Strategy choices in simple and complex addition: Contributions of working memory and counting knowledge for children with mathematical disability

Groups of first-grade (mean age = 82 months), third-grade (mean age = 107 months), and fifth-grade (mean age = 131 months) children with a learning disability in mathematics (MD, n = 58) and their normally achieving peers (n = 91) were administered tasks that assessed their knowledge of counting principles, working memory, and the strategies used to solve simple (4+3) and complex (16+8) addition problems. In all grades, the children with MD showed a working memory deficit, and in first grade, the children with MD used less sophisticated strategies and committed more errors while solving simple and complex addition problems. The group differences in strategy usage and accuracy were related, in part, to the group difference in working memory and to group and individual differences in counting knowledge. Across grade-level and group, the switch from simple to complex addition problems resulted in a shift in the mix of problem-solving strategies. Individual differences in the strategy mix and in the strategy shift were related, in part, to individual differences in working memory capacity and counting knowledge.     

Cognitive Analysis of Children's Mathematics Difficulties

The aim of the study is to investigate the informal and formal mathematical knowledge of children suffering from "mathematics difficulty" (MD). The research involves comparisons among three groups: fourth-grade children performing poorly in mathematics but normal in intelligence; fourth-grade peers matched for intelligence but experiencing no apparent difficulties in mathematics; and a randomly selected group of third graders. These children were individually presented with a large number of tasks designed to measure key mathematical concepts and skills. The findings suggest that: (1) MD children are not seriously deficient in key informal mathematical concepts and skills; (2) MD children seem to have elementary concepts of base ten notation but experience difficulty in related enumeration skills, particularly when large numbers are involved; (3) MD children's calculational errors often result from common error strategies; (4) MD children display severe difficulty in recalling common addition facts; and (5) in the area of problem solving, MD children are capable of "insightful" solutions and can solve simple forms of word problems, but experience difficulty with complex word problems. MD children are in many respects similar to normal, younger peers; an hypothesis of "essential cognitive normality" is advanced. The only and dramatic exception occurs in the area of number facts. While clinical experience corroborates this finding, its explanation is not evident.     

Brief report: Response inhibition and processing speed in children with motor difficulties and developmental coordination disorder

A previous study reported that children with poor motor skills, classified as having motor difficulties (MD) or Developmental Coordination Disorder (DCD), produced more errors in a motor response inhibition task compared to typically developing (TD) children but did not differ in verbal inhibition errors. The present study investigated whether these groups differed in the length of time they took to respond in order to achieve these levels of accuracy, and whether any differences in response speed could be explained by generally slow information processing in children with poor motor skills. Timing data from the Verbal Inhibition Motor Inhibition test were analyzed to identify differences in performance between the groups on verbal and motor inhibition, as well as on processing speed measures from standardized batteries. Although children with MD and DCD produced more errors in the motor inhibition task than TD children, the current analyses found that they did not take longer to complete the task. Children with DCD were slower at inhibiting verbal responses than TD children, while the MD group seemed to perform at an intermediate level between the other groups in terms of verbal inhibition speed. Slow processing speed did not account for these group differences. Results extended previous research into response inhibition in children with poor motor skills by explicitly comparing motor and verbal responses, and suggesting that slow performance, even when accurate, may be attributable to an inefficient way of inhibiting responses, rather than slow information processing speed per se.     

Numerical and arithmetical cognition: a longitudinal study of process and concept deficits in children with learning disability

Based on the stability and level of performance on standard achievement tests in first and second grade (mean age in first grade = 82 months), children with IQ scores in the low-average to high-average range were classified as learning disabled (LD) in mathematics (MD), reading (RD), or both (MD/RD). These children (n = 42), a group of children who showed variable achievement test performance across grades (n = 16), and a control group of academically normal peers (n = 35) were administered a series of experimental and psychometric tasks. The tasks assessed number comprehension and production skills, counting knowledge, arithmetic skills, working memory, the ease of activation of phonetic representations of words and numbers, and spatial abilities. The children with variable achievement test performance did not differ from the academically normal children in any cognitive domain, whereas the children in the LD groups showed specific patterns of cognitive deficit, above and beyond the influence of IQ. Discussion focuses on the similarities and differences across the groups of LD children.     

