss these results in the context of measurement fidelity and theory-building in numerical cognition.

The role of math anxiety in the relationship between approximate number system and math performance in young children

AbstractAlthough interest in mathematics anxiety in young children (6–11 years) has increased recently, the results of previous studies on the role of mathematics anxiety in the relationship between number sense and math performance are inconsistent. The study's main objective was to establish whether math anxiety is a mediator or a moderator in the relationship between number line estimation (NLE) and mathematical outcomes. A cross‐sectional, correlational design was used. A total of 234 Turkish public primary school children (third and fourth graders) participated in the study. The following hypotheses were tested: (1) math anxiety mediates the relationship between the approximate number system (ANS) (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency); (2) math anxiety moderates the relationship between ANS (an easier and more difficult task of ANS) and math performance (math achievement and arithmetic fluency). The results mostly suggest no mediation effect of math anxiety between NLE and math performance in young children. The results also suggest that increases in difficulty level in NLE (increases of errors in NLE) is related to more anxiety regarding mathematics. Furthermore, moderation analyses indicated that the acuity of mental representation of numbers in high math‐anxious children is a key factor contributing to math performance. Developmental aspects of the obtained results are also discussed in the paper.

Vertical versus horizontal Spatial-Numerical Associations (SNA): A processing advantage for the vertical dimension.

Humans have associations between numbers and physical space on both horizontal and vertical dimensions, called Spatial-Numerical Associations (SNAs). Several studies have considered the hypothesis of there being a dominant orientation by examining on which dimension people are more accurate and efficient at responding during various directional SNA tasks. However, these studies have difficulty differentiating between a person's efficiency at accessing mental representations of numbers in space, and the efficiency at which they exercise motor control functions, particularly bilateral ones, when manifesting a response during an explicit directional SNA task. In this study we use a conflict test employing combined explicit magnitude and spatial directional processing in which pairs of numbers are placed along the diagonal axes and response accuracy/efficiency are considered across the horizontal and vertical dimensions simultaneously. Participants indicated which number in each pair was largest using a joystick that only required unilateral input. The experiment was run in English using Arabic numerals. Results showed that directional SNAs have a vertical rather than horizontal dominance. A moderating factor was also found during post-hoc analysis, where response efficiency, but not accuracy, is conditional on a person's native language being oriented the same as the language of the experiment, left to right. The dominance of the vertical orientation suggests adopting more vertical display formats for numbers may provide situational advantages, particularly for explicit magnitude comparisons, with some domains like flight controls and the stock market already using these in some cases.

Response bias in numerosity perception at early judgments and systematic underestimation.

Mental number representation relies on mapping numerosity based on nonsymbolic stimuli to symbolic magnitudes. It is known that mental number representation builds on a logarithmic scale, and thus numerosity decisions result in underestimation. In the current study, we investigated the temporal dynamics of numerosity perception in four experiments by employing the response-deadline SAT procedure. We presented random number of dots and required participants to make a numerosity judgment by comparing the perceived number of dots to 50. Using temporal dynamics in numerosity perception allowed us to observe a response bias at early decisions and a systematic underestimation at late decisions. In all three experiments, providing feedback diminished the magnitude of underestimation, whereas in Experiment 3 the absence of feedback resulted in greater underestimation errors. These results were in accordance with the findings that suggested feedback is necessary for the calibration of the mental number representation.

Neuromuscular Diseases Affect Number Representation and Processing: An Exploratory Study.

Spinal muscular atrophy (SMA) and Duchenne muscular dystrophy (DMD) both are rare genetic neuromuscular diseases with progressive loss of motor ability. The neuromotor developmental course of those diseases is well documented. In contrast, there is only little evidence about characteristics of general and specific cognitive development. In both conditions the final motor outcome is characterized by an inability to move autonomously: children with SMA never accomplish independent motoric exploration of their environment, while children with DMD do but later lose this ability again. These profound differences in developmental pathways might affect cognitive development of SMA vs. DMD children, as cognition is shaped by individual motor experiences. DMD patients show impaired executive functions, working memory, and verbal IQ, whereas only motor ability seems to be impaired in SMA. Advanced cognitive capacity in SMA may serve as a compensatory mechanism for achieving in education, career progression, and social satisfaction. This study aimed to relate differences in basic numerical concepts and arithmetic achievement in SMA and DMD patients to differences in their motor development and resulting sensorimotor and environmental experiences. Horizontal and vertical spatial-numerical associations were explored in SMA/DMD children ranging between 6 and 12 years through the random number generation task. Furthermore, arithmetic skills as well as general cognitive ability were assessed. Groups differed in spatial number processing as well as in arithmetic and domain-general cognitive functions. Children with SMA showed no horizontal and even reversed vertical spatial-numerical associations. Children with DMD on the other hand revealed patterns in spatial numerical associations comparable to healthy developing children. From the embodied Cognition perspective, early sensorimotor experience does play a role in development of mental number representations. However, it remains open whether and how this becomes relevant for the acquisition of higher order cognitive and arithmetic skills.

