Basic numerical capacities and prevalence of developmental dyscalculia: the Havana Survey

The association of enumeration and number comparison capacities with arithmetical competence was examined in a large sample of children from 2nd to 9th grades. It was found that efficiency on numerical capacities predicted separately more than 25% of the variance in the individual differences on a timed arithmetical test, and this occurred for both younger and older learners. These capacities were also significant predictors of individual variations in an untimed curriculum-based math achievement test and on the teacher scores of math performance over developmental time. Based on these findings, these numerical capacities were used for estimating the prevalence and gender ratio of basic numerical deficits and developmental dyscalculia (DD) over the grade range defined above (N = 11,652 children). The extent to which DD affects the population with poor ability on calculation was also examined. For this purpose, the prevalence and gender ratio of arithmetical dysfluency (AD) were estimated in the same cohort. The estimated prevalence of DD was 3.4%, and the male:female ratio was 4:1. However, the prevalence of AD was almost 3 times as high (9.35%), and no gender differences were found (male:female ratio = 1.07:1). Basic numerical deficits affect 4.54% of school-age population and affect more boys than girls (2.4:1). The differences between the corresponding estimates were highly significant (α < .01). Based on these contrastive findings, it is concluded that DD, defined as a defective sense of numerosity, could be a distinctive disorder that affects only a portion of children with AD.     

Laterality briefed: laterality modulates performance in a numerosity-congruity task

It is widely agreed that irrelevant numerical values are automatically activated. However, automatic and intentional activations may give rise to different numerical representations. We examined processing of symbolic and non-symbolic (i.e., numerosity) representations asking whether they differ in automatic and intentional processing. Participants were presented with two-dimensional displays containing repetitions of a digit and were asked to report, in different blocks, whether the digit or numerosity was smaller or larger than 5. Incongruent trials differed either in laterality between the relevant and irrelevant dimensions (i.e., the location of both dimensions in reference to the midpoint 5) or in numerical distance between dimensions. Congruency affected performance regardless of symbolic or non-symbolic presentation. For incongruent trials, laterality (not distance) affected performance, again regardless of presentation. This implies that automaticity does not mean similar processing of relevant and irrelevant dimensions. Specifically, the relevant dimension is processed elaborately whereas the irrelevant dimension is processed crudely.     

Differences in the acuity of the Approximate Number System in adults: The effect of mathematical ability

It is largely admitted that processing numerosity relies on an innate Approximate Number System (ANS), and recent research consistently observed a relationship between ANS acuity and mathematical ability in childhood. However, studies assessing this relationship in adults led to contradictory results. In this study, adults with different levels of mathematical expertise performed two tasks on the same pairs of dot collections, based either on numerosity comparison or on cumulative area comparison. Number of dots and cumulative area were congruent in half of the stimuli, and incongruent in the other half. The results showed that adults with higher mathematical ability obtained lower Weber fractions in the numerical condition than participants with lower mathematical ability. Further, adults with lower mathematical ability were more affected by the interference of the continuous dimension in the numerical comparison task, whereas conversely higher-expertise adults showed stronger interference of the numerical dimension in the continuous comparison task. Finally, ANS acuity correlated with arithmetic performance. Taken together, the data suggest that individual differences in ANS acuity subsist in adulthood, and that they are related to mathematical ability.     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

In touch with numbers: Embodied and situated effects in number magnitude comparison

The idea of embodied numerosity denotes that seemingly abstract number concepts (e.g., magnitude) are rooted in bodily experiences and situated action. In the present study we evaluated whether there is an embodied representation of the place–value structure of the Arabic number system and if so whether this representation is influenced by situated aspects. In a two-digit number magnitude comparison task participants had to directly touch the larger of two numbers. Importantly, pointing responses were systematically biased toward the decade digit of the target number. Additionally, this leftward bias towards the tens digit was reduced in unit–decade incompatible number pairs. Thereby, we demonstrated an influence of place–value processing on manual pointing movement. Our results therefore corroborate the notion of an embodied representation of the place–value structure of Arabic numbers which is modulated by situated aspects.     