Discovery and maintenance of the many‐to‐one counting strategy in 4‐year‐olds: A microgenetic study

This study investigated the development of the many‐to‐one counting strategy in 4‐ year‐old children. In the first experiment, 52 children participated. Their development with respect to two kinds of tasks, a hidden‐items task and a needed‐items task, was studied over four sessions. Children (n = 28) who accurately used the many‐to‐one strategy in Session 4 also participated in the second experiment. These children were presented with more difficult hidden‐ and needed‐items tasks. It was found that children often produced the strategy for the first time on tasks with relatively few items. Most children then kept producing it, even if they initially did not obtain much profit from its use because of counting errors. Increasing task difficulty resulted in children making more counting errors or reverting to invalid strategies depending on the nature of the new task.     

Counting strategies and semantic analysis as applied to class inclusion

This study examined strategic and semantic aspects of the answers given by preschool children to class inclusion problems. The Piagetian logical model for class inclusion was contrasted with an alternative, problem processing model in three experiments. A major component of the alternative model is an enumeration strategy which is advantageous for learning reliable counting skills. The counting strategy was found to explain the inclusion errors of young children better than did the logic of the task. It was also found that young children understand the semantics of inclusion but are unable to coordinate their semantic knowledge with their counting strategy. Methodologically, one of the experiments suggested a fruitful extension of task analysis (Simon, 1969) to experimental design.     

A componential analysis of an early learning deficit in mathematics

This study was designed to assess strategy choice and information-processing differences in normal and mathematically disabled first and second grade children. Twenty-three normal and 29 learning disabled (LD) children solved 40 computer-presented simple addition problems. Strategies, and their associated solution times, used in problem solving were recorded on a trial-by-trial basis and each was classified in accordance with the distributions of associations model of strategy choices. Based on performance in a remedial education course, as indexed by achievement test scores, the LD sample was reclassified into a LD-improved group and an LD-no-change group. No substantive differences comparing the normal and LD-improved groups occurred in the distribution of strategy choices, strategy characteristics (e.g., error rates), or rate of information processing. The performance characteristics of the LD-no-change group, as compared to the two remaining groups, included frequent counting and memory retrieval errors, frequent use of an immature computational strategy, poor strategy choices, and a variable rate of information processing. These performance characteristics were discussed in terms of the strategy choice model and in terms of potential long-term memory and working memory capacity deficits. In addition, implications for remedial education in mathematics were discussed.     

Syntactic competence and reading ability in children

The effect of syntactic context on auditory word identification and on the ability to detect and correct syntactic errors in speech was examined in severely reading disabled children and in good and poor readers selected from the normal distribution of fourth graders. The poor readers were handicapped when correct reading required analysis of the sentence context. However, their phonological decoding ability was intact. Identification of words was less affected by syntactic context in the severely disabled readers than in either the good or poor readers. Moreover, the disabled readers were inferior to good readers in judging the syntactical integrity of spoken sentences and in their ability to correct the syntactically aberrant sentences. Poor readers were similar to good readers in the identification and judgment tasks, but inferior in the correction task. The results suggest that the severely disabled readers were inferior to both good and poor readers in syntactic awareness, and in ability to use syntactic rules, while poor readers were equal to good readers in syntactic awareness but were relatively impaired in using syntactic knowledge productively.     

Predictors of early numeracy: Is there a place for mistakes when learning about number?

