Activation of number facts in bilinguals

Two experiments examined the effect of the presentation format of numbers—digits versus word format in the first and in the second languages of bilinguals—on mental arithmetic. Speed of number-fact retrieval and the presence of interference produced by numbers that were either numerically close to or associatively related to the correct answers of stored arithmetic problems (e.g., 2+5 and 7×8) were compared across formats. The verification of true problems was increasingly slower and less accurate from the digit condition to the second-language condition. Interference was produced by both types of incorrect answers in the digit and first-language conditions, whereas in the second-language condition, it was constrained to answers that were numerically close to correct answers. Together, the results suggest that the retrieval of arithmetic facts and the automatic spreading of activation within the network of numerical facts are not only language-sensitive, but format-sensitive in general.     

Convergence between attention variables and factors of psychometric intelligence in older adults

The purpose of the present study was to examine the hypothesis that individual differences on measures of attention would converge with select factors of psychometric intelligence, especially fluid intelligence and short-term acquisition and retrieval. A sample of 83 elderly adults $\text{(}\text{X}\text{= 71}\text{years}\text{)}$ was administered a battery of 17 psychometric ability tests. Tests were selected to mark four psychometric ability factors (Cattell and Horn's dimensions of fluid and crystallized intelligence, short-term acquisition and retrieval, and perceptual speed). Also, seven tasks representing four aspects of attention—decoding processes, selective attention, attention switching, and concentration—were administered. A confirmatory factor analysis was conducted to examine the relationships among the four psychometric ability factors and 11 variables obtained from the attention tasks. Results were only partially consistent with the hypothesized pattern of convergence. Two attention measures had significant loadings on a fluid-type intelligence factor, and one had a marginally significant loading on a short-term memory factor. In general, the greatest convergence occured between attention variables and the ability factor of Perceptual Speed. Results were discussed with respect to previous research on psychometric abilities and cognitive processes, the theory of fluid-crystallized intelligence, and their implications for understanding intellectual aging.     

Applied Problem Solving in Children with ADHD: The Mediating Roles of Working Memory and Mathematical Calculation

The difficulties children with ADHD experience solving applied math problems are well documented; however, the independent and/or interactive contributions of cognitive processes underlying these difficulties are not fully understood and warrant scrutiny. The current study examines two primary cognitive processes integral to children’s ability to solve applied math problems: working memory (WM) and math calculation skills (i.e., the ability to utilize specific facts, skills, or processes related to basic math operations stored in long-term memory). Thirty-six boys with ADHD-combined presentation and 33 typically developing (TD) boys aged 8–12 years old were administered multiple counterbalanced tasks to assess upper (central executive [CE]) and lower level (phonological [PH STM] and visuospatial [VS STM] short-term memory) WM processes, and standardized measures of mathematical abilities. Bias-corrected, bootstrapped mediation analyses revealed that CE ability fully mediated between-group differences in applied problem solving whereas math calculation ability partially mediated the relation. Neither PH STM nor VS STM was a significant mediator. When modeled together via serial mediation analysis, CE in tandem with math calculation ability fully mediated the relation, explained 79% of the variance, and provided a more parsimonious explication of applied mathematical problem solving differences among children with ADHD. Results suggest that interventions designed to address applied math difficulties in children with ADHD will likely benefit from targeting basic knowledge of math facts and skills while simultaneously promoting the active interplay of these skills with CE processes.     

Effects of cognitive training on primary mental ability structure

We report results of the first empirical test, as far as we know, of the assumption of structural invariance of latent constructs from pretest to posttest in cognitive training research on the elderly. In all, 401 participants in the Seattle Longitudinal Study, over 62 years old, received a 5-hr test battery at pre- and posttest that included 16 ability tests, marking the five primary abilities of Spatial Orientation, Inductive Reasoning, Numerical Ability, Verbal Ability, and Perceptual Speed. A total of 229 of our subjects received 5 hr of individual training on either Spatial Orientation or Inductive Reasoning. Restricted factor analysis with the LISREL algorithm tested the hypothesis of measurement equivalence across test occasions, separately for the control subjects and for each of the training groups. When ability-specific cognitive training intervenes, no structural change is observed for abilities not subject to intervention. However, slight shifts occurred in the optimal regression weights for the different markers for the training target abilities.     