Functional brain organization for number processing in pre‐verbal infants

Humans are born with the ability to mentally represent the approximate numerosity of a set of objects, but little is known about the brain systems that sub‐serve this ability early in life and their relation to the brain systems underlying symbolic number and mathematics later in development. Here we investigate processing of numerical magnitudes before the acquisition of a symbolic numerical system or even spoken language, by measuring the brain response to numerosity changes in pre‐verbal infants using functional near‐infrared spectroscopy (fNIRS). To do this, we presented infants with two types of numerical stimulus blocks: number change blocks that presented dot arrays alternating in numerosity and no change blocks that presented dot arrays all with the same number. Images were carefully constructed to rule out the possibility that responses to number changes could be due to non‐numerical stimulus properties that tend to co‐vary with number. Interleaved with the two types of numerical blocks were audio‐visual animations designed to increase attention. We observed that number change blocks evoked an increase in oxygenated hemoglobin over a focal right parietal region that was greater than that observed during no change blocks and during audio‐visual attention blocks. The location of this effect was consistent with intra‐parietal activity seen in older children and adults for both symbolic and non‐symbolic numerical tasks. A distinct set of bilateral occipital and middle parietal channels responded more to the attention‐grabbing animations than to either of the types of numerical stimuli, further dissociating the specific right parietal response to number from a more general bilateral visual or attentional response. These results provide the strongest evidence to date that the right parietal cortex is specialized for numerical processing in infancy, as the response to number is dissociated from visual change processing and general attentional processing.     

The contribution of working memory to children's mathematical word problem solving

The study explored the contribution of working memory to mathematical word problem solving in children. A total of 69 children in grades 2, 3 and 4 were given measures of mathematical problem solving, reading, arithmetical calculation, fluid IQ and working memory. Multiple regression analyses showed that three measures associated with the central executive and one measure associated with the phonological loop contributed unique variance to mathematical problem solving when the influence of reading, age and IQ were controlled for in the analysis. In addition, the animal dual‐task, verbal fluency and digit span task continued to contribute unique variance when the effects of arithmetical calculation in addition to reading, fluid IQ, and age were controlled for. These findings demonstrate that the phonological loop and a number of central executive functions (shifting, co‐ordination of concurrent processing and storage of information, accessing information from long‐term memory) contribute to mathematical problem solving in children. Copyright © 2006 John Wiley & Sons, Ltd.     

Can we predict mathematical learning disabilities from symbolic and non‐symbolic comparison tasks in kindergarten? Findings from a longitudinal study

Background. The ability to compare numbers, as the most basic form of number sense, has been related to arithmetical achievement. Aims. The current study addressed the predictive value of non‐symbolic and symbolic (number word (NW) and Arabic number (AN)) comparison for arithmetics by means of a longitudinal design. Sample. Sixteen children with mathematical disabilities (MD), 64 low achievers (LA), and 315 typical achieving (TA) children were followed from kindergarten till grade 2. Method. The association of comparison skills with arithmetical skills in grades l and 2 was studied. The performances of MD, LA and TA children were compared. Results. Regression analyses showed that non‐symbolic skills in kindergarten were predictively related to arithmetical achievement 1 year later and fact retrieval 2 years later. AN comparison was predictively related to procedural calculation 2 years later. In grade 2, there was an association between both symbolic tasks and arithmetical achievement. Children with MD already had deficits in non‐symbolic and symbolic AN comparison in kindergarten, whereas in grade 2 the deficits in processing symbolic information remained. Conclusions. The combination of non‐symbolic and symbolic deficits represents a risk of developing MD.     

Is 26 + 26 smaller than 24 + 28? Estimating the approximate magnitude of repeated versus different numbers

It has recently been suggested that regardless of the dimension at hand (i.e., numerosity, length, time), similar operational mechanisms are involved in the comparison process based on approximate magnitude representation. One piece of evidence for this hypothesis lies in the presence of similar behavioral effects for any comparison (i.e., the distance effect). In the case of length comparison, the comparison process can be biased by summation toward either an underestimation or an overestimation: The sum of equal-size stimuli is underestimated, whereas the sum of different-size stimuli is overestimated. Relying on the hypothesis that similar operational mechanisms underlie the comparison process of any magnitude, we aim at extending these findings to another magnitude dimension. A number comparison task with digit numbers was used in the two experiments reported presently. The objective was to investigate whether summation also biases magnitude representation of numerical and symbolic information. The results provided evidence that the summation bias can also apply to numerical magnitude comparison, since the sum of repeated numbers (26 + 26) was underestimated whereas the sum of different numbers (24 + 28) was overestimated. We propose that these effects could be accounted for by a heuristic linking cognitive effort and magnitude estimation.     

