Cognitive mechanisms in numerical processing: evidence from acquired dyscalculia.

This article discusses cognitive neuropsychological research on acquired dyscalculia (i.e., impaired numerical processing resulting from brain damage), surveying issues of current interest, and illustrating the ways in which analyses of acquired deficits can contribute to an understanding of normal processing. I first review the logic whereby inferences concerning normal cognition are drawn from patterns of impaired performance. I then consider research exploring the general functional architecture of the cognitive numerical processing mechanisms, and finally turn to studies aimed at probing the internal structure and functioning of individual processing components.     

Behavioral assessment and treatment of acquired visuoperceptual disorders

Visuoperceptual deficits are common sequelae of damage to either hemisphere of the brain, but are typically more pronounced following injuries involving the right cerebral hemisphere. Common visuoperceptual disorders include visual field cuts, hemi-inattention and hemi-spatial neglect, hemi-perceptual deficits, and gaze and visual pursuit disturbances. A number of behavioral interventions have been developed to teach patients to compensate for acquired visual deficits. Studies addressing assessment and treatment issues in this area are reviewed, and future directions for research are outlined.     

Theory-based categorization under speeded conditions

It is widely accepted that similarity influences rapid categorization, whereas theories can influence only more leisurely category judgments. In contrast, we argue that it is not the type of knowledge used that determines categorization speed, but rather the complexity of the categorization processes. In two experiments, participants learned four categories of items, each consisting of three causally related features. Participants gave more weight to cause features than to effect features, even under speeded response conditions. Furthermore, the time required to make judgments was equivalent, regardless of whether participants were using causal knowledge or base-rate information. We argue that both causal knowledge and base-rate information, once precompiled during learning, can be used at roughly the same speeds during categorization, thus demonstrating an important parallel between these two types of knowledge.     

Numerical distance effect in developmental dyscalculia

Children in third and fourth grades suffering from developmental dyscalculia (DD) and typically developing children were asked to compare numbers to a standard. In two separate blocks, they were asked to compare a number between 1 and 9 to 5, or a two-digit number between 10 and 99 to 55. In the single-digit comparisons, DD children were comparable to the control group in reaction time but showed a difference in error rates. In the two-digit number comparisons, DD children presented a larger distance effect than the controls. In addition, they were more influenced by the problem size than controls were. Assuming an analog representation of quantities, this suggests that quantities are less differentiated in those with DD than in typically developing children.     

Foundational numerical capacities and the origins of dyscalculia

One important cause of very low attainment in arithmetic (dyscalculia) seems to be a core deficit in an inherited foundational capacity for numbers. According to one set of hypotheses, arithmetic ability is built on an inherited system responsible for representing approximate numerosity. One account holds that this is supported by a system for representing exactly a small number (less than or equal to four4) of individual objects. In these approaches, the core deficit in dyscalculia lies in either of these systems. An alternative proposal holds that the deficit lies in an inherited system for sets of objects and operations on them (numerosity coding) on which arithmetic is built. I argue that a deficit in numerosity coding, not in the approximate number system or the small number system, is responsible for dyscalculia. Nevertheless, critical tests should involve both longitudinal studies and intervention, and these have yet to be carried out.     

Acquired procedural dyscalculia associated to a left parietal lesion in a child.

We report the case of an 11-year-old boy who developed an anarithmetia in association with a left temporo-parietal tumor. His oral and written language were normal as well as his ability to judge magnitudes, process numbers, read operation signs and retrieve number facts. He had a specific difficulty in performing the procedures of subtraction, especially when it involved borrowing. These skills had been mastered before the present illness. This case shows that the components of calculation can be dissociated by brain lesions sustained during childhood, while arithmetic abilities are being acquired, thus reinforcing findings from developmental dyscalculias, that suggest a modular organisation of those skills during development.     

