Lara D. Nugent

Development of Number Line Representations in Children With Mathematical Learning Disability

Children with a mathematical learning disability (MLD, n = 19) and low achieving (LA, n = 43) children were identified using mathematics achievement scores below the 11th percentile and between the 11th and 25th percentiles, respectively. A control group of typically achieving (TA, n = 50) children was also identified. Number line and speed of processing tasks were administered in 1st and 2nd grade and a working memory battery in 1st grade. In both grades, the MLD children were less accurate in their number line placements and more reliant on a natural number-magnitude representational system to make these placements than were TA children. The TA children were more reliant on the school-taught linear system in both grades. The performance of the LA children was similar to that of the MLD children in first grade and to the TA children in second. The central executive component of working memory contributed to across-grade improvements in number line performance and to group differences in this performance.

TWO SEPARATE VERBAL PROCESSING RATES CONTRIBUTING TO SHORT-TERM MEMORY SPAN

Previous research indicates that verbal memory span, the number of words people can remember and immediately repeat, is related to the fastest rate at which they can pronounce the words. This relation, in turn, has been attributed to a general or global rate of information processing that differs among individuals and changes with age. However, the experiments described in this article showed that the rates of 2 processes (rapid articulation and the retrieval of words from short-term memory) are related to memory span but not to each other. Memory span depends on a profile of processing rates in the brain, not only a global rate. Moreover, there appears to be only a partial overlap between the rate variables that change with age and those that differ among individuals.

The Role of Attention in the Development of Short-Term Memory: Age Differences in the Verbal Span of Apprehension

In previous studies of memory span, participants have attended to the stimuli while they were presented, and therefore have had the opportunity to use a variety of mnemonic strategies. In the main portion of the present study, participants (first- and fourth-grade children, and adults; 24 per age group) carried out a visual task while hearing lists of spoken digits and received a post-list digit recall cue only occasionally, for some lists. Under these conditions, list information presumably must be extracted from a passively held store such as auditory sensory memory. The results suggest that each individual has a core memory capacity limit that can be observed clearly in circumstances in which it cannot be supplemented by mnemonic strategies, and that the capacity limit appears to increase with age during childhood. Other, attention-demanding processes also contribute to memory for attended lists.

Adolescents' functional numeracy is predicted by their school entry number system knowledge.

One in five adults in the United States is functionally innumerate; they do not possess the mathematical competencies needed for many modern jobs. We administered functional numeracy measures used in studies of young adults’ employability and wages to 180 thirteen-year-olds. The adolescents began the study in kindergarten and participated in multiple assessments of intelligence, working memory, mathematical cognition, achievement, and in-class attentive behavior. Their number system knowledge at the beginning of first grade was defined by measures that assessed knowledge of the systematic relations among Arabic numerals and skill at using this knowledge to solve arithmetic problems. Early number system knowledge predicted functional numeracy more than six years later (ß = 0.195, p = .0014) controlling for intelligence, working memory, in-class attentive behavior, mathematical achievement, demographic and other factors, but skill at using counting procedures to solve arithmetic problems did not. In all, we identified specific beginning of schooling numerical knowledge that contributes to individual differences in adolescents’ functional numeracy and demonstrated that performance on mathematical achievement tests underestimates the importance of this early knowledge.

Competence with fractions predicts gains in mathematics achievement.

Competence with fractions predicts later mathematics achievement, but the codevelopmental pattern between fractions knowledge and mathematics achievement is not well understood. We assessed this codevelopment through examination of the cross-lagged relation between a measure of conceptual knowledge of fractions and mathematics achievement in sixth and seventh grades (N=212). The cross-lagged effects indicated that performance on the sixth grade fractions concepts measure predicted 1-year gains in mathematics achievement (ß=.14, p<.01), controlling for the central executive component of working memory and intelligence, but sixth grade mathematics achievement did not predict gains on the fractions concepts measure (ß=.03, p>.50). In a follow-up assessment, we demonstrated that measures of fluency with computational fractions significantly predicted seventh grade mathematics achievement above and beyond the influence of fluency in computational whole number arithmetic, performance on number fluency and number line tasks, central executive span, and intelligence. Results provide empirical support for the hypothesis that competence with fractions underlies, in part, subsequent gains in mathematics achievement.