It is one thing to be able to count and share items proficiently, but it is another thing to know how counting and sharing establish and identify quantity. The aim of the study was to identify which measures of numerical knowledge predict children's success on simple number problems, where counting and set equivalence are at issue. Seventy‐two 5‐year‐olds were given a battery of nine tasks on each of three sessions (at 3‐monthly intervals). Tasks measured procedural proficiency, conceptual understanding (using an error‐detection paradigm) and the ability to compare sets using number knowledge. Procedural skills remained fairly stable over the 6‐month period, and preceded children's ability to detect another's violations to those procedures. Regression analysis revealed that children who are sensitive to procedural errors in another's counting and sharing are more likely to recognize the significance of cardinal numbers for set comparisons. We suggest that although children's conceptual understanding of well‐rehearsed routines is often limited, conceptual insight might be achieved by setting tasks that require reflection rather than practice.     

学前儿童近似数量系统敏锐度与符号数学能力的关系

选取杭州市122名学前儿童(3~6岁)为被试,以点数比较任务及点数异同任务测量幼儿的近似数量系统敏锐度,以数数测验、基数测验、符号数字知识测验及简单计算来测量幼儿的符号数学能力,以此考察学前儿童近似数量系统敏锐度的发展及与符号数学能力的关系。结果发现:(1)随年龄增长,学前儿童的近似数量加工的敏锐度逐渐提高;(2)点数比较任务与点数异同任务均适合测量学前儿童近似数量系统敏锐度,但儿童完成点数比较任务的正确率要高于点数异同任务的正确率;(3)在抑制控制、短时记忆、工作记忆和言语测验成绩被控制后,根据点数比较任务计算的韦伯系数能显著预测学前儿童的基数和符号数字知识测验分数,总正确率能显著预测学前儿童的数数、基数、符号数字知识测验分数;(4)点数异同任务中只有点数不同试次下的正确率能显著预测学前儿童的符号数字知识测验分数。     

Reading,syntactic, orthographic,and working memory skills of bilingual Arabic-English speaking Canadian children

This study assessed the reading, language, and memory skills of 56 bilingual Arab-Canadian children age's 9-14. English was their main instructional language, and Arabic was the language spoken at home. All children attended a Heritage Language Program in Toronto where they were taught to read and write Arabic. The children were administered word and pseudo-word reading, language, and working memory tests in English and Arabic. The majority of the children showed at least adequate proficiency in both languages. There was a significant relationship between the acquisition of word and pseudo-word reading working memory, and syntactic awareness skills in the two languages. The poor readers in Arabic had lower scores on all linguistic tasks, except the visual task. There were no significant differences between bilingual English Arabic children and monolingual English-speaking children on the reading, language, and memory tasks. However, bilingual English Arabic children who had reading problems in English had higher scores on English pseudo-word reading and spelling tasks than monolingual English-speaking children with reading disabilities, probably because of positive transfer from the regular nature of Arabic orthography. In this case, bilingualism does not appear to have negative consequences for the development of language reading skills in both languages—Arabic and English—despite the different nature of the two orthographies.     

The role of relational reasoning in children's addition concepts

The study addresses the relational reasoning of different‐aged children and how addition reasoning is related to problem‐solving skills within addition and to reasoning skills outside addition. Ninety‐two 5‐ to 8‐year‐olds were asked to solve a series of conceptually related and unrelated addition problems, and the speed and accuracy of all self‐reported strategies were used to monitor their addition performance. Children were also given a series of general relational reasoning tasks to assess their ability to solve problems based on thematic, causal and visual relations. The results revealed that, while children were able to reason about commutativity relations, recognition of relations based on additive composition was rare. Furthermore, children's ability to reason about addition concepts increased with age and problem‐solving proficiency. Reasoning about addition concepts was related to performance on the thematic, causal and visual reasoning tasks for older children but not for younger children. Overall, the findings suggest that while children's early knowledge of addition relations is domain specific, as children develop in their broader reasoning abilities these developments enhance their addition reasoning.     

