A chronometric study of mental addition in profoundly deaf children

In attempting to account for inferior mathematical attainment among deaf children of all ages, some researchers have proposed that observed deficits may be partly the result of an absence of vocalization and subvocalization in the acquisition and execution of arithmetic by the deaf. When performing mental arithmetic, hearing children, it is claimed, rely on covert counting mechanisms based on internalized speech, while their deaf counterparts, due to a lack of articulatory prowess, are unable to utilize such mechanisms and instead are forced to adopt other less effective methods based on associative retrieval of arithmetical facts from long-term memory. The present study sought to throw light on these assertions. Ten prelingually profoundly deaf 12-13-year-olds, and 10 hearing controls were required to solve the 100 simple addition combinations. Analysis of response times revealed strong similarities between the two groups, with all children appearing to employ mechanisms based on covert counting. Implications of the present results are discussed in the light of previous findings.     

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

Children's (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 children's 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 children's 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 children's strategy development.     

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.     

Can we predict mathematical learning disabilities from symbolic and non‐symbolic comparison tasks in kindergarten? Findings from a longitudinal study

Background. The ability to compare numbers, as the most basic form of number sense, has been related to arithmetical achievement. Aims. The current study addressed the predictive value of non‐symbolic and symbolic (number word (NW) and Arabic number (AN)) comparison for arithmetics by means of a longitudinal design. Sample. Sixteen children with mathematical disabilities (MD), 64 low achievers (LA), and 315 typical achieving (TA) children were followed from kindergarten till grade 2. Method. The association of comparison skills with arithmetical skills in grades l and 2 was studied. The performances of MD, LA and TA children were compared. Results. Regression analyses showed that non‐symbolic skills in kindergarten were predictively related to arithmetical achievement 1 year later and fact retrieval 2 years later. AN comparison was predictively related to procedural calculation 2 years later. In grade 2, there was an association between both symbolic tasks and arithmetical achievement. Children with MD already had deficits in non‐symbolic and symbolic AN comparison in kindergarten, whereas in grade 2 the deficits in processing symbolic information remained. Conclusions. The combination of non‐symbolic and symbolic deficits represents a risk of developing MD.     

Speech timing and working memory in profoundly deaf children after cochlear implantation

Thirty-seven profoundly deaf children between 8- and 9-years-old with cochlear implants and a comparison group of normal-hearing children were studied to measure speaking rates, digit spans, and speech timing during digit span recall. The deaf children displayed longer sentence durations and pauses during recall and shorter digit spans compared to the normal-hearing children. Articulation rates, measured from sentence durations, were strongly correlated with immediate memory span in both normal-hearing and deaf children, indicating that both slower subvocal rehearsal and scanning processes may be factors that contribute to the deaf children's shorter digit spans. These findings demonstrate that subvocal verbal rehearsal speed and memory scanning processes are not only dependent on chronological age as suggested in earlier research by. Instead, in this clinical population the absence of early auditory experience and phonological processing activities before implantation appears to produce measurable effects on the working memory processes that rely on verbal rehearsal and serial scanning of phonological information in short-term memory.     

How does a child solve 7 + 8? Decoding brain activity patterns associated with counting and retrieval strategies

Cognitive development and learning are characterized by diminished reliance on effortful procedures and increased use of memory-based problem solving. Here we identify the neural correlates of this strategy shift in 7-9-year-old children at an important developmental period for arithmetic skill acquisition. Univariate and multivariate approaches were used to contrast brain responses between two groups of children who relied primarily on either retrieval or procedural counting strategies. Children who used retrieval strategies showed greater responses in the left ventrolateral prefrontal cortex; notably, this was the only brain region which showed univariate differences in signal intensity between the two groups. In contrast, multivariate analysis revealed distinct multivoxel activity patterns in bilateral hippocampus, posterior parietal cortex and left ventrolateral prefrontal cortex regions between the two groups. These results demonstrate that retrieval and counting strategies during early learning are characterized by distinct patterns of activity in a distributed network of brain regions involved in arithmetic problem solving and controlled retrieval of arithmetic facts. Our findings suggest that the reorganization and refinement of neural activity patterns in multiple brain regions plays a dominant role in the transition to memory-based arithmetic problem solving. Our findings further demonstrate how multivariate approaches can provide novel insights into fine-scale developmental changes in the brain. More generally, our study illustrates how brain imaging and developmental research can be integrated to investigate fundamental aspects of neurocognitive development.     

I recognize your face, but I can't remember your name: A question of expertise?

In four experiments on the identification of familiar faces we reassessed a robust performance pattern—namely, the temporal advantage for retrieving biographical facts as compared to recalling proper names, which has been interpreted as reflecting a serial ordering of the access to semantic and name information. Evidence for recent parallel accounts had been provided by Scanlan and Johnston (1997) who reported an advantage for name retrieval in children. Here we replicated the findings of Scanlan and Johnston but also showed that the naming advantage disappears, and performance is very similar to that of adults when stimuli and tasks are used that are familiar to children. Conversely, we also demonstrated an advantage for name retrieval in adults when highly unfamiliar semantic facts were associated with the faces. Together these findings suggest that there is no fundamental difference in the cognitive architectures of children and adults. The experiments indicate that the relative speed of naming and semantic fact retrieval depends on the expertise with the semantic facts to be retrieved. Implications for models of face identification and naming are discussed.     

