Cognitive correlates of mathematical achievement in children with cerebral palsy and typically developing children

Background . Remarkably few studies have investigated the nature and origin of learning difficulties in children with cerebral palsy (CP). Aims . To investigate math achievement in terms of word‐problem solving ability in children with CP and controls. Because of the potential importance of reading for word‐problem solving, we investigated reading as well. Sample . Children with CP attending either special (n= 41) or mainstream schools (n= 16) and a control group of typically developing children in mainstream schools (n= 16). Method . Group differences in third grade math and reading, controlled for IQ, were tested with analyses of co‐variance (ANCOVAs). Hierarchical regression was used to investigate cognitive correlates of third grade math and reading. Predictors included verbal and non‐verbal IQ measured in first grade, components of working memory (WM) and executive function (EF) measured in second grade, and arithmetic fact fluency and reading measured in third grade. Results . Children with CP in special schools performed significantly worse than their peers on word‐problem solving and reading. There was a trend towards worse performance in children with CP in mainstream schools compared to typically developing children. Conclusions . Impairments of non‐verbal IQ and WM updating predicted future difficulties in both word‐problem solving and reading. Impairments of visuospatial sketchpad and inhibition predicted future word‐problem, but not reading difficulty. Conversely, deficits of phonological loop predicted reading but not word‐problem difficulty. Concurrent arithmetic fact fluency and reading ability were both important for word‐problem solving ability. These results could potentially help to predict which children are likely to develop specific learning difficulties, facilitating early intervention.     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.     

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.     

语音意识的不同成分在汉语儿童英语阅读学习中的作用

研究一测查了74名小学三、五年级儿童辨别、删除汉语和英语音节、首音-韵脚、音位等不同语音成分的能力以及英语单词阅读,考察语音意识不同成分与英语阅读学习的关系及母语语音意识的作用途径.研究二测查了83名英语阅读较差和73名英语阅读一般及以上儿童的英语语音删除和单词认读能力,考察阅读水平对于语音意识作用的调节效应.结果表明:(1)英语首音-韵脚意识对英语阅读具有显著的独立贡献;(2)汉语首音-韵脚意识和声调意识分别对英语单词认读和假词拼读具有显著的独立贡献,二者通过英语首音-韵脚意识的中介发挥作用;(3)阅读水平具有显著的调节作用.首音-韵脚意识是正常儿童阅读的有效预测变量,而音节意识是低水平儿童阅读的有效预测变量.上述结果与有关语音意识各成分在英语为母语儿童阅读学习中作用的研究结果不同,提示第二语言的学习具有特殊性,母语经验影响着个体第二语言学习的过程.     

基于关系-表征复杂性模型的数学应用题表征能力测验

基于关系-表征复杂性模型,从每道应用题涉及集合关系的嵌套程度角度事前分析其关系复杂性,编制了难度序列变化的应用题测验,以考察问题表征能力。采用该测验测查了四至七年级共165名学生,考察事前分析的合理性及表征水平随年龄的变化。结果表明:(1)事前分析对两个事后难度指标(错误率和Rasch模型分析的任务难度)的解释率分别为73.7%、78.7%;该测验得分与测查思维水平层次变化的SOLO分类测验上的得分有较高相关(r=0.65)。(2)四年级的应用题表征水平显著低于五、六、七年级,其他三个年级差异不显著;而且随着问题关系复杂性的增加,年级差异增大。这说明基于关系-表征复杂性模型的事前分析是合理的,据此编制的测验能够测查表征水平随年龄的变化。     

