The relation between spatial skill and early number knowledge: The role of the linear number line

[Correction Notice: An Erratum for this article was reported in Vol 48(5) of Developmental Psychology (see record 2012-11771-001). The grey boxes around the faces in Figure 2 are missing. The correct version is presented in the erratum.] Spatial skill is highly related to success in math and science (e.g., Casey, Nuttall, Pezaris, & Benbow, 1995). However, little work has investigated the cognitive pathways by which the relation between spatial skill and math achievement emerges. We hypothesized that spatial skill plays a crucial role in the development of numerical reasoning by helping children to create a spatially meaningful, powerful numerical representation-the linear number line. In turn, a strong linear number representation improves other aspects of numerical knowledge such as arithmetic estimation. We tested this hypothesis using 2 longitudinal data sets. First, we found that children's spatial skill (i.e., mental transformation ability) at the beginning of 1st and 2nd grades predicted improvement in linear number line knowledge over the course of the school year. Second, we found that children's spatial skill at age 5 years predicted their performance on an approximate symbolic calculation task at age 8 and that this relation was mediated by children's linear number line knowledge at age 6. The results are consistent with the hypothesis that spatial skill can improve children's development of numerical knowledge by helping them to acquire a linear spatial representation of numbers. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Sensori-motor spatial training of number magnitude representation

An adequately developed spatial representation of number magnitude is associated with children's general arithmetic achievement. Therefore, a new spatial-numerical training program for kindergarten children was developed in which presentation and response were associated with a congruent spatial numerical representation. In particular, children responded by a full-body spatial movement on a digital dance mat in a magnitude comparison task. This spatial-numerical training was more effective than a non-spatial control training in enhancing children's performance on a number line estimation task and a subtest of a standardized mathematical achievement battery (TEDI-MATH). A mediation analysis suggested that these improvements were driven by an improvement of children's mental number line representation and not only by unspecific factors such as attention or motivation. These results suggest a benefit of spatial numerical associations. Rather than being a merely associated covariate, they work as an independently manipulated variable which is functional for numerical development.     

Number games, magnitude representation, and basic number skills in preschoolers

The effect of 3 intervention board games (linear number, linear color, and nonlinear number) on young children's (mean age = 3.8 years) counting abilities, number naming, magnitude comprehension, accuracy in number-to-position estimation tasks, and best-fit numerical magnitude representations was examined. Pre- and posttest performance was compared following four 25-min intervention sessions. The linear number board game significantly improved children's performance in all posttest measures and facilitated a shift from a logarithmic to a linear representation of numerical magnitude, emphasizing the importance of spatial cues in estimation. Exposure to the number card games involving nonsymbolic magnitude judgments and association of symbolic and nonsymbolic quantities, but without any linear spatial cues, improved some aspects of children's basic number skills but not numerical estimation precision.     

The relationship between the shape of the mental number line and familiarity with numbers in 5- to 9-year old children: evidence for a segmented linear model

This experiment aimed to expand previous findings on the development of mental number representation. We tested the hypothesis that children's familiarity with numbers is directly reflected by the shape of their mental number line. This mental number line was expected to be linear as long as numbers lay within the range of numbers children were familiar with. Five- to 9-year-olds (N=78) estimated the positions of numbers on an external number line and additionally completed a counting assessment mirroring their familiarity with numbers. A segmented regression model consisting of two linear segments described number line estimations significantly better than a logarithmic or a simple linear model. Moreover, the change point between the two linear segments, indicating a change of discriminability between numbers, was significantly correlated with children's familiar number range. Findings are discussed in terms of the accumulator model, assuming a linear mental representation with scalar variability.     

Semantic numerical representation in blind subjects: The role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.     

