Cognitive predictors of children's development in mathematics achievement: A latent growth modeling approach.

Research has identified various domain-general and domain-specific cognitive abilities as predictors of children's individual differences in mathematics achievement. However, research into the predictors of children's individual growth rates, namely between-person differences in within-person change in mathematics achievement is scarce. We assessed 334 children's domain-general and mathematics-specific early cognitive abilities and their general mathematics achievement longitudinally across four time-points within the first and second grades of primary school. As expected, a constellation of multiple cognitive abilities contributed to the children's starting level of mathematical success. Specifically, latent growth modeling revealed that WM abilities, IQ, counting skills, nonsymbolic and symbolic approximate arithmetic and comparison skills explained individual differences in the children's initial status on a curriculum-based general mathematics achievement test. Surprisingly, however, only one out of all the assessed cognitive abilities was a unique predictor of the children's individual growth rates in mathematics achievement: their performance in the symbolic approximate addition task. In this task, children were asked to estimate the sum of two large numbers and decide if this estimated sum was smaller or larger compared to a third number. Our findings demonstrate the importance of multiple domain-general and mathematics-specific cognitive skills for identifying children at risk of struggling with mathematics and highlight the significance of early approximate arithmetic skills for the development of one's mathematical success. We argue the need for more research focus on explaining children's individual growth rates in mathematics achievement.

The developmental onset of symbolic approximation: beyond nonsymbolic representations, the language of numbers matters.

Symbolic (i.e., with Arabic numerals) approximate arithmetic with large numerosities is an important predictor of mathematics. It was previously evidenced to onset before formal schooling at the kindergarten age (Gilmore et al., 2007) and was assumed to map onto pre-existing nonsymbolic (i.e., abstract magnitudes) representations. With a longitudinal study (Experiment 1), we show, for the first time, that nonsymbolic and symbolic arithmetic demonstrate different developmental trajectories. In contrast to Gilmore et al.'s (2007) findings, Experiment 1 showed that symbolic arithmetic onsets in grade 1, with the start of formal schooling, not earlier. Gilmore et al. (2007) had examined English-speaking children, whereas we assessed a large Dutch-speaking sample. The Dutch language for numbers can be cognitively more demanding, for example, due to the inversion property in numbers above 20. Thus, for instance, the number 48 is named in Dutch "achtenveertig" (eight and forty) instead of "forty eight." To examine the effect of the language of numbers, we conducted a cross-cultural study with English- and Dutch-speaking children that had similar SES and math achievement skills (Experiment 2). Results demonstrated that Dutch-speaking kindergarteners lagged behind English-speaking children in symbolic arithmetic, not nonsymbolic and demonstrated a working memory overload in symbolic arithmetic, not nonsymbolic. Also, we show for the first time that the ability to name two-digit numbers highly correlates with symbolic approximate arithmetic not nonsymbolic. Our experiments empirically demonstrate that the symbolic number system is modulated more by development and education than the nonsymbolic system. Also, in contrast to the nonsymbolic system, the symbolic system is modulated by language.

Longitudinal development of number line estimation and mathematics performance in primary school children.

Children's ability to relate number to a continuous quantity abstraction visualized as a number line is widely accepted to be predictive of mathematics achievement. However, a debate has emerged with respect to how children's placements are distributed on this number line across development. In the current study, different models were applied to children's longitudinal number placement data to get more insight into the development of number line representations in kindergarten and early primary school years. In addition, longitudinal developmental relations between number line placements and mathematical achievement, measured with a national test of mathematics, were investigated using cross-lagged panel modeling. A group of 442 children participated in a 3-year longitudinal study (ages 5-8 years) in which they completed a number-to-position task every 6 months. Individual number line placements were fitted to various models, of which a one-anchor power model provided the best fit for many of the placements at a younger age (5 or 6 years) and a two-anchor power model provided better fit for many of the children at an older age (7 or 8 years). The number of children who made linear placements also grew with age. Cross-lagged panel analyses indicated that the best fit was provided with a model in which number line acuity and mathematics performance were mutually predictive of each other rather than models in which one ability predicted the other in a non-reciprocal way. This indicates that number line acuity should not be seen as a predictor of math but that both skills influence each other during the developmental process.