Activation of Real-World Knowledge in the Solution of Word Problems

This article describes two studies that examine factors influencing children's access to real-world knowledge during the solution of word problems. In the first study, based on work in Brazil by Carraher, Carraher, and Schliemann (1987), children were asked to solve arithmetic problems presented in three contexts: (a) as word problems, (b) in simulated store situations, and (c) as symbolic computations. Brazilian children were both more successful and more likely to use mental, informal strategies when solving word problems than when solving symbolic computations. We did not find the same results with our U.S. sample; no effects of context were found in either strategy use or success. Comparison of U.S. and Brazilian children's responses suggested that children may tend to access real-world content when the numbers in a word problem match the problem content, and a second study was conducted to test this interpretation. Children were presented with word problems in which the problem content either matched or did not match the numbers in the problem. It was found that when the numbers matched the problem content, children were more successful in solving the problems and more likely to access their domain knowledge during problem solution, as evidenced by the strategies they used to solve problems in the matched condition. These findings suggest ways in which activation of real-world knowledge might be facilitated during the solution of word problems in school.     

Patterns of problem‐solving in children's literacy and arithmetic

Patterns of problem‐solving among 5‐to‐7 year‐olds' were examined on a range of literacy (reading and spelling) and arithmetic‐based (addition and subtraction) problem‐solving tasks using verbal self‐reports to monitor strategy choice. The results showed higher levels of variability in the children's strategy choice across Years 1 and 2 on the arithmetic (addition and subtraction) than literacy‐based tasks (reading and spelling). However, across all four tasks, the children showed a tendency to move from less sophisticated procedural‐based strategies, which included phonological strategies for reading and spelling and counting‐all and finger modelling for addition and subtraction, to more efficient retrieval methods from Years 1 to 2. Distinct patterns in children's problem‐solving skill were identified on the literacy and arithmetic tasks using two separate cluster analyses. There was a strong association between these two profiles showing that those children with more advanced problem‐solving skills on the arithmetic tasks also showed more advanced profiles on the literacy tasks. The results highlight how different‐aged children show flexibility in their use of problem‐solving strategies across literacy and arithmetical contexts and reinforce the importance of studying variations in children's problem‐solving skill across different educational contexts.     

A computer simulation of children’s arithmetic word-problem solving

ARITHPRO is a computer simulation of children’s arithmetic word-problem solving behavior. It is an instantiation of a recently proposed cognitive model of the knowledge and procedures required to solve such problems. The program solves word problems by (1) comprehending the story text in which the problem is embedded, (2) comprehending numerical information as sets of objects, (3) building superstructures from these sets, thereby specifying their logical relations, and (4) using a counting procedure to derive the answer to the problem. This report describes ARITHPRO and its architecture and knowledge base. A few comparisons of ARITHPRO’s performance with that of children are also provided.     

Language and modeling word problems in mathematics among bilinguals

The study was conducted to determine whether the language of math word problems would affect how Filipino-English bilingual problem solvers would model the structure of these word problems. Modeling the problem structure was studied using the problem-completion paradigm, which involves presenting problems without the question. The paradigm assumes that problem solvers can infer the appropriate question of a word problem if they correctly grasp its problem structure. Arithmetic word problems in Filipino and English were given to bilingual students, some of whom had Filipino as a first language and others who had English as a first language. The problem-completion data and solution data showed similar results. The language of the problem had no effect on problem-structure modeling. The results were discussed in relation to a more circumscribed view about the role of language in word problem solving among bilinguals. In particular, the results of the present study showed that linguistic factors do not affect the more mathematically abstract components of word problem solving, although they may affect the other components such as those related to reading comprehension and understanding.     

Working memory components that predict word problem solving: Is it merely a function of reading,calculation, and fluid intelligence?

The purpose of this study was to assess whether the differential effects of working memory (WM) components (the central executive, phonological loop, and visual–spatial sketchpad) on math word problem-solving accuracy in children (N?=?413, ages 6–10) are completely mediated by reading, calculation, and fluid intelligence. The results indicated that all three WM components predicted word problem solving in the nonmediated model, but only the storage component of WM yielded a significant direct path to word problem-solving accuracy in the fully mediated model. Fluid intelligence was found to moderate the relationship between WM and word problem solving, whereas reading, calculation, and related skills (naming speed, domain-specific knowledge) completely mediated the influence of the executive system on problem-solving accuracy. Our results are consistent with findings suggesting that storage eliminates the predictive contribution of executive WM to various measures Colom, Rebollo, Abad, & Shih (Memory & Cognition, 34: 158-171, 2006). The findings suggest that the storage component of WM, rather than the executive component, has a direct path to higher-order processing in children.     