The role of absolute and relative amounts of time in forgetting within immediate memory: The case of tone-pitch comparisons

Many investigators of auditory sensory memory have assumed that memory loss during a retention interval of some seconds results from a process (such as decay) that depends on the absolute amount of time that has elapsed since presentation of the sound. An alternative possibility, brought to light by studies on immediate verbal memory, is that it is the relative, rather than the absolute, amount of time that matters. We examine these factors in a tone-comparison study by varying not only the retention interval between two tones to be compared, but also the interpair interval. Relative time played a role, but absolute time also appeared to be important. Several concepts of “decay” are considered in relation to the results.

There Are Two Word-Length Effects in Verbal Short-Term Memory: Opposed Effects of Duration and Complexity

In the word-length effect (WLE), lists of shorter words are better recalled than lists of longer words This effect is fundamental to decay-based theories of verbal short-term memory, such as the phonological loop theory (Baddeley, 1986) The WLE has been attributed to the time taken to articulate words, not their structure, a critical point in the debate between decay and interference theories However, word duration and complexity have previously been confounded In this article, we show that the traditional WLE comprises two opposed effects an advantage for words spoken more quickly (short words in terms of duration) and an advantage for words with more elements (long words in terms of complexity) We also report two interactions a disadvantage for a midlist change in duration and an advantage for a midlist change in complexity These results contradict simple decay-based theories and establish the importance of interference in short-term memory We discuss whether decay is also required

Independent contributions of the central executive, intelligence, and in-class attentive behavior to developmental change in the strategies used to solve addition problems.

Childrens (N=275) use of retrieval, decomposition (e.g., 7=4+3 and thus 6+7=6+4+3), and counting to solve additional problems was longitudinally assessed from first grade to fourth grade, and intelligence, working memory, and in-class attentive behavior was assessed in one or several grades. The goal was to assess the relation between capacity of the central executive component of working memory, controlling for intelligence and in-class attentive behavior, and grade-related changes in childrens use of these strategies. The predictor on intercept effects from multilevel models revealed that children with higher central executive capacity correctly retrieved more facts and used the most sophisticated counting procedure more frequently and accurately than their lower capacity peers at the beginning of first grade, but the predictor on slope effects indicated that this advantage disappeared (retrieval) or declined in importance (counting) from first grade to fourth grade. The predictor on slope effects also revealed that from first grade to fourth grade, children with higher capacity adopted the decomposition strategy more quickly than other children. The results remained robust with controls for childrens sex, race, school site, speed of encoding Arabic numerals and articulating number words, and mathematics achievement in kindergarten. The results also revealed that intelligence and in-class attentive behavior independently contributed to childrens strategy development.

Is There a Temporal Basis of the Word Length Effect? A Response to Service (1998):

Service (1998) carried out a study of the word length effect with Finnish pseudowords in which short and long pseudowords were identical except for the inclusion of certain phonemes differing only in pronunciation length, a manipulation that is impossible in English. She obtained an effect of phonemic complexity but little or no word duration effect per se — a discrepancy from the expectations generated by the well-known working memory model of Baddeley (1986). In the present study using English words, we controlled for phonemic complexity differences by using the same words for the short- and long-word sets, but with instructions inducing shorter or longer pronunciation of the words. We obtained substantial word duration effects. Concerns raised by Service are addressed, and we conclude that both duration and complexity are likely to contribute to the word length effect in serial recall.

Mathematical cognition in intellectually precocious first graders.

Forty-six intellectually precocious (M age = 74 months) and 250 intellectually typical (M age = 75 months) children were administered a standardized working memory battery, speed of processing measures, and tasks that assessed skill at number line estimation and strategies used to solve simple and complex addition problems. Precocious children had an advantage over same-age peers for all components of working memory, and used a more mature mix of strategies to solve addition problems and to make number line estimates; there were no group differences for speed of processing. Many of the advantages of the precocious children on the number line and addition strategy tasks were significantly reduced or eliminated when group differences in working memory were controlled. Individual differences analyses revealed that each of the three components of working memory contributed to different aspects of skilled performance on the mathematics tasks.