Book reviews

Results from past studies show that the central executive, as conceptualised in the working memory model of Baddeley and Hitch (), plays an important role in simple mental arithmetic. According to this model the central executive is viewed as a unitary system. Recent research, however, suggests that the “central executive” can be fractionated in more autonomous executive functions, such as inhibition, response selection, planning, and input monitoring. In four experiments we studied the role of two possible executive functions (input monitoring and response selection) in solving simple sums. Subjects solved one‐digit sums (e.g., 5 + 7) in a single‐task condition as well as in dual‐task conditions. The results show that secondary tasks, which require a choice, impair the calculation of simple sums. On the other hand, an increased degree of input monitoring in the secondary tasks does not seem to impair the calculation of the sums. These findings show that response selection is strongly involved in simple arithmetic and may be one of the core executive processes in mental arithmetic.     

Does solution of mental arithmetic problems such as 2 + 6 and 3 x 8 rely on the process of "Memory updating"?

Two experiments are reported that used the selective interference paradigm to study whether, besides response selection, the process of memory updating is involved in simple mental arithmetic. Participants were asked to solve simple sums (e.g., 2 + 6, Experiment 1) or simple products (e.g., 3 X 8, Experiment 2) in a single-task control condition and in three dual-task conditions with a selective interference task, simple reactions, choice reactions, or delayed choice reactions. The role of memory updating was estimated on the basis of the difference in impairment due to the choice reaction time and the delayed choice reaction time task, whereas the difference in impairment between the simple reaction time and the choice reaction time task indicates the role of response selection. While replicating previous results concerning response selection (Deschuyteneer & Vandierendonck, 2005, in press), the study showed that memory updating is strongly involved in solving simple mental arithmetic sums and products.     

Strategy use in mental subtraction determines central executive load

A dual task method was used to examine the relationship between strategy use and working memory load during subtraction problem solving. Undergraduates mentally solved subtraction problems alone and while performing secondary tasks that involved the central executive of working memory. Analyses revealed that a central executive task involving response selection and input monitoring (CRT-R task) interfered more with subtraction problem solving than a task that involved only input monitoring (SRT-R task). Additional analyses showed that the CRT-R task interfered more when participants used a nonretrieval (counting) strategy than a retrieval strategy. These findings suggest that the response selection subcomponent of the central executive is involved during both retrieval-based and non-retrieval-based simple subtraction problem solving but is involved more during the latter.     

Mental multiplication and working memory

This paper is concerned with the role of working memory resources in mental multiplication. In two experiments a dual-task paradigm was used. In the first experiment neutral tapping was contrasted to three modalityspecific secondary tasks: Irrelevant speech and articulatory suppression were used to disrupt the phonological loop and a visuo-spatial tapping was used to disrupt the visuo-spatial sketchpad. Multiplication sums needed to be solved mentally and results needed to be spoken aloud. Sums varied in difficulty (easy, e.g., 3 x 4 =, difficult, e.g., 8 x 17 =). Results from the first experiment revealed declines in performance on difficult sums under articulatory suppression but no interference effect for easy sums. To investigate the role of central executive processes, a second experiment extended the range of interference conditions to a central executive interference task (random letter generation). Now articulatory suppression and random generation caused a decrease of performance on difficult sums. In addition, performance on easy sums was negatively impacted by random letter generation as well. We infer that solving complex multiplication sums demands phonological loop and central executive processes, whereas retrieving numerical facts in solving simple multiplication sums requires only central executive processes. We found no evidence of modality-specific access to numerical facts stored in long-term memory.     

工作记忆中央执行对儿童算术认知策略的影响

运用“双重任务”范式,以113名小学二年级儿童为被试,考察了儿童工作记忆的中央执行对算术认知策略表现的影响。结果表明:中央执行对儿童算术认知策略的选择产生了针对性的影响,集中表现在外部策略、支持性策略和猜测放弃策略上。中央执行干扰使策略执行的正确率下降、反应时增加。在简单加法任务上,中央执行影响了外部策略和支持性策略的选择频次;在复杂任务中,这种影响体现在外部策略和放弃策略上。无论是简单任务还是复杂任务,中央执行的干扰都造成了策略整体执行效果的下降。     

High-performers use the phonological loop less to process mental arithmetic during working memory tasks