When the mental number line involves a delay: the writing of numbers by children of different arithmetical abilities

Our study focused on number transcoding in children. It investigated how 9-year-olds with and without arithmetical disabilities wrote Arabic digits after they had heard them as number words. Planning time before writing each digit was registered. Analyses revealed that the two groups differed not only in arithmetical abilities but also in verbal and reading abilities. Children with arithmetical disabilities were overall slower in planning Arabic digits than were control children with normal arithmetical abilities. In addition, they showed a number size effect for numbers smaller than 10, suggesting a semantically mediated route in number processing. Control children did not need more planning time for large numbers (e.g., 8) than for small numbers (e.g., 3), suggesting a direct nonsemantic route. For both two- and three-digit numbers, both groups of children showed a number size effect, although the effect was smaller each time for control children. The presence of the stronger number size effect for children with arithmetical disabilities was seen as a delay in the development of quick and direct transcoding. The relation between transcoding problems and arithmetical disabilities is discussed. A defect in the linking of numerical symbols to analog numerical representations is proposed as an explanation for the transcoding problems found in some children.     

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.     

Errors in children's subtraction

Many of the errors that occur in children' subtraction are due to the use of incorrect strategies rather than to the incorrect recall of number facts. A production system is presented for performing written subtraction which is consistent with an earlier analysis of the nature of such a cognitive skill. Most of the incorrect strategies used by schoolchildren can be accounted for in a principled way by simple changes in the production system, such as the omission of individual rules or the inclusion of rules appropriate to other arithmetical tasks. The production system model is evaluated against a corpus of over 1500 subtraction problems done by 10-year olds and is shown to account for about two-thirds of the (nonnumber fact) errors. It also provides an alternative, simpler interpretation of the subtraction errors analysed by Brown and Burton (1978). Some implications for teaching are discussed.     

Working memory as a predictor of written arithmetical skills in children: The importance of central executive functions

Background . The study was conducted in an attempt to further our understanding of how working memory contributes to written arithmetical skills in children. Aim . The aim was to pinpoint the contribution of different central executive functions and to examine the contribution of the two subcomponents of children's written arithmetical skills. Sample and method . A total of 141 third‐ and fourth‐graders were administered arithmetical tasks and measures of working memory, fluid IQ and reading. Regression analysis was used to examine the relationship between working memory and written arithmetical skills. Results . Three central executive measures (counting span, trail making and verbal fluency) and one phonological loop measure (Digit Span) were significant and predictors of arithmetical performance when the influence of reading, age and IQ was controlled for in the analysis. Conclusions . The present findings demonstrate that working memory, in general, and the central executive, in particular, contribute to children's arithmetical skills. It was hypothesized that monitoring and coordinating multiple processes, and accessing arithmetical knowledge from long‐term memory, are important central executive functions during arithmetical performance. The contribution of the phonological loop and the central executive (concurrent processing and storage of numerical information) indicates that children aged 9–10 years primarily utilize verbal coding strategies during written arithmetical performance.     

Operational efficiency and the growth of short-term memory span

Four studies are reported which link developmental increases in memory span with developmental increases in operational speed or efficiency. In the first, a linear relationship is demonstrated between increases in word span and increases in speed of word repetition. In the second, adults and 6-year-olds are equated on speed of word repetition, by manipulating word familiarity. It is shown that their word spans are no longer different under these conditions. Similar findings are then reported for a test of M space called “Counting Span.” First, a linear relationship is demonstrated between increases in Counting Span and increases in speed of counting. Next, adults and 6-year-olds are equated on speed of counting, by forcing adults to count in an unfamiliar language. It is shown that their counting spans are no longer different under these conditions. The conclusion is that developmental increases in memory span do not result from increases in total processing space. Rather, with development, basic operations become faster and more efficient. This means that they require less processing space, and that more space becomes available for storage as a result.     

The development of strategy use in elementary school children: working memory and individual differences

The current study tested the development of working memory involvement in children's arithmetic strategy selection and strategy efficiency. To this end, an experiment in which the dual-task method and the choice/no-choice method were combined was administered to 10- to 12-year-olds. Working memory was needed in retrieval, transformation, and counting strategies, but the ratio between available working memory resources and arithmetic task demands changed across development. More frequent retrieval use, more efficient memory retrieval, and more efficient counting processes reduced the working memory requirements. Strategy efficiency and strategy selection were also modified by individual differences such as processing speed, arithmetic skill, gender, and math anxiety. Short-term memory capacity, in contrast, was not related to children's strategy selection or strategy efficiency.     