数量加工单方向干扰时间知觉的认知与神经机制

近年来,数量加工对时间知觉的单方向干扰效应受到广泛关注,然而关于该效应的认知和神经机制的研究却缺乏深度的文献梳理和理论分析。本文从抽象数量、数字符号等认知加工如何干扰时间知觉的角度,系统总结了近期数量加工单方向影响时间知觉的研究进展;并在对相关文献加以梳理的基础上,详细分析了该效应可能的认知和神经机制,并提出若干未来研究的设想。     

发展性计算障碍儿童的数认知

发展性计算障碍作为一种特异性学习障碍,影响儿童的数学成就与数学学习观念,因而引起了多学科研究的关注和重视。文章分析了发展性计算障碍领域在数认知方面的已有研究,揭示了发展性计算障碍的数认知缺陷及内部差异,从一般认知加工障碍与数字加工模块障碍两个角度阐述了发展性计算障碍的可能成因,最后对当前研究进行了小结与展望,认为数认知缺陷的复杂关系、亚类型特征、成因之间的相互作用与机制还需进一步研究     

Spontaneous focusing on numerosity as a domain-specific predictor of arithmetical skills

The aim of this 2 year longitudinal study was to explore whether children’s individual differences in spontaneous focusing on numerosity (SFON) in kindergarten predict arithmetical and reading skills 2 years later in school. Moreover, we investigated whether the positive relationship between SFON and mathematical skills is explained by children’s individual differences in spontaneous focusing on a non-numerical aspect. The participants were 139 Finnish-speaking children. The results show that SFON tendency in kindergarten is a significant domain-specific predictor of arithmetical skills, but not reading skills, assessed at the end of Grade 2. In addition, the relationship between SFON and number sequence skills in kindergarten is not explained by children’s individual differences in their focusing on a non-numerical aspect that is, spatial locations.     

Relational and Representational Aspects of Early Number Development

Two experiments examined the role of representations of numerosity in children's reasoning about relations between two quantities. In the first experiment, 3- and 4-year-old children were able to solve matching-to-sample problems on the basis of both the numerosity of a focal set (number matches) and the correspondence relation between two sets (relational matches); but in a conflict condition, the younger children used only the relational information. In the second experiment, both the 3- and the 4-year-olds showed very high levels of success in inferring numerical equivalence from commutativity relations between two pairs of sets. Children were successful even when one of the sets in each pair was covered, so that children could not directly enumerate the quantities to be compared. Both findings support the notion that relational reasoning originates independently of processes for representing numerosity.     

Eye gaze patterns reveal how we reason about fractions

ABSTRACT

Fractions are defined by numerical relationships, and comparing two fractions’ magnitudes requires within-fraction (holistic) and/or between-fraction (componential) relational comparisons. To better understand how individuals spontaneously reason about fractions, we collected eye-tracking data while they performed a fraction comparison task with conditions that promoted or obstructed different types of comparisons. We found evidence for both componential and holistic processing in this mixed-pairs task, consistent with the hybrid theory of fraction representation. Additionally, making within-fraction eye movements on trials that promoted a between-fraction comparison strategy was associated with slower responses. Finally, participants who performed better on a non-numerical test of reasoning took longer to respond to the most difficult fraction trials, which suggests that those who had greater facility with non-numerical reasoning attended more to numerical relationships. These findings extend prior research and support the continued investigation into the mechanistic links between numerical and non-numerical reasoning.     

Associations of non‐symbolic and symbolic numerical magnitude processing with mathematical competence: a meta‐analysis

Many studies have investigated the association between numerical magnitude processing skills, as assessed by the numerical magnitude comparison task, and broader mathematical competence, e.g. counting, arithmetic, or algebra. Most correlations were positive but varied considerably in their strengths. It remains unclear whether and to what extent the strength of these associations differs systematically between non‐symbolic and symbolic magnitude comparison tasks and whether age, magnitude comparison measures or mathematical competence measures are additional moderators. We investigated these questions by means of a meta‐analysis. The literature search yielded 45 articles reporting 284 effect sizes found with 17,201 participants. Effect sizes were combined by means of a two‐level random‐effects regression model. The effect size was significantly higher for the symbolic (r = .302, 95% CI [.243, .361]) than for the non‐symbolic (r = .241, 95% CI [.198, .284]) magnitude comparison task and decreased very slightly with age. The correlation was higher for solution rates and Weber fractions than for alternative measures of comparison proficiency. It was higher for mathematical competencies that rely more heavily on the processing of magnitudes (i.e. mental arithmetic and early mathematical abilities) than for others. The results support the view that magnitude processing is reliably associated with mathematical competence over the lifespan in a wide range of tasks, measures and mathematical subdomains. The association is stronger for symbolic than for non‐symbolic numerical magnitude processing. So symbolic magnitude processing might be a more eligible candidate to be targeted by diagnostic screening instruments and interventions for school‐aged children and for adults.     