Theory-based Bayesian models of inductive learning and reasoning

Inductive inference allows humans to make powerful generalizations from sparse data when learning about word meanings, unobserved properties, causal relationships, and many other aspects of the world. Traditional accounts of induction emphasize either the power of statistical learning, or the importance of strong constraints from structured domain knowledge, intuitive theories or schemas. We argue that both components are necessary to explain the nature, use and acquisition of human knowledge, and we introduce a theory-based Bayesian framework for modeling inductive learning and reasoning as statistical inferences over structured knowledge representations.     

一个新的测量过程框架——对引入认知加工模型的再思考

从认知任务分析出发的测验设计,核心在于从认知加工的角度对项目底层作答机制作出解释。其中,能够刻画任务难度的模型最有可能与心理计量模型相结合。因而,具备成熟难度法则的各种小型理论或通用理论可以进入测量过程,实现从纯粹误差结构控制的测量到内容导引的测量的转变。在这样一个框架之下,理论对任务上作答过程的理解是否恰当,刺激特性变量与任务难度关系的揭示(即难度法则)是否准确,可以在测量的过程中进行证伪。项目反应理论还未及很好回答的效度问题,可望在这一拓展的框架中获得圆满解决。本文为新的测量过程框架提供了一个示意图。     

基于理论观的类别研究进展

介绍了西方类别研究中基于理论的观点及其研究进展。该理论强调知识背景对所形成的类别表征的影响,主张类别知识的组织是多范畴的,不同的学习任务可以导致基于原型或样例的类别表征,类别知识的运用反过来也会影响原有的类别表征。按时间顺序对理论观的产生、主要观点和经典研究进行了总结。     

Core information processing deficits in developmental dyscalculia and low numeracy

There are two different conceptions of the innate basis for numerical abilities. On the one hand, it is claimed that infants possess a 'number module' that enables them to construct concepts of the exact numerosities of sets upon which arithmetic develops (e.g. Butterworth, 1999; Gelman & Gallistel, 1978). On the other hand, it has been proposed that infants are equipped only with a sense of approximate numerosities (e.g. Feigenson, Dehaene & Spelke, 2004), upon which the concepts of exact numerosities are constructed with the aid of language (Carey, 2004) and which forms the basis of arithmetic (Lemer, Dehaene, Spelke & Cohen, 2003). These competing proposals were tested by assessing whether performance on approximate numerosity tasks is related to performance on exact numerosity tasks. Moreover, performance on an analogue magnitude task was tested, since it has been claimed that approximate numerosities are represented as analogue magnitudes. In 8-9-year-olds, no relationship was found between exact tasks and either approximate or analogue tasks in normally achieving children, in children with low numeracy or in children with developmental dyscalculia. Low numeracy was related not to a poor grasp of exact numerosities, but to a poor understanding of symbolic numerals.     

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.     

Working memory deficits in developmental dyscalculia: The importance of serial order

Although a number of studies suggests a link between working memory (WM) storage capacity of short-term memory and calculation abilities, the nature of verbal WM deficits in children with developmental dyscalculia (DD) remains poorly understood. We explored verbal WM capacity in DD by focusing on the distinction between memory for item information (the items to be retained) and memory for order information (the order of the items within a list). We hypothesized that WM for order could be specifically related to impaired numerical abilities given that recent studies suggest close interactions between the representation of order information in WM and ordinal numerical processing. We investigated item and order WM abilities as well as basic numerical processing abilities in 16 children with DD (age: 8–11 years) and 16 typically developing children matched on age, IQ, and reading abilities. The DD group performed significantly poorer than controls in the order WM condition but not in the item WM condition. In addition, the DD group performed significantly slower than the control group on a numerical order judgment task. The present results show significantly reduced serial order WM abilities in DD coupled with less efficient numerical ordinal processing abilities, reflecting more general difficulties in explicit processing of ordinal information.     

