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

The present study examined the influence of features differing across problems (problem size and operation) and across individuals (gender, amount of daily arithmetic practice, calculator use, and arithmetic skill) on simple arithmetic performance. Regression analyses were used to investigate the role of these variables in both strategy selection and strategy efficiency. Results show that more skilled and highly practiced students used memory retrieval more often and executed their strategies more efficiently than did less skilled and practiced students. Furthermore, calculator use correlated with both retrieval and procedural strategy efficiency but not with strategy selection. Only very small associations with gender were observed, with boys retrieving slightly faster than girls. Implications of the present findings for models of mental arithmetic are discussed.     

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.     

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.     

Do multiplication and division strategies rely on executive and phonological working memory resources?

The role of executive and phonological working memory resources in simple arithmetic was investigated in two experiments. Participants had to solve simple multiplication problems (e.g., 4 x 8; Experiment 1) or simple division problems (e.g., 42 / 7; Experiment 2) under no-load, phonological-load, and executive-load conditions. The choice/no-choice method was used to investigate strategy execution and strategy selection independently. Results for strategy execution showed that executive working memory resources were involved in direct memory retrieval of both multiplication and division facts. Executive working memory resources were also involved in the use of nonretrieval strategies. Phonological working memory resources, on the other hand, tended to be involved in nonretrieval strategies only. Results for strategy selection showed no effects of working memory load. Finally, correlation analyses showed that both strategy execution and strategy selection correlated with individual-difference variables, such as gender, math anxiety, associative strength, calculator use, arithmetic skill, and math experience.     

High working memory capacity favours the use of finger counting in six-year-old children

In this study, we show that 6-year-old children with high working memory capacity are more likely to use their fingers in an addition task than children with lower capacity. Moreover and as attested by a strong correlation between finger counting and accuracy in the arithmetic task, finger counting appears to be a very efficient strategy. Therefore, discovering the finger counting strategy seems to require a large amount of working memory resources, which could lack in low-span children. Furthermore, when children with low working memory capacities use their fingers to solve addition problems, they more often use the laborious counting-all strategy than children with higher capacities who use more elaborated procedures such as the Min strategy. Consequently, we suggest that explicit teaching of finger counting during the first years of schooling should be promoted because it could help less gifted children to overcome their difficulties in arithmetic.     

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.     

Counting on working memory in simple arithmetic when counting is used for problem solving

A concurrent-task methodology was used to investigate relations between the availability of aspects of working memory resources and both strategy selection and execution while simple addition equations (e.g., 4 + 3 = 8) were being verified. Consistent with prior research in which production trials have been used, undergraduates selected a variety of procedures other than retrieval. Availability of working memory resources did not generally affect strategy selection. Disrupting central executive and phonological aspects of memory affected strategy execution, but only when min counting was used to solve the problems. These and other features of the results suggest that availability of working memory resources does not contribute to individual differences in strategy selection and time to execute retrieval processes.     

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.     

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.     

工作记忆成分的年龄相关差异对算术策略运用的预测效应

采用选择/无选范式,借助工作记忆成套测验,在两位数乘法估算问题中探讨了工作记忆系统各成分对不同年龄段个体算术策略运用的预测效应。结果显示:(1)工作记忆的不同成分与年龄之间存在明显的相关。表现为,除视空模板成分外,其他各成分得分随着年龄增长而呈现上升趋势;(2)估算策略运用中,年龄与策略选择显著相关,表现为随着年龄增长,策略选择表现明显提高;(3)估算策略运用中,不同年龄个体的工作记忆不同成分和策略选择表现出不同的联系,中央执行均显示出显著的预测效应,语音环路和视空模板的预测效应均不显著。不同年龄个体的工作记忆不同成分对策略执行的预测效应均不显著。上述发现对于深刻理解工作记忆系统在算术认知策略运用中的作用机制具有重要理论含义。     

工作记忆负荷和自动化提取对小学生加法心算策略执行效果的影响

本研究选取43名小学四年级学生(18名男生和25名女生)为实验被试,探究了工作记忆负荷和自动化提取对复杂加法心算策略效果的影响.结果显示:(1)工作记忆负荷对复杂加法心算策略的影响显著,即一项加法心算策略所需的工作记忆负荷越小,该策略的执行效果越好;(2)自动化提取对加法心算策略的影响显著,即一项加法心算策略所需自动化提取的程度越高,该策略的执行效果越好;(3)工作记忆负荷和自动化提取对加法心算策略效果的交互作用显著,表现为在自动化提取水平较高的情况下,工作记忆负荷的大小对心算策略执行效果的影响差异不显著;而在自动化提取水平较低的情况下,工作记忆负荷小的心算策略的执行效果显著优于工作记忆负荷大的心算策略的执行效果.     

Evolution of cognitive processes for solving simple additions during the first three school years

The development of a group of children's cognitive strategies forn solving simple additions was studied by analyzing verbal reports given after each problem (I+J) was solved. The evolution of the cognitive processes involved a gradual shift from more primitive and less demanding strategies (in which, e.g., the child's fingers served as memory aid) to reconstructive memory processes (in which e.g., the answer was derived in a counting process in working memory) to retrieval processes (in which the answer was obtained form long term memory search). During the first semester of the first school year 36 percent of the problems (I⁺J≤13, I≠J, I≠0, I≠1, J≠1,) could not be answered, 40 percent of the solutions were obtained in the most frequent processes utilizing external meory aid and 16 percent in reconstructive memory processes. When in the second semester of the third school year, the same children solved th same problems by utilixing the followitn most frequent strategies; 31 percent long term memory retrieval, 38 percent reconstructive memory processes and 19 percent in processes utilizing external memory aid. If a problem was solved by using a given strategy this strategy was often most likely to have been used bt the child on the occasion before and to be used during the following semester as well. For long-term memory solutions this tendency was strongest and for other strategies it was coupled with a gradual shift towards strategies with increasing sophistication in terms of memory representation.     