Strategic development in exact calculation: group and individual differences in four achievement subtypes

This longitudinal study sought to identify developmental changes in strategy use between 5 and 7 years of age when solving exact calculation problems. Four mathematics and reading achievement subtypes were examined at four time points. Five strategies were considered: finger counting, verbal counting, delayed retrieval, automatic retrieval, and derived fact retrieval. Results provided unique insights into children's strategic development in exact calculation at this early stage. Group analysis revealed relationships between mathematical and/or reading difficulties and strategy choice, shift, and adaptiveness. Use of derived fact retrieval by 7 years of age distinguished children with mathematical difficulties from other achievement subtypes. Analysis of individual differences revealed marked heterogeneity within all subtypes, suggesting (inter alia) no marked qualitative distinction between our two mathematical difficulty subtypes.     

The production and verification tasks in mental addition: An empirical comparison

We respond to A. Baroody's comment (1984, Developmental Review, 4, 148–156) with an empirical comparison of the production and verification tasks. With the exception of performance at the first grade level, the two tasks yield essentially identical conclusions. The results of an adjunct task, in which the rate of mental counting was assessed, suggest that children as young as second grade are relying on memory retrieval to a significant degree. In contrast to Baroody's speculation, there appear to be no widespread difficulties associated with results from the verification task. Furthermore, the task permits a more analytic examination of performance and underlying mental process than is afforded by the production task. We conclude by reiterating the empirical support for a model of fact retrieval, and suggesting that accessibility of the arithmetic facts is the basic factor which underlies both fact retrieval and procedural knowledge performance.     

A microgenetic study of simple division

How simple division strategies develop over a short period of time was examined with a microgenetic study. Grade 5 students (mean age = 10 years, 3 months) solved simple division problems in 8 weekly sessions. Performance improved with faster and more accurate responses across the study. Consistent with R. S. Siegler's (1996) overlapping waves model, strategies varied in their use. Direct retrieval increased, retrieval of multiplication facts remained steady, and addition facts, derived facts, and special tricks marginally decreased. Consistent with previous research, multiplication fact retrieval was the most common strategy, although it was slower and more error prone than direct retrieval. Strategy variability within and across individuals was striking across all of the sessions and underscores Siegler's (1996) assertion that development is in a constant transitional state. (PsycINFO Database Record (c) 2008 APA, all rights reserved).     

The codevelopment of skill at and preference for use of retrieval-based processes for solving addition problems: individual and sex differences from first to sixth grades

The ability to retrieve basic arithmetic facts from long-term memory contributes to individual and perhaps sex differences in mathematics achievement. The current study tracked the codevelopment of preference for using retrieval over other strategies to solve single-digit addition problems, independent of accuracy, and skilled use of retrieval (i.e., accuracy and reaction time [RT]) from first to sixth grades inclusive (N=311). Accurate retrieval in first grade was related to working memory capacity and intelligence, and it predicted a preference for retrieval in second grade. In later grades, the relation between skill and preference changed such that preference in one grade predicted accuracy and RT in the next grade as RT and accuracy continued to predict future gains in preference. In comparison with girls, boys had a consistent preference for retrieval over other strategies and had faster retrieval speeds, but the sex difference in retrieval accuracy varied across grades. Results indicate that ability influences early skilled retrieval, but both practice and skill influence each other in a feedback loop later in development and provide insights into the source of the sex difference in problem-solving approaches.     

Spatial congruency between stimulus presentation and response key arrangements in arithmetic fact retrieval

It is known that number and space representations are connected to one another in numerical and arithmetic abilities. Numbers are represented using the metaphor of a mental number line, oriented along horizontal and vertical space. This number line also seems to be linked to mental arithmetic, which is based partly on arithmetic fact retrieval. It seems that number representation and mental arithmetic are linked together. The present study tested the effect of spatial contextual congruency between stimulus presentation and response key arrangements in arithmetic fact retrieval, using number-matching and addition verification tasks. For both tasks in Experiment 1, a contextual congruency effect was present horizontally (i.e., horizontal presentation of stimuli and horizontal response key alignments) but not vertically (i.e., vertical presentation of stimuli but horizontal response key alignments). In Experiment 2, both tasks showed a contextual congruency effect for both spatial conditions. Experiment 1 showed that the interference and distance effects were found in the horizontal condition, probably because of the spatial congruency between stimulus presentation and response key arrangements. This spatial congruency could be related to the activation of the horizontal number line. Experiment 2 showed similar interference and distance effects for both spatial conditions, suggesting that the congruency between stimulus presentation and response alignment could facilitate the retrieval of arithmetic facts. This facilitation could be related to the activation of both horizontal and vertical number lines. The results are discussed in light of the possible role of a mental number line in arithmetic fact retrieval.     

Components of verbal learning in children: Analysis by selective reminding

Free recall verbal learning by 5- and 8-year-old children was analyzed by selectively reminding them only of items not recalled on the preceding trial (instead of continuing to present the entire list before each recall trial) to show learning by retrieval from long-term storage without presentation. Concurrent analysis of long-term storage, consistent and random retrieval from long-term storage, and recall from short-term storage indicates that, while 5-year-olds showed slower acquisition than 8-year-olds, lower recall by 5-year-olds also was due to less effective retrieval from longterm storage. Repeated retrieval, without any further presentation after an item has been recalled just once, indicates that lower recall by 9-year-old children than by adults also reflects retrieval difficulty, since these children showed storage and retention of almost as many items as adults by eventual spontaneous retrieval without further presentation.