This study investigated the effects of three working memory components—the central executive, phonological loop, and visuospatial sketchpad—on performance differences in complex mental arithmetic between individuals. Using the dual-task method, we examined how performance during two-digit addition was affected by load on the central executive (random tapping condition), phonological loop (articulatory suppression condition), and visuospatial sketchpad (spatial tapping condition) compared to that under no load (control condition) in high- and low-performers of complex mental arithmetic in Experiment 1. Low-performers showed an increase in errors under the random tapping and articulatory suppression conditions, whereas high-performers showed an increase of errors only under the random tapping condition. In Experiment 2, we conducted similar experiments on only the high-performers but used a shorter presentation time of each number. We found the same pattern for performing complex mental arithmetic as seen in Experiment 1. These results indicate that high-performers might reduce their dependence on the phonological loop, because the central executive enables them to choose a strategy in which they use less working memory capacity.     

Input monitoring and response selection as components of executive control in pro-saccades and anti-saccades

Several studies have shown that anti-saccades, more than pro-saccades, are executed under executive control. It is argued that executive control subsumes a variety of controlled processes. The present study tested whether some of these underlying processes are involved in the execution of anti-saccades. An experiment is reported in which two such processes were parametrically varied, namely input monitoring and response selection. This resulted in four selective interference conditions obtained by factorially combining the degree of input monitoring and the presence of response selection in the interference task. The four tasks were combined with a primary task which required the participants to perform either pro-saccades or anti-saccades. By comparison of performance in these dual-task conditions and performance in single-task conditions, it was shown that anti-saccades, but not pro-saccades, were delayed when the secondary task required input monitoring or response selection. The results are discussed with respect to theoretical attempts to deconstruct the concept of executive control.     

The role of phonological and executive working memory resources in simple arithmetic strategies

The current study investigated the role of the central executive and the phonological loop in arithmetic strategies to solve simple addition problems (Experiment 1) and simple subtraction problems (Experiment 2). The choice/no-choice method was used to investigate strategy execution and strategy selection independently. The central executive was involved in both retrieval and procedural strategies, but played a larger role in the latter than in the former. Active phonological processes played a role in procedural strategies only. Passive phonological resources, finally, were only needed when counting was used to solve subtraction problems. No effects of working memory load on strategy selection were observed.     

中央执行负荷对成人估算策略运用的影响

随机选取128名大学生为被试, 运用选择/无选法研究范式, 考察了不同中央执行负荷对估算策略运用的影响。结果发现：(1)中央执行负荷不影响策略分布; (2)策略运用条件、中央执行负荷影响策略执行。主次一致任务, 负荷对策略执行反应时的影响随负荷强度增大而增大, 对策略执行精确度影响不大; 而对主次不一致任务, 低负荷对策略执行反应时及精确度影响都不明显; (3)策略运用条件、中央执行负荷影响策略选择。负荷强度对策略选择反应时起重要作用, 只有当次级任务负荷高时, 干扰作用才明显; (4)成人的策略选择适应性受负荷强度的影响。无负荷条件下个体策略适应性更好。     

Individual differences in cognitive arithmetic

Unities in the processes involved in solving arithmetic problems of varying operations have been suggested by studies that have used both factor-analytic and information-processing methods. We designed the present study to investigate the convergence of mental processes assessed by paper-and-pencil measures defining the Numerical Facility factor and component processes for cognitive arithmetic identified by using chronometric techniques. A sample of 100 undergraduate students responded to 320 arithmetic problems in a true-false reaction-time (RT) verification paradigm and were administered a battery of ability measures spanning Numerical Facility, Perceptual Speed, and Spatial Relations factors. The 320 cognitive arithmetic problems comprised 80 problems of each of four types: simple addition, complex addition, simple multiplication, and complex multiplication. The information-processing results indicated that regression models that included a structural variable consistent with memory network retrieval of arithmetic facts were the best predictors of RT to each of the four types of arithmetic problems. The results also verified the effects of other elementary processes that are involved in the mental solving of arithmetic problems, including encoding of single digits and carrying to the next column for complex problems. The relation between process components and ability measures was examined by means of structural equation modeling. The final structural model revealed a strong direct relation between a factor subsuming efficiency of retrieval of arithmetic facts and of executing the carry operation and the traditional Numerical Facility factor. Furthermore, a moderate direct relation between a factor subsuming speed of encoding digits and decision and response times and the traditional Perceptual Speed factor was also found. No relation between structural variables representing cognitive arithmetic component processes and ability measures spanning the Spatial Relations factor was found. Results of the structural modeling support the conclusion that information retrieval from a network of arithmetic facts and execution of the carry operation are elementary component processes involved uniquely in the mental solving of arithmetic problems. Furthermore, individual differences in the speed of executing these two elementary component processes appear to underlie individual differences on ability measures that traditionally span the Numerical Facility factor.(ABSTRACT TRUNCATED AT 400 WORDS)     

