The influence of problem features and individual differences on strategic performance in simple arithmetic

Proactive Control Mediates the Relationship Between Working Memory and Math Ability in Early Childhood

Based on the dual mechanisms of control (DMC) theory, there are two distinct mechanisms of cognitive control, proactive and reactive control. Importantly, accumulating evidence indicates that there is a developmental shift from predominantly using reactive control to proactive control during childhood, and the engagement of proactive control emerges as early as 5–7 years old. However, less is known about whether and how proactive control at this early age stage is associated with children’s other cognitive abilities such as working memory and math ability. To address this issue, the current study recruited 98 Chinese children under 5–7 years old. Among them, a total of 81 children (mean age = 6.29 years) contributed useable data for the assessments of cognitive control, working memory, and math ability. The results revealed that children at this age period predominantly employed a pattern of proactive control during an AX-Continuous Performance Task (AX-CPT). Moreover, the proactive control index estimated by this task was positively associated with both working memory and math performance. Further regression analysis showed that proactive control accounted for significant additional variance in predicting math performance after controlling for working memory. Most interestingly, mediation analysis showed that proactive control significantly mediated the association between working memory and math performance. This suggests that as working memory increases so does proactive control, which may in turn improve math ability in early childhood. Our findings may have important implications for educational practice.

Dissociating Arithmetic Operations in the Parietal Cortex Using 1 Hz Repetitive Transcranial Magnetic Stimulation: The Importance of Strategy Use

The triple-code model (TCM) of number processing suggests the involvement of distinct parietal cortex areas in arithmetic operations: the bilateral horizontal segment of the intraparietal sulcus (hIPS) for arithmetic operations that require the manipulation of numerical quantities (e.g., subtraction) and the left angular gyrus (AG) for arithmetic operations that require the retrieval of answers from long-term memory (e.g., multiplication). Although neuropsychological, neuroimaging, and brain stimulation studies suggest the dissociation of these operations into distinct parietal cortex areas, the role of strategy (online calculation vs. retrieval) is not yet fully established. In the present study, we further explored the causal involvement of the left AG for multiplication and left hIPS for subtraction using a neuronavigated repetitive transcranial magnetic stimulation (rTMS) paradigm. Stimulation sites were determined based on an fMRI experiment using the same tasks. To account for the effect of strategy, participants were asked whether they used retrieval or calculation for each individual problem. We predicted that the stimulation of the left AG would selectively disrupt the retrieval of the solution to multiplication problems. On the other hand, stimulation of the left hIPS should selectively disrupt subtraction. Our results revealed that left AG stimulation was detrimental to the retrieval and online calculation of solutions for multiplication problems, as well as, the retrieval (but not online calculation) of the solutions to subtraction problems. In contrast, left hIPS stimulation had no detrimental effect on both operations regardless of strategy.

The Effect of Central Executive Load on Fourth and Sixth Graders' Use of Arithmetic Strategies

In the present study, we set out to investigate whether and how central executive load constrains the strategies that children use during arithmetic processing. Using a dual-task paradigm accompanied by the choice/no-choice method, we tested 233 children (115 6^th graders, 118 4^th graders). Results showed that the impact of central executive load on reaction times and accuracy scores related to strategy use increased with the magnitude of the demands of the central executive, with central executive load playing an important role in strategy use. Sixth graders performed better than 4^th graders in the application of appropriate strategies. Children's adaptability with respect to strategy choice was affected by the type and magnitude of the central executive load; children showed better adaptability under the no-load condition and the inconsistent/low load condition than under conditions with greater load. Grade level affected children's adaptability with respect to strategy choice, with 6^th graders exhibiting significantly better performance than 4^th graders. These results confirm that the development of central executive skills contributes to children's overall strategy use and adaptability. These findings have important implications for understanding the category specificity of central executive working memory in arithmetic cognition and the mechanisms of strategy development in childhood.