Working memory and intrusions of irrelevant information in a group of specific poor problem solvers

An important body of evidence has shown that reading comprehension ability is related to working memory and, in particular, to the success in Daneman and Carpenter's (1980) reading and listening span test. This research tested a similar hypothesis for arithmetic word problems, since, in order to maintain and process the information, they require working memory processes. A group of children possessing average vocabulary but poor arithmetic problem-solving skills was compared with a group of good problem solvers, matched for vocabulary, age, and socioeconomic status. Poor problem solvers presented lower recall and a greater number of intrusion errors in a series of tasks testing working memory and memory for problems. The results obtained over a series of six experimental phases, conducted during a 2-school-year period, offer evidence in favor of the hypotheses that groups of poor problem solvers may have poor performance in a working memory test requiring inhibition of irrelevant information (Hypothesis 1), but not in other short-term memory tests (Hypothesis 2), that this difficulty is related to poor recall of critical information and greater recall of to-be-inhibited information (Hypothesis 3), that poor problem solvers also have difficulty in remembering only relevant information included in arithmetic word problems (Hypothesis 4) despite the fact that they are able to identify relevant information (Hypothesis 5). The results show that problem-solving ability is related to the ability of reducing the memory accessibility of nontarget and irrelevant information.     

Arithmetic word problem solving: evidence for a magnitude-based mental representation

Previous findings have suggested that number processing involves a mental representation of numerical magnitude. Other research has shown that sensory experiences are part and parcel of the mental representation (or “simulation”) that individuals construct during reading. We aimed at exploring whether arithmetic word-problem solving entails the construction of a mental simulation based on a representation of numerical magnitude. Participants were required to solve word problems and to perform an intermediate figure discrimination task that matched or mismatched, in terms of magnitude comparison, the mental representations that individuals constructed during problem solving. Our results showed that participants were faster in the discrimination task and performed better in the solving task when the figures matched the mental representations. These findings provide evidence that an analog magnitude-based mental representation is routinely activated during word-problem solving, and they add to a growing body of literature that emphasizes the experiential view of language comprehension.     

Arithmetic word problem solving: a Situation Strategy First framework

Before instruction, children solve many arithmetic word problems with informal strategies based on the situation described in the problem. A Situation Strategy First framework is introduced that posits that initial representation of the problem activates a situation-based strategy even after instruction: only when it is not efficient for providing the numerical solution is the representation of the problem modified so that the relevant arithmetic knowledge might be used. Three experiments were conducted with Year 3 and Year 4 children. Subtraction, multiplication and division problems were created in two versions involving the same wording but different numerical values. The first version could be mentally solved with a Situation strategy (Si version) and the second with a Mental Arithmetic strategy (MA version). Results show that Si-problems are easier than MA-problems even after instruction, and, when children were asked to report their strategy by writing a number sentence, equations that directly model the situation were predominant for Si-problems but not for MA ones. Implications of the Situation Strategy First framework regarding the relation between conceptual and procedural knowledge and the development of arithmetic knowledge are discussed.     

Working memory components as predictors of children's mathematical word problem solving

This study determined the working memory (WM) components (executive, phonological loop, and visual–spatial sketchpad) that best predicted mathematical word problem-solving accuracy of elementary school children in Grades 2, 3, and 4 (N = 310). A battery of tests was administered to assess problem-solving accuracy, problem-solving processes, WM, reading, and math calculation. Structural equation modeling analyses indicated that (a) all three WM components significantly predicted problem-solving accuracy, (b) reading skills and calculation proficiency mediated the predictive effects of the central executive system and the phonological loop on solution accuracy, and (c) academic mediators failed to moderate the relationship between the visual–spatial sketchpad and solution accuracy. The results support the notion that all components of WM play a major role in predicting problem-solving accuracy, but basic skills acquired in specific academic domains (reading and math) can compensate for some of the influence of WM on children’s mathematical word problem solving.     