The role of working memory in the carry operation of mental arithmetic: number and value of the carry

Two experiments were conducted to investigate the role of phonological and executive working-memory components in the carry operation in mental arithmetic. We manipulated the number of carry operations, as previous research had done, but also the value that had to be carried. Results of these experiments show that in addition to the number of carry operations, the value of the carry is also an important variable determining the difficulty of arithmetical sums. Furthermore, both variables (number and value) interacted with each other in such a way that the combination of multiple carries and values of carries larger than one resulted in more difficult problems irrespective of the presence of a working-memory load. The findings with respect to working-memory load suggest that mainly the central executive is important in handling the number of carry operations as well as the value that has to be carried. The implications of the present findings for our views on mental arithmetic and its reliance on working memory are discussed.     

The role of working memory in the carry operation of mental arithmetic: Number and value of the carry

Two experiments were conducted to investigate the role of phonological and executive working-memory components in the carry operation in mental arithmetic. We manipulated the number of carry operations, as previous research had done, but also the value that had to be carried. Results of these experiments show that in addition to the number of carry operations, the value of the carry is also an important variable determining the difficulty of arithmetical sums. Furthermore, both variables (number and value) interacted with each other in such a way that the combination of multiple carries and values of carries larger than one resulted in more difficult problems irrespective of the presence of a working-memory load. The findings with respect to working-memory load suggest that mainly the central executive is important in handling the number of carry operations as well as the value that has to be carried. The implications of the present findings for our views on mental arithmetic and its reliance on working memory are discussed.     

Number line estimation and complex mental calculation: Is there a shared cognitive process driving the two tasks?

It is widely accepted that different number-related tasks, including solving simple addition and subtraction, may induce attentional shifts on the so-called mental number line, which represents larger numbers on the right and smaller numbers on the left. Recently, it has been shown that different number-related tasks also employ spatial attention shifts along with general cognitive processes. Here we investigated for the first time whether number line estimation and complex mental arithmetic recruit a common mechanism in healthy adults. Participants’ performance in two-digit mental additions and subtractions using visual stimuli was compared with their performance in a mental bisection task using auditory numerical intervals. Results showed significant correlations between participants’ performance in number line bisection and that in two-digit mental arithmetic operations, especially in additions, providing a first proof of a shared cognitive mechanism (or multiple shared cognitive mechanisms) between auditory number bisection and complex mental calculation.     

Separate roles for executive and phonological components of working memory in mental arithmetic

A dual-task methodology was used to investigate the roles played by executive and phonological aspects of working memory in mental arithmetic. Experiment 1 showed that suppression of articulation impaired the ability to add a pair of briefly presented three-digit numbers. Suppression had no effect when the need to store temporarily was minimized by making the numbers visible throughout calculation. Experiment 2 showed that disrupting executive processes by requiring concurrent performance of a Trails task impaired the ability to add numbers that remained permanently visible. Performance on the Trails task deteriorated as the number of carry operations in the addition increased. Experiment 3 showed that this decline in Trails performance was not simply due to the extra time taken by carrying. These and other features of the results suggest that the carrying component of mental arithmetic places substantial demands on executive processes, whereas the need to retain problem information is met by the phonological loop. The results are consistent with an interpretation of executive processes according to which there is a limit on the capacity to inhibit strongly primed routine operations.     

