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.     

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.     

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.     

Expediting arithmetic automaticity: Do inefficient computation methods induce spontaneous testing effects?

Before achieving arithmetic automaticity (direct answer recall), learners self-compute answers or use calculators/tables. Mechanisms supporting arithmetic automaticity are of interest because automaticity predicts math achievement and employment outcomes. Future recall is enhanced when, during study, learners attempt to recall the to-be-learned information (“testing effect”). However, learners may not attempt retrieval when another strategy is available (computation, re-reading). We expected learners to more frequently attempt recall when the alternative strategy was inefficient. Participants learned novel arithmetic problems, and could recall answers (if able) or use a computation method, which varied by condition: (i) self-computation; (ii) calculator-computation; and (iii) using a calculator with scrambled keys. Final recall ability was highest after the condition with the most inefficient non-retrieval method (scrambled calculator). We assumed recall attempts preceded computation, but also considered parallel execution. As evidence against parallel retrieval, answer recall rarely interrupted computation (mental/calculator) after it began. Thus, learners should make pre-emptive recall attempts.     

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.     

Cognitive arithmetic across cultures.

Canadian university students either of Chinese origin (CC) or non-Asian origin (NAC) and Chinese university students educated in Asia (AC) solved simple-arithmetic problems in the 4 basic operations (e.g., 3 + 4, 7 - 3, 3 x 4, 12 divided by 3) and reported their solution strategies. They also completed a standardized test of more complex multistep arithmetic. For complex arithmetic, ACs outperformed both CCs and NACs. For simple arithmetic, however, ACs and CCs were equal and both performed better than NACs. The superior simple-arithmetic skills of CCs relative to NACs implies that extracurricular culture-specific factors rather than differences in formal education explain the simple-arithmetic advantage for Chinese relative to non-Asian North American adults. NAC's relatively poor simple-arithmetic performance resulted both from less efficient retrieval skills and greater use of procedural strategies. Nonetheless, all 3 groups reported using procedures for the larger simple subtraction and division problems, confirming the importance of procedural knowledge in skilled adults' performance of elementary mathematics.     

Group differences in adult simple arithmetic: Good retrievers, not-so-good retrievers, and perfectionists

We used the choice/no-choice methodology in two experiments to examine patterns of strategy selection and execution in groups of undergraduates. Comparisons between choice and no-choice trials revealed three groups. Some participants good retrievers) were consistently able to use retrieval to solve almost all arithmetic problems. Other participants (perfectionists) successfully used retrieval substantially less often in choice-allowed trials than when strategy choices were prohibited. Not-so-good retrievers retrieved correct answers less often than the other participants in both the choice-allowed and no-choice conditions. No group differences emerged with respect to time needed to search and access answers from long-term memory; however, not-so-good retrievers were consistently slower than the other subgroups at executing fact-retrieval processes that are peripheral to memory search and access. Theoretical models of simple arithmetic, such as the Strategy Choice and Discovery Simulation (Shrager & Siegler, 1998), should be updated to include the existence of both perfectionist and not-so-good retriever adults.     

When plausibility judgments supersede fact retrieval: The example of the odd-even effect on product verification

In the present study, we examined the conditions that favor the plausibility judgment strategy over the retrieval strategy when we verify some statements. In particular, we examined the effect of odd-even status of numbers on subjects' verification of single-digit arithmetic problems. In Experiment 1, we explored how factors such as problem difficulty and stimulus onset asynchrony (SOA) influence this effect in adults. In Experiment 2, we present evidence that this odd-even effect is also present in elementary school children, although it varies with the age of the children, the difficulty of the problems, and the SOA. We argue that the odd-even information is helpful in arithmetic verification tasks with difficult problems early in the verification processes and that the time course of these effects differs across ages. The present results are consistent with the view that the plausibility strategy is preferred over the retrieval strategy at an early stage of processing and with information that is not quickly accessible. Finally, we discuss the implications of the present experiments for understanding of single-digit arithmetic and for understanding the more general issue of how people coordinate use of multiple strategies.     

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.     

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.     

数学焦虑影响大学生/儿童估算策略运用的对比研究

选取66名大学生和40名小学四年级儿童,运用选择/无选法研究范式,采用自编乘法估算材料考察了数学焦虑对成人和四年级儿童算术策略运用的不同影响.研究结果显示:(1)数学焦虑仅影响成人在自由选择条件中策略选择的正确率.(2)数学焦虑影响儿童的策略分布、策略执行的正确率及最佳选择条件中策略选择的正确率.(3)成人和儿童策略选择的适应性(以有选条件下的策略使用正确率为指标)均受到数学焦虑的影响,低数学焦虑者的适应性明显更好.文章讨论了数学焦虑影响成人与儿童算术策略运用的不同潜在机制.     

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.     

Alzheimer's disease disrupts arithmetic fact retrieval processes but not arithmetic strategy selection

Three groups of healthy younger adults, healthy older adults, and probable AD patients, performed an addition/number comparison task. They compared 128 couples of additions and numbers (e.g., 4 + 9 15) and had to identify the largest item for each problem by pressing one of two buttons located under each item. Manipulations of problem characteristics (i.e., problem difficulty and splits between correct sums and proposed numbers) enabled us to examine strategy selection and specific arithmetic fact retrieval processes. Results showed that arithmetic facts retrieval processes, which were spared with aging, were impaired in AD patients. However, AD patients were able to switch between strategies across trials according to problem characteristics as well as healthy older adults, and less systematically than healthy younger adults. We discuss implications of these findings for further understanding AD-related differences in arithmetic in particular, and problem solving in general.     