心算的加工机制：来自认知神经科学的研究

心算加工分编码（表征）、运算（或提取）和反应三个阶段,这三个阶段相互影响。不同输入形式的数字表征在顶叶的不同区域完成。算术知识提取主要与左脑顶内沟有关,但当心算变得更复杂时而需要具体运算时,左脑额叶下部出现明显激活。所有与心算有关的脑区涉及大脑前额皮层和颞顶枕联合皮层的综合作用,并总体表现为左脑优势,但估算、珠心算以及某些具有特殊心算能力的人的心算还依赖视空间表征,这与右脑额顶区和楔前叶的活动有关     

Cognitive representations and processes in arithmetic: inferences from the performance of brain-damaged subjects.

In this article, we present data from two brain-damaged patients with calculation impairments in support of claims about the cognitive mechanisms underlying simple arithmetic performance. We first present a model of the functional architecture of the cognitive calculation system based on previous research. We then elaborate this architecture through detailed examination of the patterns of spared and impaired performance of the two patients. From the patients' performance we make the following theoretical claims: that some arithmetic facts are stored in the form of individual fact representations (e.g., 9 x 4 = 36), whereas other facts are stored in the form of a general rule (e.g., 0 x N = 0); that arithmetic fact retrieval is mediated by abstract internal representations that are independent of the form in which problems are presented or responses are given; that arithmetic facts and calculation procedures are functionally independent; and that calculation algorithms may include special-case procedures that function to increase the speed or efficiency of problem solving. We conclude with a discussion of several more general issues relevant to the reported research.     

The Role of Executive Function in Arithmetic Problem-Solving Processes: A Study of Third Graders

This study investigated the roles of different executive function (EF) components (inhibition, shifting, and working memory) in 2-step arithmetic word problem solving. A sample of 139 children aged 8 years old and regularly attending the 3rd grade of primary school were tested on 6 EF tasks measuring different EF components, a reading task and a reading comprehension task, an arithmetic facts task evaluating basic knowledge of calculation, and three 2-step arithmetic word problems. Multiple hierarchical regression analyses were conducted to investigate the roles of the different EF components in the various phases of the problem-solving process. The results showed that EF affects the various phases of problem solving differently over and above calculation knowledge and reading abilities. The implications of these findings are discussed in relation to further understanding the role of cognitive skills in mathematical problem solving and in relation to instructional approaches that may increase children’s performance on 2-step arithmetic word problems.     

Mental addition in third,fourth, and sixth graders

Research on mental arithmetic has suggested that young children use a counting algorithm for simple mental addition, but that adults use memory retrieval from an organized representation of addition facts. To determine the age at which performance shifts from counting to retrieval, children in grades 3, 4, and 6 were tested in a true/false verification task. Reaction time patterns suggested that third grade is a transitional age with respect to memory structure for addition—half of these children seemed to be counting and half retrieving from memory. Results from fourth and sixth graders implicated retrieval quite strongly, as their results resembled adult RTs very closely. Fourth graders' processing, however, was easily disrupted when false problems were presented. The third graders' difficulties are not due to an inability to form mental representations of number; all three grades demonstrated a strong split effect, indicative of a simpler mental representation of numerical information than is necessary for addition. The results were discussed in the context of memory retrieval versus counting models of mental arithmetic, and the increase across age in automaticity of retrieval processes.     

Retrieval-induced forgetting of arithmetic facts but not rules

Retrieval of a multiplication fact (2×6 =12) can disrupt retrieval of its addition counterpart (2+6=8). We investigated whether this retrieval-induced forgetting effect applies to rule-governed arithmetic facts (i.e., 0×N=0, 1×N=N). Participants (n=40) practised rule-governed multiplication problems (e.g., 1×4, 0×5) and multiplication facts (e.g., 2×3, 4×5) for four blocks and then were tested on the addition counterparts (e.g., 1+4, 0+5, 2+3, 4+5) and control additions. Increased addition response times and errors relative to controls occurred only for problems corresponding to multiplication facts, with no problem-specific effects on addition counterparts of rule-governed multiplications. In contrast, the rule-governed 0+N problems provided evidence of generalisation of practice across items, whereas the fact-based 1+N problems did not. These findings support the theory that elementary arithmetic rules and facts involve distinct memory processes, and confirmed that previous, seemly inconsistent findings of RIF in arithmetic owed to the inclusion or exclusion of rule-governed problems.     