Twenty-four or four-and-twenty: Language modulates cross-modal matching for multidigit numbers in children and adults

Does number–word structure have a long-lasting impact on transcoding? Contrary to English, German number words comprise decade–unit inversion (e.g., vierundzwanzig is literally translated as four-and-twenty). To investigate the mental representation of numbers, we tested the effect of visual and linguistic–morphological characteristics on the development of verbal–visual transcoding. In a longitudinal cross-linguistic design, response times (RTs) in a number-matching experiment were analyzed in Grade 2 (119 German-speaking and 179 English-speaking children) and in Grade 3 (131 German-speaking and 160 English-speaking children). To test for long-term effects, the same experiment was given to 38 German-speaking and 42 English-speaking adults. Participants needed to decide whether a spoken number matched a subsequent visual Arabic number. Systematic variation of digits in the nonmatching distractors allowed comparison of three different transcoding accounts (lexicalization, visual, and linguistic–morphological). German speakers were generally slower in rejecting inverted number distractors than English speakers. Across age groups, German speakers were more distracted by Arabic numbers that included the correct unit digit, whereas English speakers showed stronger distraction when the correct decade digit was included. These RT patterns reflect differences in number–word morphology. The individual cost of rejecting an inverted distractor (inversion effect) predicted arithmetic skills in German-speaking second-graders only. The moderate relationship between the efficiency to identify a matching number and arithmetic performance could be observed cross-linguistically in all age groups but was not significant in German-speaking adults. Thus, findings provide consistent evidence of a persistent impact of number–word structure on number processing, whereas the relationship with arithmetic performance was particularly pronounced in young children.

How do working memory and inhibition contribute to the SNARC effect in Chinese school-aged children?

The SNARC effect is a phenomenon that reflects a close correspondence between the mental representation of numbers and physical space. In this study, we examined the development of the SNARC effect among Chinese school children aged 8- to 10- years old, and the relationships between the SNARC effect, working memory and inhibition. The SNARC effect was assessed with the magnitude comparison task, while working memory and inhibition were respectively assessed with tasks in both spatial frame and verbal frame. The SNARC effect was not observed among 8-year-old children but was present in children at 9 years old and 10 years old. Moreover, correlation and regression analyses demonstrated that working memory and inhibition in the spatial frame, rather than in the verbal frame, contributed to the magnitude SNARC effect.

The interplay between spatial ordinal knowledge, linearity of number-space mapping, and arithmetic skills

The ability to map numbers onto space has been widely investigated with the number line (NL) task. Accurate (linear) placement is typically preceded by a developmental phase in which children assign more space to small numbers and compress large numbers to the right part of the line, thereby resembling a biased log-like mapping. Here, we exploited a task that separately assesses direction, order and accuracy of spatial mapping (DOS task) to investigate the origin of the biased NL mapping in a sample of first and second graders. Children with reduced ordinal knowledge in the DOS task showed a more biased NL mapping, which was formally assessed using the leading mathematical models of the NL task. However, the log-like biased mapping did not emerge in the DOS task, thereby showing a lack of generalisation across different number-space mapping tasks. Both ordinal knowledge in the DOS task and linearity in the NL task related to arithmetic fluency beyond domain-general cognitive factors. We conclude that the NL task is a meter of children’s arithmetic skills rather than an expression of the mental representation of numbers.

O que as pesquisas em educação matemática apontam sobre estimativas numéricas de crianças?