Semantic numerical representation in blind subjects: the role of vision in the spatial format of the mental number line

Does vision play a role in the elaboration of the semantic representation of small and large numerosities, notably in its spatial format? To investigate this issue, we decided to compare in the auditory modality the performance of congenitally and early blind people with that of a sighted control group, in two number comparison tasks (to 5 and to 55) and in one parity judgement task. Blind and sighted participants presented exactly the same distance and SNARC (Spatial Numerical Association of Response Codes) effects, indicating that they share the same semantic numerical representation. In consequence, our results suggest that the spatial dimension of the numerical representation is not necessarily attributable to the visual modality and that the absence of vision does not preclude the elaboration of this representation for 1-digit (Experiment 1) and 2-digit numerosities (Experiment 2). Moreover, as classical semantic numerical effects were observed in the auditory modality, the postulate of the amodal nature of the mental number line for both small and large magnitudes was reinforced.     

Expanding on the mental number line: Zero is perceived as the "smallest"

The representation of 0 in healthy adults was studied with the physical comparison task. Automatic processing of numbers, as indicated by the size congruity effect, was used for detecting the basic numerical representations stored in long-term memory. The size congruity effect usually increases with numerical distance between the physically compared numbers. This increase was attenuated for comparisons to 0 or 1 (but not to 2) when they were perceived as the smallest number in the set. Furthermore, the size congruity effect was enlarged in these cases. These results indicate an end effect in automatic processing of numbers and suggest that 0, or 1 in the absence of 0, is perceived as the smallest entity on the mental number line. The implications of these findings are discussed with regard to models of number representation. (PsycINFO Database Record (c) 2012 APA, all rights reserved).     

Children's mappings of large number words to numerosities

Previous studies have suggested that children's learning of the relation between number words and approximate numerosities depends on their verbal counting ability, and that children exhibit no knowledge of mappings between number words and approximate numerical magnitudes for number words outside their productive verbal counting range. In the present study we used a numerical estimation task to explore children's knowledge of these mappings. We classified children as Level 1 counters (those unable to produce a verbal count list up to 35), Level 2 counters (those who were able to count to 35 but not 60) and Level 3 counters (those who counted to 60 or above) and asked children to estimate the number of items on a card. Although the accuracy of children's estimates depended on counting ability, children at all counting skill levels produced estimates that increased linearly in proportion to the target number, for numerosities both within and beyond their counting range. This result was obtained at the group level (Experiment 1) and at the level of individual children (Experiment 2). These findings provide evidence that even the least skilled counters do exhibit some knowledge of the form of the mapping between large number words and approximate numerosities.     

Sex differences in number line estimation: The role of numerical estimation

Sex differences in mathematical performance have frequently been examined over the last decades indicating an advantage for males especially when numerical problems cannot be solved by (classroom‐)learnt strategies and/or estimation. Even in basic numerical tasks such as number line estimation, males were found to outperform females – with sex differences argued to emerge from different solution strategies applied by males and females. We evaluated the latter using two versions of the number line estimation task: a bounded and an unbounded task version. Assuming that women tend more strongly to apply known procedures, we expected them to be at a particular disadvantage in the unbounded number line estimation task, which is less prone to be solved by specific strategies such as proportion judgement but requires numerical estimation. Results confirmed more pronounced sex differences for unbounded number line estimation with males performing significantly more accurately in this task version. This further adds to recent evidence suggesting that estimation performance in the bounded task version may reflect solution strategies rather than numerical estimation. Additionally, it indicates that sex differences regarding the spatial representation of number magnitude may not be universal, but associated with spatial–numerical estimations in particular.     

The representational space of numerical magnitude: illusions of length

In recent years, a growing amount of evidence concerning the relationships between numerical and spatial representations has been interpreted, by and large, in favour of the mental number line hypothesis--namely, the analogue continuum where numbers are spatially represented (Dehaene, 1992; Dehaene, Piazza, Pinel, & Cohen, 2003). This numerical representation is considered the core of number meaning and, accordingly, needs to be accessed whenever numbers are semantically processed. The present study explored, by means of a length reproduction task, whether besides the activation of lateralized spatial codes, numerical processing modulates the mental representation of a horizontal spatial extension. Mis-estimations of length induced by Arabic numbers are interpreted in terms of a cognitive illusion, according to which the elaboration of magnitude information brings about an expansion or compression of the mental representation of spatial extension. These results support the hypothesis that visuo-spatial resources are involved in the representation of numerical magnitude.     