Learning to Solve Compare Word Problems: The Effect of Example Format and Generating Self-Explanations

A series of 3 experiments was conducted to examine factors that influence learning to solve 2-step arithmetic word problems by studying worked examples. Experiment 1 compared studying worked examples with conventional problem solving. Third graders presented with worked examples showed superior test performance to those required to solve conventional problems. Experiments 2 and 3 contrasted split-attention and integrated worked examples and investigated the influence of generating self-explanations. Significant split-attention effects were observed. Children presented with integrated worked examples outperformed those presented with split-source examples. Self-explanations further elucidated the distinction between integrated and split-source worked examples, but there was no significant effect of asking learners to generate self-explanations. Implications for word-problem-solving instruction are discussed.     

Working memory and phonological processing as predictors of children’s mathematical problem solving at different ages

The study explored the contribution of working memory (WM) to mathematical problem solving in younger (8-year-old) and older (11-year-old) children. The results showed that (1) significant age-related differences in WM performance were maintained when measures of phonological processing (i.e., digit naming speed, short-term memory, phonological deletion) were partialed from the analysis; (2) WM predicted solution accuracy of word problems independently of measures of problem representation, knowledge of operations and algorithms, phonological processing, fluid intelligence, reading, and math skill; and (3) a second-order WM factor was correlated with problem solving, suggesting that a general or executive system underlies age-related performance. The results were interpreted as support for the notion that the executive system was an important predictor of age-related changes in problem solving beyond the contribution of math and reading skills, and this system operates independently of the phonological system and domain-specific knowledge in predicting solution accuracy.     

Number-word sequence skill and arithmetic performance

In two studies, the role of the number‐word sequence skill for arithmetic performance was investigated. In the first, children between 4 and 8 years of age were asked to count forward and backward on the number‐word sequence and to solve arithmetic problems followed by post‐solution interviews about solution procedures. The results demonstrated that the number‐word sequence skill predicted both number of problems solved and strategy to solve the problems. In Study 2 it was found that solving doubles (e.g., 2 + 2 = ?) problems served as a link between the number‐word sequence skill and the number of arithmetic problems solved. The findings suggest that counting on the number‐word sequence may be an early solution procedure and that, with increasing counting skill, the child may detect regularities in the number‐word sequence that can be used to form new and more accurate strategies for solving arithmetic problems.     

Understanding and using principles of arithmetic: operations involving negative numbers

Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers ( Dixon, Deets, & Bangert, 2001 ). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation. Participants ( N = 44) completed two tasks. In the Evaluation task, participants rated how well sets of equations were solved. Some sets violated principles of arithmetic and others did not. Participants rated non-violation sets higher than violation sets for two different principles for subtraction with a negative number. In the Word Problem task, participants read word problems and set up equations that could be used to solve them. Participants who displayed greater knowledge of principles of arithmetic with a negative number were more likely to set up equations that involved negative numbers. Thus, participants' knowledge of arithmetic principles was related to their problem representations.     

Contributions of linguistic,quantitative, and spatial attention skills to young children's math versus reading: Same,different, or both?

The study used Bayesian and Frequentist methods to investigate whether the roles of linguistic, quantitative, and spatial attention skills are distinct in children's acquisition of reading and math. A sample of 175 Chinese kindergarteners was tested with measures of linguistic skills (phonological awareness and phonological memory), quantitative knowledge (number line task, symbolic digit comparison, and non-symbolic number estimation), spatial attention skills (visual span, mental rotation, and visual search), word reading, and calculation. After statistically controlling for age and nonverbal intelligence, phonological awareness and digit comparison performance explained unique variance in both math and reading. Moreover, number line estimation was specifically important for math, while phonological memory was specifically essential for reading. These findings highlight the possibility of developing early screening tools with different cognitive measures for children at risk of learning disabilities in reading and/or math.