Symbolic and nonsymbolic number comparison in children with and without dyscalculia

Developmental dyscalculia (DD) is a pervasive difficulty affecting number processing and arithmetic. It is encountered in around 6% of school-aged children. While previous studies have mainly focused on general cognitive functions, the present paper aims to further investigate the hypothesis of a specific numerical deficit in dyscalculia. The performance of 10- and 11-year-old children with DD characterised by a weakness in arithmetic facts retrieval and age-matched control children was compared on various number comparison tasks. Participants were asked to compare a quantity presented in either a symbolic (Arabic numerals, number words, canonical dots patterns) or a nonsymbolic format (noncanonical dots patterns, and random sticks patterns) to the reference quantity 5. DD children showed a greater numerical distance effect than control children, irrespective of the number format. This favours a deficit in the specialised cognitive system underlying the processing of number magnitude in children with DD. Results are discussed in terms of access and representation deficit hypotheses.     

Dyslexia and dyscalculia: Two learning disorders with different cognitive profiles

This study tests the hypothesis that dyslexia and dyscalculia are associated with two largely independent cognitive deficits, namely a phonological deficit in the case of dyslexia and a deficit in the number module in the case of dyscalculia. In four groups of 8- to 10-year-olds (42 control, 21 dyslexic, 20 dyscalculic, and 26 dyslexic/dyscalculic), phonological awareness, phonological and visual-spatial short-term and working memory, naming speed, and basic number processing skills were assessed. A phonological deficit was found for both dyslexic groups, irrespective of additional arithmetic deficits, but not for the dyscalculia-only group. In contrast, deficits in processing of symbolic and nonsymbolic magnitudes were observed in both groups of dyscalculic children, irrespective of additional reading difficulties, but not in the dyslexia-only group. Cognitive deficits in the comorbid dyslexia/dyscalculia group were additive; that is, they resulted from the combination of two learning disorders. These findings suggest that dyslexia and dyscalculia have separable cognitive profiles, namely a phonological deficit in the case of dyslexia and a deficient number module in the case of dyscalculia.     

Developmental trajectory of number acuity reveals a severe impairment in developmental dyscalculia

Developmental dyscalculia is a learning disability that affects the acquisition of knowledge about numbers and arithmetic. It is widely assumed that numeracy is rooted on the “number sense”, a core ability to grasp numerical quantities that humans share with other animals and deploy spontaneously at birth. To probe the links between number sense and dyscalculia, we used a psychophysical test to measure the Weber fraction for the numerosity of sets of dots, hereafter called number acuity. We show that number acuity improves with age in typically developing children. In dyscalculics, numerical acuity is severely impaired, with 10-year-old dyscalculics scoring at the level of 5-year-old normally developing children. Moreover, the severity of the number acuity impairment predicts the defective performance on tasks involving the manipulation of symbolic numbers. These results establish for the first time a clear association between dyscalculia and impaired “number sense”, and they may open up new horizons for the early diagnosis and rehabilitation of mathematical learning deficits.     

Cognitive mechanisms in number processing and calculation: evidence from dyscalculia

This article presents a framework for the cognitive analysis of number processing and calculation. Within this framework the primary objective is the development of a model that is sufficiently detailed to serve as a basis for explaining the number-processing/calculation performance of both normal and cognitively impaired subjects. First a general model of the cognitive mechanisms for number processing and calculation is outlined. It is shown that patterns of impairments observed in brain-damaged patients support the major assumptions of the model and that the model provides a theoretically motivated framework for interpreting the deficits. A single case is then discussed in some detail, to demonstrate that through detailed analyses of impaired performance the preliminary model can be elaborated to specify not only the general architecture of the number-processing and calculation systems, but also the inner workings of specific components and the consequences of damage to these components. The article concludes with a discussion of several general issues arising from the presented arguments.     

基于理论的类别观和类别形成中的知识效应

1 引言 类别学习和概念形成,既是人类思维活动的起点,也是人类组织和贮存知识的基本方式。自上世纪50年代Bruner关于人工概念形成的经典研究之后,分类及分别学习逐渐成为认知心理学家关注的重要问题之一。但是以往的分类研究多以事物的感知特征为出发点,强调自下而上的加工(基于相似性的加工);而在基于理论的类别观被提出以后,特别是二十世纪九十年代以来,分类活动中自上而下的加工(基于知识的加工)开始受到重视,知识效应问题成为概念心理学领域的新热点。这方面的研究进展有望为深入理解概念和类别形成的机制提供新的契机。