The development of numerosity estimation: Evidence for a linear number representation early in life

Several studies investigating the development of approximate number representations used the number-to-position task and reported evidence for a shift from a logarithmic to a linear representation of numerical magnitude with increasing age. However, this interpretation as well as the number-to-position method itself has been questioned recently. The current study tested 5- and 8-year-old children on a newly established numerosity production task to examine developmental changes in number representations and to test the idea of a representational shift. Modelling of the children's numerical estimations revealed that responses of the 8-year-old children approximate a simple positive linear relation between estimated and actual numbers. Interestingly, however, the estimations of the 5-year-old children were best described by a bilinear model reflecting a relatively accurate linear representation of small numbers and no apparent magnitude knowledge for large numbers. Taken together, our findings provide no support for a shift of mental representations from a logarithmic to a linear metric but rather suggest that the range of number words which are appropriately conceptualised and represented by linear analogue magnitude codes expands during development.     

负数的空间表征机制

本研究采用快速数字大小分类范式,每次试验呈现一个数字,要求被试快速判断即时呈现的数字大于或小于-5（或5）,探讨负数在心理数字线上的表征方向问题。实验一将负数（-1～-9）和正数（1～9）分两组分别呈现;实验二将正负数混合呈现,仅对负数进行反应。结果表明,负数按照其绝对值大小表征在心理数字线上,绝对值小的负数表征在心理数字线的左侧,绝对值大的负数表征在心理数字线的右侧。该结果支持系统进化论假说     

The mental number line electrified: brain potentials in a numerical flanker task

It has been suggested that the mental representation of numbers is spatial in nature such that numbers are ordered on a mental number line. In the present investigation we use a variant of the Eriksen flanker task requiring a magnitude decision (smaller or larger than 5) for a central target number by pressing a response button with the right or left hand. The target number is flanked by irrelevant distracters that are either identical to the target, different from the target but biasing the same response, or different from the target and biasing a different response. Response latencies and event-related brain potentials were obtained in a group of healthy adults. Besides the typical response congruency effects on response latency and the N2 component of the ERP, we observed several other effects. First, numerical distance of the target to the standard 5 influenced decision latencies and amplitude and latency of the P3 component with smaller distances leading to longer decision latencies, longer P3 latencies and smaller P3 amplitudes. Second, smaller numerical distance between target and distracters led to faster decisions for response congruent and to slower decisions for response-incongruent trials. For response-incongruent trials P3 amplitude was small/large and P3 latency was long/short for small/large distances. These findings underscore the spatial character of number representation and further show that the relation between targets and distracters, although task irrelevant, is assessed automatically with facilitatory and inhibitory effects driven by spatial distance on the mental number line.     

数形联觉——数字空间表征研究“新”取向

数字空间表征是人类对数字进行表征的重要方式。数形联觉（number-form synesthesia）是一种数字可以有意识地引起空间知觉的独特现象,与此类似的是非联觉者中广泛存在的无意识的心理数字线（mental number line）现象。两者在行为和脑机制上存在着很多重叠,也存在着值得思考的差异。数形联觉的研究能够提供实质性的行为和脑机制数据,用以解决数字空间表征研究中出现的问题,加强对于数字空间表征的理解;也为更加全面深入地开展进一步研究提供了新的启示,成为数字空间表征研究中值得推崇的新取向。     