Trata-se de uma revisão bibliográfica acerca do conhecimento produzido sobre estimativa numérica nas últimas três décadas, com objetivo de destacar a sua importância enquanto habilidade matemática. Observou-se que estão sendo discutidos cinco aspectos principais: as suas possíveis bases mentais, a sua relação com a representação mental dos números, o uso de estratégias e as maneiras de avaliar o seu desempenho (julgamento de quantidades e estimativa na reta numérica). Os resultados indicam que os sujeitos possuem diferentes formas de representação numérica mental e de estratégias de resolução de tarefas envolvendo estimativa numérica, demonstrando que o processo estratégico é dinâmico e construído pelo sujeito de acordo com os fatores envolvidos na situação apresentada. Conclui-se que a estimativa numérica é considerada, pelos estudos revisados, uma habilidade matemática necessária para o sucesso na vida adulta, devendo, portanto, ter espaço de discussão na escola.

A shared code for Braille and Arabic digits revealed by cross-modal priming in sighted Braille readers

Quantities can be represented by different formats (e.g. symbolic or non-symbolic) and conveyed via different modalities (e.g. tactile or visual). Despite different priming curves: V-shape and step-shape for place and summation coded representation, respectively, the occurrence of priming effect supports the notion of different format overlap on the same mental number line. However, little is known about tactile-visual overlap of symbolic numerosities i.e. Braille numbers to Arabic digits on the magnitude number representation. Here, in a priming experiment, we tested a unique group of sighted Braille readers to investigate whether tactile Braille digits would activate a place-coding type of mental number representation (V-shape), analogous to other symbolic formats. The primes were either tactile Braille digits presented on a Braille display or number words presented on a computer screen. The targets were visually presented Arabic digits, and subjects performed a naming task. Our results reveal a V-shape priming function for both prime formats: tactile Braille and written words representing numbers, with strongest priming for primes of identical value (e.g. “four” and “4”), and a symmetrical decrease of priming strength for neighboring numbers, which indicates that the observed priming is due to identity priming. We thus argue that the magnitude information is processed according to a shared phonological code, independent of the input modality.

Numbers around Descartes: A preregistered study on the three-dimensional SNARC effect

The Spatial-Numerical Association of Response Codes (SNARC) effect suggests that numbers are represented along a horizontal left-to-right oriented, mental number line, with small numbers on the left and large numbers on the right. Much less evidence exists for vertical (down-to-up) and sagittal (near-to-far) SNARC effects. This might be due to the employment of different experimental paradigms among studies and to the, sometimes, inexact definition of the vertical and sagittal axes. We investigated for the first time the SNARC effect along the horizontal, vertical, and sagittal axes, by means of a classic SNARC task. Our results suggest the presence of three equally-strong SNARC effects. Our findings can be considered as evidence in favor of a three-dimensional, mental representation of numbers, in the form of a mental number space, defined by Cartesian coordinates.

Number Representations Drive Number-Line Estimates.

The number-line task has been extensively used to study the mental representation of numbers in children. However, studies suggest that proportional reasoning provides a better account of children's performance. Ninety 4- to 6-year-olds were given a number-line task with symbolic numbers, with clustered dot arrays that resembled a perceptual scaling task, or with spread-out dot arrays that involved numerical estimation. Children performed well with clustered dot arrays, but poorly with symbolic numbers and spread-out dot arrays. Performances with symbolic numbers and spread-out dot arrays were highly correlated and were related to counting skill; neither was true for clustered dot arrays. Overall, results provide evidence for the role of mental representation of numbers in the symbolic number-line task.

Spontaneous partitioning and proportion estimation in children’s numerical judgments

What do numerical estimates tell us about developing an understanding of number? One theory is that bounded number line estimation (NLE) tasks reveal a “representational shift” from logarithmically to linearly organized mental representations of number over development. According to a different theoretical framework, developmental change in estimation reflects changes in children’s numerical knowledge and their ability to make appropriate relative judgments. Empirical support for this “proportion estimation” framework includes the fact that quantitative models of proportion estimation describe signature patterns of estimation bias. A recent study argued against this latter theory by suggesting that patterns of curvature in number line placements are simply artifacts of a task procedure in which participants receive explicit information about the location of the numerical midpoint. We tested this claim in two experiments with children aged 6–8 years (Experiment 1: N = 47; Experiment 2: N = 104). Results demonstrated that the proportion estimation framework provides a good explanation of children’s number line placement in the absence of explicit midpoint cues, that explicit cues to the midpoint are associated with more frequent use of middle reference points in young children, and that children can use a middle reference point spontaneously in the absence of explicit cues (with this tendency increasing with age). These findings provide novel support for the idea that psychophysical models of proportion estimation successfully account for numerical estimates across development regardless of whether spatial and numerical midpoint cues are provided as part of the NLE task.