Working memory and number line representations in single-digit addition: Approximate versus exact, nonsymbolic versus symbolic.

How do kindergarteners solve different single-digit addition problem formats? We administered problems that differed solely on the basis of two dimensions: response type (approximate or exact), and stimulus type (nonsymbolic, i.e., dots, or symbolic, i.e., Arabic numbers). We examined how performance differs across these dimensions, and which cognitive mechanism (mental model, transcoding, or phonological storage) underlies performance in each problem format with respect to working memory (WM) resources and mental number line representations. As expected, nonsymbolic problem formats were easier than symbolic ones. The visuospatial sketchpad was the primary predictor of nonsymbolic addition. Symbolic problem formats were harder because they either required the storage and manipulation of quantitative symbols phonologically or taxed more WM resources than their nonsymbolic counterparts. In symbolic addition, WM and mental number line results showed that when an approximate response was needed, children transcoded the information to the nonsymbolic code. When an exact response was needed, however, they phonologically stored numerical information in the symbolic code. Lastly, we found that more accurate symbolic mental number line representations were related to better performance in exact addition problem formats, not the approximate ones. This study extends our understanding of the cognitive processes underlying children's simple addition skills.

Working memory in nonsymbolic approximate arithmetic processing: a dual-task study with preschoolers.

Preschool children have been proven to possess nonsymbolic approximate arithmetic skills before learning how to manipulate symbolic math and thus before any formal math instruction. It has been assumed that nonsymbolic approximate math tasks necessitate the allocation of Working Memory (WM) resources. WM has been consistently shown to be an important predictor of children's math development and achievement. The aim of our study was to uncover the specific role of WM in nonsymbolic approximate math. For this purpose, we conducted a dual-task study with preschoolers with active phonological, visual, spatial, and central executive interference during the completion of a nonsymbolic approximate addition dot task. With regard to the role of WM, we found a clear performance breakdown in the central executive interference condition. Our findings provide insight into the underlying cognitive processes involved in storing and manipulating nonsymbolic approximate numerosities during early arithmetic.

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.

How do children deal with inconsistencies in text? An eye fixation and self-paced reading study in good and poor reading comprehenders.

In two experiments, we investigated comprehension monitoring in 10-12 years old children differing in reading comprehension skill. The children's self-paced reading times (Experiment 1) and eye fixations and regressions (Experiment 2) were measured as they read narrative texts in which an action of the protagonist was consistent or inconsistent with a description of the protagonist's character given earlier. The character description and action were adjacent (local condition) or separated by a long filler paragraph (global condition). The self-paced reading data (Experiment 1), the initial reading and rereading data (Experiment 2), together with the comprehension question data (both experiments), are discussed within the situation model framework and suggest that poor comprehenders find difficulty in constructing a richly elaborated situation model. Poor comprehenders presumably fail to represent character information in the model as a consequence of which they are not able to detect inconsistencies in the global condition (in which the character information is lost from working memory). The patterns of results rule out an explanation in terms of impaired situation model updating ability.

Quality of arithmetic education for children with cerebral palsy.

The aim of this exploratory study was to investigate the quality of arithmetic education for children with cerebral palsy. The use of individual educational plans, amount of arithmetic instruction time, arithmetic instructional grouping, and type of arithmetic teaching method were explored in three groups: children with cerebral palsy (CP) in special (CP-special; n = 41) and mainstream schools (CP-mainstream; n = 16) and a control group in mainstream schools (n = 16). The majority of individual educational plans did not include well-formulated arithmetic goals and many were not based on optimal assessment. Special schools scheduled much less arithmetic instruction time. Many CP-mainstream children received individualized instruction, which may help to explain why their arithmetic performance did not differ from controls. Remedial arithmetic teaching methods used in special schools did not seem to be optimal, but more research is required. Suggestions to improve arithmetic education to children with CP were discussed.