元认知监测与算术知识制约小学儿童心算策略运用能力的发展:一年纵向考察

为探究元认知监测与算术知识对儿童心算策略运用能力的影响如何随个体发展而变化，采用计算机任务与纸笔测量的方法，对85名小学三、五年级儿童进行了历时一年的纵向追踪研究。研究发现：（1）两组儿童的元认知监测和算术知识均呈增长趋势，算术知识的增长速度五年级显著快于三年级，且元认知监测增长速度与算术知识增长速度显著相关；（2）两组儿童中，元认知监测与算术知识增长速度更快的个体策略执行反应时与错误率的减少速度也更快；（3）五年级儿童的算术知识在元认知监测影响策略选择发展中起着完全中介作用。     

算术策略运用能力的年龄差异：元认知监测与算术知识的作用

采用选择/无选法,以估算与精确心算为研究任务,考察了元认知监测与算术知识影响个体算术计算策略选择与执行的年龄相关差异。129名不同年龄儿童与成人被试参加实验。结果发现：（1）算术知识对儿童及成人的估算复杂策略有促进作用,并对提升他们心算策略运用的速度和减少错误有作用;（2）元认知监测显著影响儿童的估算策略选择,能够促进使用最佳策略;（3）算术知识在估算及心算策略执行的年龄差异方面起完全中介作用,元认知监测则在估算策略选择的年龄发展中起部分中介作用;（4）算术知识对元认知监测在估算及心算策略执行上的作用起完全中介作用,而对估算的策略选择则不存在中介作用,这表明元认知监测在估算策略选择上具有举足轻重的地位。     

Counting on working memory in arithmetic problem solving

Mental calculation is an important everyday skill involving access to well-learned procedures, problem solving, and working memory. Although there is an active literature on acquiring concepts and procedures for mental arithmetic, relatively little is known about the role of working memory in this task. This paper reports two experiments in which dual-task methodology is used to study the role of components of working memory in mental addition. In Experiment 1, mental addition of auditorily presented two-digit numbers was significantly disrupted by concurrent random letter generation and, to a lesser extent, by concurrent articulatory suppression, but was unimpaired by concurrent hand movement or by presentation of irrelevant pictures. Although the number of errors increased with two of the dual tasks, the incorrect responses tended to be quite close to the correct answer. In Experiment 2, the numbers for addition were presented visually. Here again, random generation produced the largest disruption of mental arithmetic performance, while a smaller amount of disruption was observed for articulatory suppression, hand movement, and unattended auditorily presented two-digit numbers. The overall levels of performance were better and the absolute size of the disruptive effects shown with visual presentation was very small compared with those found for auditory presentation. This pattern of results is consistent with a role for a central executive component of working memory in performing the calculations required for mental addition and in producing approximately correct answers. Visuospatial resources in working memory may also be involved in approximations. The data support the view that the subvocal rehearsal component of working memory provides a means of maintaining accuracy in mental arithmetic, and this matches a similar conclusion derived from previous work on counting. The general implications for the role of working memory in arithmetic problem solving will be discussed.     

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.     

Working memory, strategy execution, and strategy selection in mental arithmetic

A total of 72 participants estimated products of complex multiplications of two-digit operands (e.g., 63 × 78), using two strategies that differed in complexity. The simple strategy involved rounding both operands down to the closest decades (e.g., 60 × 70), whereas the complex strategy required rounding both operands up to the closest decades (e.g., 70 × 80). Participants accomplished this estimation task in two conditions: a no-load condition and a working-memory load condition in which executive components of working memory were taxed. The choice/no-choice method was used to obtain unbiased strategy execution and strategy selection data. Results showed that loading working-memory resources led participants to poorer strategy execution. Additionally, participants selected the simple strategy more often under working-memory load. We discuss the implications of the results to further our understanding of variations in strategy selection and execution, as well as our understanding of the impact of working-memory load on arithmetic performance and other cognitive domains.     

Working memory,strategy execution,and strategy selection in mental arithmetic

A total of 72 participants estimated products of complex multiplications of two-digit operands (e.g., 63 × 78), using two strategies that differed in complexity. The simple strategy involved rounding both operands down to the closest decades (e.g., 60 × 70), whereas the complex strategy required rounding both operands up to the closest decades (e.g., 70 × 80). Participants accomplished this estimation task in two conditions: a no-load condition and a working-memory load condition in which executive components of working memory were taxed. The choice/no-choice method was used to obtain unbiased strategy execution and strategy selection data. Results showed that loading working-memory resources led participants to poorer strategy execution. Additionally, participants selected the simple strategy more often under working-memory load. We discuss the implications of the results to further our understanding of variations in strategy selection and execution, as well as our understanding of the impact of working-memory load on arithmetic performance and other cognitive domains.