Children’s understandings of counting: Detection of errors and pseudoerrors by kindergarten and primary school children

In this study, the development of comprehension of essential and nonessential aspects of counting is examined in children ranging from 5 to 8 years of age. Essential aspects, such as logical rules, and nonessential aspects, including conventional rules, were studied. To address this, we created a computer program in which children watched counting errors (abstraction and order irrelevance errors) and pseudoerrors (with and without cardinal value errors) occurring during a detection task. The children judged whether the characters had counted the items correctly and were asked to justify their responses. In general, our data show that performance improved substantially with age in terms of both error and pseudoerror detection; furthermore, performance was better with regard to errors than to pseudoerrors as well as on pseudoerror tasks with cardinal values versus those without cardinal values. In addition, the children’s justifications, for both the errors and pseudoerrors, made possible the identification of conventional rules underlying the incorrect responses. A particularly relevant trend was that children seem to progressively ignore these rules as they grow older. Nevertheless, this process does not end at 8 years of age given that the conventional rules of temporal and spatial adjacency were present in their judgments and were primarily responsible for the incorrect responses.     

Phonological processing in reading Chinese among normally achieving and poor readers

Ninety-four children in grades one to six (7 to 12 years old) in Hong Kong were individually administered a Chinese reading test consisting of 80 characters that varied in frequency (high or low) and orthographic structure (simple or complex). Phonological processing tasks, including short-term memory, pseudo-character recognition, and tone discrimination, were also administered. During reading, younger normal and poor readers made more semantic and visual errors, whereas older and normally achieving children made more phonologically related errors. Normally achieving readers also performed at a higher level than poor readers on short-term memory, pseudo-character recognition, and tone discrimination tasks. Phonological processing apparently plays a significant part in the development of reading skills in Chinese.     

Numerical and arithmetical cognition: patterns of functions and deficits in children at risk for a mathematical disability.

Based on performance on standard achievement tests, first-grade children (mean age = 82 months) with IQ scores in the low-average to high-average range were classified as at risk for a learning disability (LD) in mathematics, reading, or both. These at-risk children (n = 55) and a control group of academically normal peers (n = 35) were administered experimental tasks that assessed number comprehension and production skills, counting knowledge, arithmetic skills, working memory, and ease of retrieving information from long-term memory. Different patterns of intact cognitive functions and deficits were found for children in the different at-risk groups. As a set, performance on the experimental tasks accounted for roughly 50% and 10% of the group differences in mathematics and reading achievement, respectively, above and beyond the influence of IQ. Performance on the experimental tasks thus provides insights into the cognitive deficits underlying different forms of LD, as well as into the sources of individual differences in academic achievement.     

Early Maternal Spatial Support for Toddlers and Math Skills in Second Grade

ABSTRACT

The aim of this study is to investigate whether maternal spatial support during two types of joint manipulative toy play tasks with 2-year-old children was longitudinally associated with math screening test scores in second grade. The interaction between spatial support and maternal education was explored as well. We also investigated predictions of a teacher rating of math performance at second grade, although these effects were less robust. Data were drawn from BONDS (The Behavior Outlook Norwegian Developmental Study), a longitudinal study of Norwegian children and their families. Participants were a subsample of 932 mothers and their 2-year-olds. Mothers were asked to help their children solve both a puzzle task and a shape-color sorting task. Mothers’ spatial support included spatial language, gestures, and placement of objects. Results showed that higher levels of spatial support during mother-child interaction tasks at 2 years of age was significantly associated with fewer math difficulties in second grade. This was the case for a puzzle task (a task associated with spatial visualization skills), but not for a shape-color sorting task (a task associated with shape and color feature discriminations). Conclusions are drawn with respect to the importance of identifying optimal parental spatial strategies associated with better math outcomes. These findings on parental facilitation of spatial skills during joint early play may be useful for future training interventions directed at parents of children at risk for poor math skills.     