Arithmetic difficulties in children with cerebral palsy are related to executive function and working memory.

BACKGROUND: Although it is believed that children with cerebral palsy are at high risk for learning difficulties and arithmetic difficulties in particular, few studies have investigated this issue. METHODS: Arithmetic ability was longitudinally assessed in children with cerebral palsy in special (n = 41) and mainstream education (n = 16) and controls in mainstream education (n = 16). Second grade executive function and working memory scores were used to predict third grade arithmetic accuracy and response time. RESULTS: Children with cerebral palsy in special education were less accurate and slower than their peers on all arithmetic tests, even after controlling for IQ, whereas children with cerebral palsy in mainstream education performed as well as controls. Although the performance gap became smaller over time, it did not disappear. Children with cerebral palsy in special education showed evidence of executive function and working memory deficits in shifting, updating, visuospatial sketchpad and phonological loop (for digits, not words) whereas children with cerebral palsy in mainstream education only had a deficit in visuospatial sketchpad. Hierarchical regression revealed that, after controlling for intelligence, components of executive function and working memory explained large proportions of unique variance in arithmetic accuracy and response time and these variables were sufficient to explain group differences in simple, but not complex, arithmetic. CONCLUSIONS: Children with cerebral palsy are at risk for specific executive function and working memory deficits that, when present, increase the risk for arithmetic difficulties in these children.

The relationship between medical impairments and arithmetic development in children with cerebral palsy.

Arithmetic ability was tested in children with cerebral palsy without severe intellectual impairment (verbal IQ >or= 70) attending special (n = 41) or mainstream education (n = 16) as well as control children in mainstream education (n = 16) throughout first and second grade. Children with cerebral palsy in special education did not appear to have fully automatized arithmetic facts by the end of second grade. Their lower accuracy and consistently slower (verbal) response times raise important concerns for their future arithmetic development. Differences in arithmetic performance between children with cerebral palsy in special or mainstream education were not related to localization of cerebral palsy or to gross motor impairment. Rather, lower accuracy and slower verbal responses were related to differences in nonverbal intelligence and the presence of epilepsy. Left-hand impairment was related to slower verbal responses but not to lower accuracy.

Prevalence of combined reading and arithmetic disabilities.

This study assesses the prevalence of combined reading and arithmetic disabilities in 799 Dutch schoolchildren using st dardized school achievement tests. Scores of arithmetic, word recognition, reading comprehension, and spelling of child in fourth and fifth grade were used. The main interest involved the co-occurrence of word recognition and arithmetic d abilities because of their possible relationship. The authors find a percentage of 7.6 for combined reading and arithme disabilities. Reading disabilities and arithmetic disabilities co-occurred more often than expected based on rates of the s arate conditions. Children with combined reading and arithmetic disabilities seem to have more generalized achievem difficulties than single-deficit groups. Different operationalizations for reading disabilities (spelling and reading comp hension measures instead of word recognition) led, in part, to selection of other children. This might imply that differ processes underlie the relationship between arithmetic and word recognition disabilities compared to the relationship arithmetic disabilities with difficulties in spelling and reading comprehension.

The effect of cerebral palsy on arithmetic accuracy is mediated by working memory, intelligence, early numeracy, and instruction time.

The development of addition and subtraction accuracy was assessed in first graders with cerebral palsy (CP) in both mainstream (16) and special education (41) and a control group of first graders in mainstream education (16). The control group out-performed the CP groups in addition and subtraction accuracy and this difference could not be fully explained by differences in intelligence. Both CP groups showed evidence of working memory deficits. The three groups exhibited different developmental patterns in the area of early numeracy skills. Children with CP in special education were found to receive less arithmetic instruction and instruction time was positively related to arithmetic accuracy. Structural equation modeling revealed that the effect of CP on arithmetic accuracy is mediated by intelligence, working memory, early numeracy, and instruction time.