Individual strategy ratings improve the control for task difficulty effects in arithmetic problem solving paradigms

Mental arithmetic is a powerful paradigm to study problem solving using neuroimaging methods. However, the evaluation of task complexity varies significantly across neuroimaging studies. Most studies have parameterized task complexity by objective features such as the number size. Only a few studies used subjective rating procedures. In fMRI, we provided evidence that strategy self-reports control better for task complexity across arithmetic conditions than objective features (Tschentscher and Hauk, 2014). Here, we analyzed the relative predictive value of self-reported strategies and objective features for performance in addition and multiplication tasks, by using a paradigm designed for neuroimaging research. We found a superiority of strategy ratings as predictor of performance above objective features. In a Principal Component Analysis on reaction times, the first component explained over 90 percent of variance and factor loadings reflected percentages of self-reported strategies well. In multiple regression analyses on reaction times, self-reported strategies performed equally well or better than objective features, depending on the operation type. A Receiver Operating Characteristic (ROC) analysis confirmed this result. Reaction times classified task complexity better when defined by individual ratings. This suggests that participants' strategy ratings are reliable predictors of arithmetic complexity and should be taken into account in neuroimaging research.

Efficiency of Arithmetic Procedures Modulates the Problem-Size Effect in Subtraction

The aim of this study was to examine whether differences in strategy selection and/or strategy efficiency can explain the modulation of the problem-size effect by arithmetic skill. More specifically, we wondered whether arithmetic skill increases the use of retrieval strategy in large problems, and/or enhances the efficiency of either retrieval or procedural strategies. The performance of highly-skilled (HS) and less highly-skilled (LS) individuals on a subtraction verification task was analyzed according to problem size and to the strategy reported on a trial-by-trial basis after each problem. The problem size effect was larger for LS individuals than for their HS peers, both in response time and in hit rate. Nevertheless, groups did not differ regarding the strategy reported for each subtraction size. As expected, problems in which retrieval strategy was reported were solved more quickly and more accurately than problems solved by procedural strategies. Responses using retrieval strategy were equally fast in the two groups, but HS individuals performed better than LS when using procedural strategies. The results therefore suggest that the differences in behavioral measures between groups might specifically be due to differences in the efficiency of procedural strategies.

The arithmetic problem size effect in children: an event-related potential study

This study used for the first time event-related potentials (ERPs) to examine the well-known arithmetic problem size effect in children. The electrophysiological correlates of this problem size effect have been well documented in adults, but such information in children is lacking. In the present study, 22 typically developing 12-year-olds were asked to solve single-digit addition problems of small (sum ≤ 10) and large problem size (sum > 10) and to speak the solution into a voice key while ERPs were recorded. Children displayed similar early and late components compared to previous adult studies on the problem size effect. There was no effect of problem size on the early components P1, N1, and P2. The peak amplitude of the N2 component showed more negative potentials on left and right anterior electrodes for large additions compared to small additions, which might reflect differences in attentional and working memory resources between large and small problems. The mean amplitude of the late positivity component which follows the N2, was significantly larger for large than for small additions at right parieto-occipital electrodes, in line with previous adult data. The ERPs of the problem size effect during arithmetic might be a useful neural marker for future studies on fact retrieval impairments in children with mathematical difficulties.

The roles of the central executive and visuospatial storage in mental arithmetic: a comparison across strategies

Previous research has demonstrated that working memory plays an important role in arithmetic. Different arithmetical strategies rely on working memory to different extents—for example, verbal working memory has been found to be more important for procedural strategies, such as counting and decomposition, than for retrieval strategies. Surprisingly, given the close connection between spatial and mathematical skills, the role of visuospatial working memory has received less attention and is poorly understood. This study used a dual-task methodology to investigate the impact of a dynamic spatial n-back task (Experiment 1) and tasks loading the visuospatial sketchpad and central executive (Experiment 2) on adults’ use of counting, decomposition, and direct retrieval strategies for addition. While Experiment 1 suggested that visuospatial working memory plays an important role in arithmetic, especially when counting, the results of Experiment 2 suggested this was primarily due to the domain-general executive demands of the n-back task. Taken together, these results suggest that maintaining visuospatial information in mind is required when adults solve addition arithmetic problems by any strategy but the role of domain-general executive resources is much greater than that of the visuospatial sketchpad.