Cognitive arithmetic: a review of data and theory.

The area of cognitive arithmetic is concerned with the mental representation of number and arithmetic, and the processes and procedures that access and use this knowledge. In this article, I provide a tutorial review of the area, first discussing the four basic empirical effects that characterize the evidence on cognitive arithmetic: the effects of problem size or difficulty, errors, relatedness, and strategies of processing. I then review three current models of simple arithmetic processing and the empirical reports that support or challenge their explanations. The third section of the review discusses the relationship between basic fact retrieval and a rule-based component or system, and considers current evidence and proposals on the overall architecture of the cognitive arithmetic system. The review concludes with a final set of speculations about the all-pervasive problem difficulty effect, still a central puzzle in the field.     

How are visuospatial working memory, executive functioning, and spatial abilities related? A latent-variable analysis.

This study examined the relationships among visuospatial working memory (WM) executive functioning, and spatial abilities. One hundred sixty-seven participants performed visuospatial short-term memory (STM) and WM span tasks, executive functioning tasks, and a set of paper-and-pencil tests of spatial abilities that load on 3 correlated but distinguishable factors (Spatial Visualization, Spatial Relations, and Perceptual Speed). Confirmatory factor analysis results indicated that, in the visuospatial domain, processing-and-storage WM tasks and storage-oriented STM tasks equally implicate executive functioning and are not clearly distinguishable. These results provide a contrast with existing evidence from the verbal domain and support the proposal that the visuospatial sketchpad may be closely tied to the central executive. Further, structural equation modeling results supported the prediction that, whereas they all implicate some degree of visuospatial storage, the 3 spatial ability factors differ in the degree of executive involvement (highest for Spatial Visualization and lowest for Perceptual Speed). Such results highlight the usefulness of a WM perspective in characterizing the nature of cognitive abilities and, more generally, human intelligence.     

样例和练习在促进解题迁移能力中的作用

通过一个2×2×2的因素实验,对96名初一学生在解题迁移能力中受样例和练习的影响进行了研究。结果表明,结合样例进行的练习促进了技能的熟练和解题能力的迁移;练习本身并不总能保证促进技能的熟练和解题能力的迁移,它至少要受三方面因素的影响第一,与在练习中是否有来自外部的指导和反馈有关;第二,与练习的任务性质有关;第三,与参与练习的个体智力和认知水平有关。     

Validity of measures of cognitive processes and general ability for learning and performance on highly complex computerized tutors: is the g factor of intelligence even more general?

Theoretical arguments and analyses from 2 studies provide compelling evidence that computerized measures of information-processing skills and abilities are highly useful supplements to more traditional, paper-based measures of general mental ability for predicting individuals' capacity to learn from and perform on highly challenging, multifaceted tutors. These tutors were designed to emulate learning and performance in complex, real-world settings. Hierarchical confirmatory factor analysis provided evidence that a general, higher order factor model with general ability at the apex could quite adequately and singularly account for the individual-differences data, both traditional and cognitive-process measures. Results are interpreted in light of the utility and generality of human cognitive abilities.     

Strategies in subtraction problem solving in children

The aim of this study was to investigate the strategies used by third graders in solving the 81 elementary subtractions that are the inverses of the one-digit additions with addends from 1 to 9 recently studied by Barrouillet and Lépine. Although the pattern of relationship between individual differences in working memory, on the one hand, and strategy choices and response times, on the other, was the same in both operations, subtraction and addition differed in two important ways. First, the strategy of direct retrieval was less frequent in subtraction than in addition and was even less frequent in subtraction solving than the recourse to the corresponding additive fact. Second, contrary to addition, the retrieval of subtractive answers is confined to some peculiar problems involving 1 as the subtrahend or the remainder. The implications of these findings for developmental theories of mental arithmetic are discussed.     

How is phonological processing related to individual differences in children’s arithmetic skills?