SNARC效应: 现状、理论与建议

SNARC效应是当对数字进行奇偶判断时,即使数的奇偶性与数的大小无关,但右手（左手）对相对大（小）的数的反应快。首先介绍SNARC效应的起源和理论解释,然后总结SNARC效应的特性,论述SNARC效应和Simon效应以及MARC 效应的关系,并对SNARC效应的脑机制进行了概述,最后提出3个有待深入研究的问题：（1）SNARC效应的加工处理机制;（2）SANRC效应的理论探索;（3）SNARC效应的本质。     

运算动量效应的理论解释及其发展性预测因素

了解运算偏差的形成与发展对探索算数运算系统的内在机制具有重要意义,早期的算数运算能力是儿童理解和进行复杂数学运算的基础。运算动量偏差是指个体在进行基本数学运算时倾向于高估加法运算结果而低估减法运算结果的一种运算偏差,主要包括三种理论解释,即注意转移假说、启发式解释和压缩解释。鉴于运算动量效应在成年群体中相对稳定却在不同发展阶段儿童中存在不一致的证据,数学能力的提高与空间注意的成熟可结合不同的理论解释来阐明儿童发展过程中运算动量效应的变化趋势。未来可以进一步整合多种研究任务以揭示运算动量效应的发展轨迹,考察数量表征系统与运算动量效应间的关联,探究运算动量效应在不同运算符号中的稳定性,探讨不同因素共同作用对运算动量效应的影响,并设计有关数学能力的干预措施以减少运算动量效应这一运算偏差。     

符号数量和非符号数量的空间表征:5岁儿童的SNARC效应和距离效应

对物理刺激的数量信息表征是符号数字表征的前提和基础,据此假设在儿童的SNARC效应发生的时序问题上,非符号数量(如面积)的空间表征早于符号数量(如阿拉伯数字)的空间表征。本研究邀请5岁幼儿完成数字比较和面积比较两类任务,结果发现在数字比较任务中没有出现SNARC效应,但却存在距离效应;在面积比较任务中出现了SNARC效应和距离效应。可以推断,在阿拉伯数字的空间表征出现之前,儿童已经能够对非符号数量信息进行空间表征。     

The development of numerical estimation: evidence for multiple representations of numerical quantity

Abstract - We examined children's and adults' numerical estimation and the representations that gave rise to their estimates. The results were inconsistent with two prominent models of numerical representation: the logarithmic-ruler model, which proposes that people of all ages possess a single, logarithmically spaced representation of numbers, and the accumulator model, which proposes that people of all ages represent numbers as linearly increasing magnitudes with scalar variability. Instead, the data indicated that individual children possess multiple numerical representations; that with increasing age and numerical experience, they rely on appropriate representations increasingly often; and that the numerical context influences their choice of representation. The results, obtained with second graders, fourth graders, sixth graders, and adults who performed two estimation tasks in two numerical contexts, strongly suggest that one cause of children's difficulties with estimation is reliance on logarithmic representations of numerical magnitudes in situations in which accurate estimation requires reliance on linear representations.     

The powers of noise-fitting: reply to Barth and Paladino

Barth and Paladino (2011) argue that changes in numerical representations are better modeled by a power function whose exponent gradually rises to 1 than as a shift from a logarithmic to a linear representation of numerical magnitude. However, the fit of the power function to number line estimation data may simply stem from fitting noise generated by averaging over changing proportions of logarithmic and linear estimation patterns. To evaluate this possibility, we used conventional model fitting techniques with individual as well as group average data; simulations that varied the proportion of data generated by different functions; comparisons of alternative models' prediction of new data; and microgenetic analyses of rates of change in experiments on children's learning. Both new data and individual participants' data were predicted less accurately by power functions than by logarithmic and linear functions. In microgenetic studies, changes in the best fitting power function's exponent occurred abruptly, a finding inconsistent with Barth and Paladino's interpretation that development of numerical representations reflects a gradual shift in the shape of the power function. Overall, the data support the view that change in this area entails transitions from logarithmic to linear representations of numerical magnitude.