The early development of numerical reasoning.

Children of age 1-4 years were found capable of engaging in numerical reasoning. Children were presented with a task in which they placed a set of objects one by one into an opaque container. An experimenter then visibly performed either an addition, a subtraction, or no transformation on the screened set. Children were then instructed to remove all objects from the container. Across two experiments, children searched for and removed the correct number of objects when set numerosity was small. Knowledge of numerical identity and knowledge of the effects of addition and subtraction transformations on numerosity were present even in children who had not yet begun to count verbally. These findings provide evidence that the emergence of numerical reasoning does not depend upon the prior development of a verbal counting ability or upon cultural experience with numbers.     

Sex differences in spatial cognition, computational fluency, and arithmetical reasoning

Alternative explanations for the male advantage in arithmetical reasoning, as measured by the ability to solve complex word problems, include a male advantage in spatial cognition and a male advantage in computational fluency. The current study was designed to test these competing hypotheses. To this end, 113 male and 123 female undergraduates were administered arithmetical computations and arithmetical reasoning tests, along with an IQ test and a test of spatial cognition. There was no sex difference on the IQ test, but males showed significantly higher mean scores on the arithmetical computations, arithmetical reasoning, and spatial cognition measures. A series of structural equation models indicated that individual differences in arithmetical reasoning were related to individual differences in IQ, spatial abilities, and computational fluency. Moreover, the results suggested that the male advantage in arithmetical reasoning is mediated by the male advantages in both computational fluency and spatial cognition.     

数学焦虑个体近似数量加工的神经机制:一项EEG研究

近似数量加工是对大数目物体数量在不依赖逐个数数前提下的估计。行为学研究提示高数学焦虑人群近似数量加工能力下降, 但神经机制未明。本研究探讨高数学焦虑个体近似数量加工的神经机制, 比较高低数学焦虑脑电活动的差异:(1)行为上无显著组间差异; (2)高数学焦虑组的P2p成分波幅增加; (3) δ频段ERS及β频段ERD无显著数量比例效应, 而低数学焦虑组在上述指标的数量比例效应显著。本研究为高数学焦虑人群近似数量加工能力下降提供了电生理学的证据。     

Behavioral evidence for format-dependent processes in approximate numerosity representation

A genuinely abstract number representation is thought to be capable of representing the numerosity of any set of discrete elements, whether they are sequentially or simultaneously presented. Recent neuroimaging studies, however, have demonstrated that different areas of intraparietal sulcus play a role in extracting numerosity across simultaneous or sequential presentation during a quantification process, suggesting the existence of a format-dependent numerical system. To test whether behavioral evidence exists for format-dependent numerical processing in adult humans, we measured the Weber fractions of numerosity discrimination for sequential stimuli, simultaneous stimuli, and cross-format stimuli with a carefully controlled experimental procedure. The results showed distinct differences between the performance in the simultaneous and sequential conditions, supporting the existence of format-dependent processes for numerosity representation. Moreover, the performance on cross-format trials differed among participants, with the exception that performance was always worse than in the simultaneous condition. Taken together, our findings suggest that numerical representation may involve a complex set of multiple stages.     

从数量表征到数表征：具身认知视角下的人类数能力获得

数能力是数学认知的基本成分。与动物所具有的基本数能力不同,人类不仅具备数量表征能力,更重要的是还拥有对数概念进行表征的数表征能力。虽然具身认知与离身认知都对数概念的表征问题进行了解释,但二者却存在明显理论分歧。具身认知观点主要从具身数量表征和数能力发展的具身认知机制两方面为人类独特数能力的获得提供了理论支撑及实证证据。这启示人们需要重视具身学习在数能力形成实践中的关键作用,重视具身数量表征在数学教学中的作用,仍需进一步揭示其内在的心理和神经基础。