While there is evidence for an association between the development of reading and arithmetic, the precise locus of this relationship remains to be determined. Findings from cognitive neuroscience research that point to shared neural correlates for phonological processing and arithmetic as well as recent behavioral evidence led to the present hypothesis that there exists a highly specific association between phonological awareness and single‐digit arithmetic with relatively small problem sizes. The present study examined this association in 37 typically developing fourth and fifth grade children. Regression analyses revealed that phonological awareness was specifically and uniquely related to arithmetic problems with a small but not large problem size. Further analysis indicated that problems with a high probability of being solved by retrieval, but not those typically associated with procedural problem‐solving strategies, are correlated with phonological awareness. The specific association between phonological awareness and arithmetic problems with a small problem size and those for which a retrieval strategy is most common was maintained even after controlling for general reading ability and phonological short‐term memory. The present findings indicate that the quality of children’s long‐term phonological representations mediates individual differences in single‐digit arithmetic, suggesting that more distinct long‐term phonological representations are related to more efficient arithmetic fact retrieval.     

Menatal addition: A test of three verification models

Three explanations of adults’ mental addition performance, a counting-based model, a direct-access model with a backup counting procedure, and a network retrieval model, were tested. Whereas important predictions of the two counting models were not upheld, reaction times (RTs) to simple addition problems were consistent with the network retrieval model. RT both increased with problem size and was progressively attenuated to false stimuli as the split (numerical difference between the false and correct sums increased. For large problems, the extreme level of split (13) yielded an RT advantage for false over true problems, suggestive of a global evaluation process operating in parallel with retrieval. RTs to the more complex addition problems in Experiment 2 exhibited a similar pattern of significance and, in regression analyses, demonstrated that complex addition (e.g., 14+12=26) involves retrieval of the simple addition components (4+2=6). The network retrieval/decision model is discussed in terms of its fit to the present data, and predictions concerning priming facilitation and inhibition are specified. The similarities between mental arithmetic results and the areas of semantic memory and mental comparisons indicate both the usefulness of the network approach to mental arithmetic and the usefulness of mental arithmetic to cognitive psychology.     

Preschoolers doing arithmetic: the concepts are willing but the working memory is weak.

The study of early mathematical development provides important insights into young children's emerging academic competencies and, potentially, a basis for adapting instructional methods. We presented nonverbal forms of two- and three-term arithmetic problems to 4-year-olds to determine (a) the extent to which certain information-processing demands make some problems more difficult than others and (b) whether preschoolers use arithmetic concepts spontaneously when solving novel problems. Children's accuracy on simple arithmetic problems (a + b and a - b) was strongly related (r2 = .88) to representational set size, the maximum number of units that need to be held in working memory to solve a given problem. Some children also showed spontaneous use of procedures based on the arithmetic principle of inversion when solving problems of the form a + b - b. These results highlight the importance of identifying information-processing and conceptual characteristics in the early development of mathematical cognition.     

Models of arithmetic fact retrieval: an evaluation in light of findings from normal and brain-damaged subjects.

Retrieval of basic arithmetic facts is a central aspect of almost any arithmetic performance. Furthermore, the arithmetic facts provide an opportunity to study memory processes in the context of a naturally occurring but circumscribed set of facts. This article examines current models of arithmetic fact retrieval in light of previously reported data from normal subjects, as well as the results from brain-damaged patients reported by Sokol, McCloskey, Cohen, and Aliminosa (1991) in the preceding article. The discussion serves to delineate the strengths and limitations of the models and, more generally, to identify important theoretical and empirical issues in the study of arithmetic fact retrieval.     

Contextualization in Perspective

In this article, we examine the hypothesis that problem comprehension and computational processes interact during the solving of an arithmetic word problem. Two experiments tested subjects on a series of addition and subtraction word problems, the content of which varied on the basis of problem type and on the magnitude of the numbers involved (problem size). Performance data are presented and analyzed in terms of solution reaction times and error patterns. Results confirmed the main effects of problem size and problem type as factors determining reaction times but failed to show any significant interaction between the two factors. These results suggest that the cognitive processes involved in understanding an arithmetic word problem and in performing the required computations are best explained by a serial processing model. The absence of an interaction between problem comprehension and computational processes questions the notion that automatized retrieval facilitates problem solving and assertions suggesting that increasing computational requirements can interfere with problem-solving performance.