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.     

Strategy use,the development of automaticity,and working memory involvement in complex multiplication

Participants practiced a set of complex multiplication problems (e.g., 3 x 18) in a pre-/postpractice design. Before, during, and after practice, the participants gave self-reports of problem-solving strategies. At prepractice, the most common strategy was a mental version of the standard multidigit algorithm, and dual tasks revealed that working memory load was high and heavier for problems solved via nonretrieval strategies. After practice, retrieval was used almost exclusively, and participant variability, automaticity level of problems (proportion of trials on which retrieval was used over the entire experiment), and error rates were significant predictors of problem-solving latencies. Practice reduced working memory involvement to minimal levels, and there was no relationship between automaticity level and working memory load. The commonalities between the present findings and findings related to automaticity development in simple arithmetic are discussed.     

数学焦虑影响儿童数学任务表现的作用机制探析

研究设计了心算加法和图形旋转/平移两类数学任务,同时采用《儿童数学焦虑问卷》、工作记忆和抑制任务,以探讨数学焦虑影响儿童数学任务表现的作用机制.213名小学五年级儿童参与研究,其中89人参与正式实验.结果发现:(1)数学焦虑与数学成绩呈显著负相关;(2)高数学焦虑组在工作记忆和两类数学任务中的得分均显著低于低数学焦虑组,而两组在抑制任务指标上差异不显著;(3)言语工作记忆和视空间工作记忆是数学焦虑影响心算加法任务的完全中介变量;视空间工作记忆是数学焦虑影响图形旋转/平移任务的完全中介变量,言语工作记忆是二者关系的调节变量.研究结果支持了加工效能理论.     

How is phonological processing related to individual differences in children’s arithmetic skills?

While there is evidence for an association between the development of reading and arithmetic, the precise locus of this relationship remains to be determined. Findings from cognitive neuroscience research that point to shared neural correlates for phonological processing and arithmetic as well as recent behavioral evidence led to the present hypothesis that there exists a highly specific association between phonological awareness and single‐digit arithmetic with relatively small problem sizes. The present study examined this association in 37 typically developing fourth and fifth grade children. Regression analyses revealed that phonological awareness was specifically and uniquely related to arithmetic problems with a small but not large problem size. Further analysis indicated that problems with a high probability of being solved by retrieval, but not those typically associated with procedural problem‐solving strategies, are correlated with phonological awareness. The specific association between phonological awareness and arithmetic problems with a small problem size and those for which a retrieval strategy is most common was maintained even after controlling for general reading ability and phonological short‐term memory. The present findings indicate that the quality of children’s long‐term phonological representations mediates individual differences in single‐digit arithmetic, suggesting that more distinct long‐term phonological representations are related to more efficient arithmetic fact retrieval.     

Working memory and children's use of retrieval to solve addition problems

This study tested the hypothesis that children with high working memory capacities solve single-digit additions by direct retrieval of the answers from long-term memory more often than do children with low working memory capacities. Counting and reading letter span tasks were administered to groups of third-grade (mean age=107 months) and fourth-grade (mean age=118 months) children who were also asked to solve 40 single-digit additions. High working memory capacity was associated with more frequent use of retrieval and faster responses in solving additions. The effect of span on the use of retrieval increased with the size of the minimum addend. The relation between working memory measures and use and speed of retrieval did not depend on the numerical or verbal nature of the working memory task. Implications for developmental theories of cognitive arithmetic and theories of working memory are discussed.     

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。     

工作记忆成分与儿童算术认知

根据Baddeley和Hitch(1974)提出的工作记忆多成分模型,研究者对工作记忆的各个成分与儿童算术认知之间的关系进行了广泛的探索。同语音环和视空间模板相比,中央执行在儿童算术认知加工中的作用显得更为关键。目前,探讨各种中央执行功能与儿童算术认知的关系,以及由此来解释算术学习困难儿童的认知成因是这个领域研究者最为关注的焦点。     

Individual capacity differences predict working memory performance and prefrontal activity following dopamine receptor stimulation

Dopamine receptors are abundant in the prefrontal cortex (PFC), a critical region involved in working memory. This pharmacological fMRI study tested the relationships between dopamine, PFC function, and individual differences in working memory capacity. Subjects performed a verbal delayed-recognition task after taking either the dopamine receptor agonist bromocriptine or a placebo. Behavioral effects of bromocriptine treatment depended on subjects’ working memory spans, with the greatest behavioral benefit for lower span subjects. After bromocriptine, PFC activity was positively correlated with a measure of cognitive efficiency (RT slope) during the probe period of the task. Less efficient subjects with slower memory retrieval rates had greater PFC activity, whereas more efficient subjects had less activity. After placebo, these measures were uncorrelated. These results support the role of dopamine in verbal working memory and suggest that dopamine may modulate the efficiency of retrieval of items from the contents of working memory. Individual differences in PFC dopamine receptor concentration may thus underlie the behavioral effects of dopamine stimulation on working memory function.