Early place-value understanding as a precursor for later arithmetic performance--a longitudinal study on numerical development

It is assumed that basic numerical competencies are important building blocks for more complex arithmetic skills. The current study aimed at evaluating this interrelation in a longitudinal approach. It was investigated whether first graders' performance in basic numerical tasks in general as well as specific processes involved (e.g., place-value understanding) reliably predicted performance in an addition task in third grade. The results indicated that early place-value understanding was a reliable predictor for specific aspects of arithmetic performance. Implications of the role of basic numerical competencies for the acquisition of complex arithmetic are discussed.

Switch costs and the operand-recognition paradigm

Experimental research in cognitive arithmetic frequently relies on participants' self-reports to discriminate solutions based on direct memory retrieval from use of procedural strategies. Given concerns about the validity and reliability of strategy reports, Thevenot et al. in Mem Cogn 35:1344-1352, (2007) developed the operand-recognition paradigm as an objective measure of arithmetic strategies. Participants performed addition or number comparison on two sequentially presented operands followed by a speeded operand-recognition task. Recognition times increased with problem size following addition but not comparison. Thevenot et al. argued that the complexity of addition strategies increases with problem size. A corresponding increase in operand-recognition time occurs because, as problem size increases, working memory contains more numerical distracters. However, because addition is substantially more difficult than comparison, and difficulty increases with problem size for addition but not comparison, their findings could be due to difficulty-related task-switching costs. We repeated Thevenot et al. (Experiment 1) but added a control condition wherein participants performed a parity (odd or even) task instead of operand recognition. We replicated their findings for operand recognition but found robust, albeit smaller, effects of addition problem size on parity judgements. The results indicate that effects of strategy complexity in the operand-recognition paradigm are confounded with task-switching effects, which complicates its application as a precise measure of strategy complexity in arithmetic.

Practice effects on strategy selection and strategy efficiency in simple mental arithmetic

Two experiments were conducted to investigate the effects of practice on strategy selection and strategy efficiency in mental arithmetic. Participants had to solve simple addition or multiplication problems, after having received 0, 3, or 6 practice sessions (Experiment 1), and before and after having received 3 practice sessions (Experiment 2). Strategy selection was measured by means of trial-by-trial strategy reports, whereas strategy efficiency was measured by means of response latencies. Results showed significant practice effects on retrieval frequency, procedural frequency, retrieval efficiency, and procedural efficiency. However, practice effects on strategy efficiency appeared to be both strategy-specific (i.e., only for procedural strategies) and operation-specific (i.e., only for multiplication problems). Implications of the present results for mathematic cognition and its modeling are discussed.

Effects of problem size, operation, and working-memory span on simple-arithmetic strategies: differences between children and adults?

Adult's simple-arithmetic strategy use depends on problem-related characteristics, such as problem size and operation, and on individual-difference variables, such as working-memory span. The current study investigates (a) whether the effects of problem size, operation, and working-memory span on children's simple-arithmetic strategy use are equal to those observed in adults, and (b) how these effects emerge and change across age. To this end, simple-arithmetic performance measures and a working-memory span measure were obtained from 8-year-old, 10-year-old, and 12-year-old children. Results showed that the problem-size effect in children results from the same strategic performance differences as in adults (i.e., size-related differences in strategy selection, retrieval efficiency, and procedural efficiency). Operation-related effects in children were equal to those observed in adults as well, with more frequent retrieval use on multiplication, more efficient strategy execution in addition, and more pronounced changes in multiplication. Finally, the advantage of having a large working-memory span was also present in children. The differences and similarities across children's and adult's strategic performance and the relevance of arithmetic models are discussed.