Stability and change in children's division strategies

Age-related changes in children's performance on simple division problems (e.g., 6/2, 72/9) were investigated by asking children in Grades 4 through 7 to solve 32 simple division problems. Differences in performance were found across grade, with younger children performing more slowly and less accurately than older children. Problem size effects were also found in that children were faster and more accurate on small problems than on large problems. Two strategies changed across age, with children in Grade 4 relying heavily on the strategy of "addition" (adding the divisor until the dividend was reached) to solve the problems and children in Grades 5 through 7 relying primarily on the strategy of "multiplication" (recasting the division problem as a multiplication problem) to solve the problems. Surprisingly, the frequency of direct retrieval (retrieving the answer directly from memory) did not increase across grade and never became the dominant strategy of choice. Reasons for why retrieval use remains infrequent and age invariant are discussed. Overall, the results suggest that division is a unique operation and that the continued study of division may have implications for further understanding of how procedural and conceptual knowledge of arithmetic develops.     

Use of indirect addition in adults' mental subtraction in the number domain up to 1,000

This study examined adults' use of indirect addition and direct subtraction strategies on multi-digit subtractions in the number domain up to 1,000. Seventy students who differed in their level of arithmetic ability solved multi-digit subtractions in one choice and two no-choice conditions. Against the background of recent findings in elementary subtraction, we manipulated the size of the subtrahend compared to the difference and only selected items with large distances between these two integers. Results revealed that adults frequently and efficiently apply indirect addition on multi-digit subtractions, yet adults with higher arithmetic ability performed more efficiently than those with lower arithmetic ability. In both groups, indirect addition was more efficient than direct subtraction both on subtractions with a subtrahend much larger than the difference (e.g., 713 - 695) and on subtractions with a subtrahend much smaller than the difference (e.g., 613 - 67). Unexpectedly, only adults with lower arithmetic ability fitted their strategy choices to their individual strategy performance skills. Results are interpreted in terms of mathematical and cognitive perspectives on strategy efficiency and adaptiveness.     

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.     

From print to sound in mature readers as a function of reader ability and two forms of orthographic regularity

Four experiments using college students as subjects provide evidence that both highly skilled and less skilled mature readers derive the names of printed words from visual access of the lexicon rather than by phonological recoding. Regularity of pronunciation (regular vs. exception words) as a variable of orthographic regularity effectiveafter visual code formation had no effect either between or within reading ability groups. Less skilled readers made more errors and were slower than highly skilled readers on both types of words. Sing-letter spatial redundancy, as a variable of orthographic regularity that influences the formation of visual codes, served to differentiate the two groups only in naming nonwords. Highly skilled readers used spatial redundancy to offset the effect of array length, whereas less skilled readers did not. Except for high-frequency words, visual access and retrieval of the pronunciation of words was significantly faster for highly skilled readers. Less skilled readers were most disadvantaged in naming nonwords, a task that requires phonological recoding. Overall results support the hypothesis that reading ability in mature readers is related to the speed of word recognition. Highly skilled readers may make more use of variables of orthographic regularity effective in the formation of visual codes.     

Application of the dual-component model of working memory to ADHD:Greater secondary memory deficit despite confounded cognitive differences

The dual-component model postulates that working memory capacity consists of two dissociable components: maintenance in primary memory (PM) and retrieval from secondary memory (SM). Recent application of this model to attention-deficit/hyperactivity disorder (ADHD) has revealed that the SM component is more deficient than the PM component across both verbal and spatial modalities. The present study attempts to strengthen this conclusion by addressing two weaknesses in the previous study. First, the present study shows that the SM component continues to be more deficient than the PM component across both modalities under conditions in which (1) all participants were instructed to use the same recall strategy (resulting in the exclusion of fewer participants); and, (2) individual differences in this strategy were controlled. Second, the present study also documents a group difference in word reading efficiency that is confounded with diagnostic status and that might have influenced estimates of PM and SM capacities in the verbal modality. However, although the SM component is more deficient than the PM component in the ADHD group, the magnitude of this interaction does not vary as a function task modality. These findings are interpreted to suggest that the pattern of WM deficiencies observed are part of a causal pathway that can lead to the symptoms of ADHD, as well as to impairments in reading (and intelligence) due to overlapping cue-dependent retrieval mechanisms. These findings provide additional support for the notion that the SM component of WM is an important and neglected target for treatment.     

不同数学焦虑成人的算术策略运用差异：ERP研究

采用事件相关电位(ERP)技术和选择/无选法范式, 在两位数加法心算和估算中, 探索高、低数学焦虑个体的算术计算策略运用及其内在机制。行为结果：数学焦虑效应在策略运用的反应时和正确率指标上的差异都不显著; 而脑电结果：高数学焦虑个体的N400波幅显著高于低数学焦虑个体; 选择条件中, 估算与心算的数学焦虑效应的N100波幅差异; 无选条件中, 高低数学焦虑个体N1-P2复合波的波幅和潜伏期差异显著。数学焦虑效应在策略编码(0~250 ms)和策略选择/执行阶段(250 ms之后)存在差异。