Working memory and access to numerical information in children with disability in mathematics

The relationship among working memory, mathematic ability, and the cognitive impairment of children with difficulties in mathematics was examined. A group of children with difficulties in mathematics (MD) was compared with a group of children with a normal level of achievement matched for vocabulary, age, and gender (N = 49). The children were required to perform a variety of working memory and short-term memory tasks that had been administered 1 year previously. Moreover, the children were asked to perform tasks designed to provide information about speed of articulation. The results suggest a general working memory deficit in children with MD, specifically in the central executive component of Baddeley's model and primarily in the ability to inhibit irrelevant information. However, the MD children were not impaired in speech rate and counting speed tasks, which mainly involve the role of the articulatory loop.     

Five-year-olds' handling of reference and description in the domains of language and mental representation

Children's concurrent success on false belief tasks and their handling of two labels for one object (e.g., bunny/rabbit) has been interpreted as demonstrating understanding about the essential features of representation. Three experiments reveal the limitations in 5-year-olds' understanding for both mental and linguistic representations. We report relatively poor performance on a task involving two labels for one object (e.g., dice/eraser) which required children to treat another's knowledge as representing only some of the feature of its real referent: Dice but not eraser. Five year olds who made errors also had difficulty handling the fact that a written word 'dice' referring to such a dice/eraser, can also be applied to a standard dice but not to a standard eraser. These children lacked metalinguistic awareness of words as entities that both refer and describe.     

Basic number processing in children with specific learning disorders: Comorbidity of reading and mathematics disorders

As well as being the hallmark of mathematics disorders, deficits in number processing have also been reported for individuals with reading disorders. The aim of the present study was to investigate separately the components of numerical processing affected in reading and mathematical disorders within the framework of the Triple Code Model. Children with reading disorders (RD), mathematics disorders (MD), comorbid deficits (RD + MD), and typically developing children (TD) were tested on verbal, visual-verbal, and nonverbal number tasks. As expected, children with MD were impaired across a broad range of numerical tasks. In contrast, children with RD were impaired in (visual-)verbal number tasks but showed age-appropriate performance in nonverbal number skills, suggesting their impairments were domain specific and related to their reading difficulties. The comorbid group showed an additive profile of the impairments of the two single-deficit groups. Performance in speeded verbal number tasks was related to rapid automatized naming, a measure of visual-verbal access in the RD but not in the MD group. The results indicate that deficits in number skills are due to different underlying cognitive deficits in children with RD compared to children with MD: a phonological deficit in RD and a deficit in processing numerosities in MD.     

Connaissance du nom et du son des lettres, habiletés métaphonémiques et capacités de décodage en grande section de maternelle

This study examined the specific implication of letter-name knowledge in reading acquisition on 5-year-old kindergarten children. Several studies underlined the importance of both phonological and orthographic skills (letters and grapho-phonological associations’ knowledge developed during reading acquisition). So, we focused on the link between those different early abilities and on their links with reading acquisition during school year. Thus, at the beginning and at the end of the year, we proposed to 40 children metaphonemic tasks (phoneme identification in initial and final position in the words), letter-sound and letter-name knowledge tasks (naming and identification) and a pseudo-words decoding task. In addition, to access letter-name knowledge, two types of measurements were proposed: an accuracy measure and a speed one. Mean number of correct responses and reaction times were collected by children and analysed too. Results show that letter-name knowledge could play a part in metaphonemic abilities development and in letter-sound associations’ discovery and acquisition, at the beginning of the year. Thanks to those skills improvement, letter-name knowledge could have an indirect impact on later children's decoding skills. Furthermore, the direct link, already observed at the beginning of the year, between letter-name knowledge and decoding skills still exists in April. At last, analyses on letter-name knowledge access time show that the more children name an important number of letters, the more they do it quickly. These results could mean that a certain level of letter-name knowledge have to be reach before children could use this knowledge automatically. All together, results confirm the importance of good letter-name knowledge in the first stages of reading acquisition. We also suggest taking into account some clues of letter-name knowledge's automatic use to better understand reading acquisition processes.