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.     

Retrieval processes in fifth graders and adults

This research was concerned with retrieval processes which underlie conceptual clustering. In a free recall task with categorized lists, fifth graders' and adults' retrieval was examined temporally by means of interword response times. List organization and retrieval cue factors were manipulated in order to assess the developmental relevance of an hypothesized three component retrieval model. The results indicated that both age groups used a qualitatively similar retrieval strategy, involving search for and decoding of higher-order memory units. Providing retrieval cues eliminated the category search component, but also increased the likelihood of exhaustive recall attempts under random presentation conditions. The results were discussed in terms of the inter-relationships between storage and retrieval strategies. Age-related differences in retrieval time were attributed to insufficient reorganization during input and failure to attempt to recall all of a category's members.     

Menatal addition: A test of three verification models

Three explanations of adults’ mental addition performance, a counting-based model, a direct-access model with a backup counting procedure, and a network retrieval model, were tested. Whereas important predictions of the two counting models were not upheld, reaction times (RTs) to simple addition problems were consistent with the network retrieval model. RT both increased with problem size and was progressively attenuated to false stimuli as the split (numerical difference between the false and correct sums increased. For large problems, the extreme level of split (13) yielded an RT advantage for false over true problems, suggestive of a global evaluation process operating in parallel with retrieval. RTs to the more complex addition problems in Experiment 2 exhibited a similar pattern of significance and, in regression analyses, demonstrated that complex addition (e.g., 14+12=26) involves retrieval of the simple addition components (4+2=6). The network retrieval/decision model is discussed in terms of its fit to the present data, and predictions concerning priming facilitation and inhibition are specified. The similarities between mental arithmetic results and the areas of semantic memory and mental comparisons indicate both the usefulness of the network approach to mental arithmetic and the usefulness of mental arithmetic to cognitive psychology.     

Priming effects of arithmetic signs in 10- to 15-year-old children

In this research, 10- to 12- and 13- to 15-year-old children were presented with very simple addition and multiplication problems involving operands from 1 to 4. Critically, the arithmetic sign was presented before the operands in half of the trials, whereas it was presented at the same time as the operands in the other half. Our results indicate that presenting the ‘x’ sign before the operands of a multiplication problem does not speed up the solving process, irrespective of the age of children. In contrast, presenting the ‘+’ sign before the operands of an addition problem facilitates the solving process, but only in 13 to 15-year-old children. Such priming effects of the arithmetic sign have been previously interpreted as the result of a pre-activation of an automated counting procedure, which can be applied as soon as the operands are presented. Therefore, our results echo previous conclusions of the literature that simple additions but not multiplications can be solved by fast counting procedures. More importantly, we show here that these procedures are possibly convoked automatically by children after the age of 13 years. At a more theoretical level, our results do not support the theory that simple additions are solved through retrieval of the answers from long-term memory by experts. Rather, the development of expertise for mental addition would consist in an acceleration of procedures until automatization.     

On knowing how well one is remembering: The persistence of strategy use during transfer

A potential cause for children's failure to transfer learning strategies was explored in the present study. A self-monitoring process was suggested to be essential for evaluating one's own level of performance and the effectiveness of various mnemonic strategies. Matched on free recall scores in this study's first sort-and-recall phase, first, third, and fifth graders (ages 7, 9, and 11 years, respectively) were assigned to one of three treatment groups or a control group. During Phase II, the treatment groups received instruction in sorting pictures according to semantic similarities in preparation for future recall. In addition, Groups 3 and 4 later received feedback indicating their Phase II improvement in performance. A cause-and-effect relationship between strategy use and enhanced recall was further suggested to Group 4. Among third graders, only those provided with feedback and strategy instruction continued to rely upon the input organization strategy when faced with transfer tasks in Phase III. Both enhanced recall and improved sorting styles were observed for these subjects in Phase III. Some first graders also showed improved sorting styles and improved recall following feedback, while fifth graders showed enhanced recall even without experimenter-provided feedback. Subject's responses to a metamemory interview provided additional support for the hypothesis that the self-monitoring of memory performance is more likely to be part of fifth graders' (than first graders') memory abilities.     

Procedural knowledge versus fact retrieval in mental arithmetic: A reply to Baroody

Based on a review of reaction time studies, a model of mental arithmetic performance which emphasizes the process of fact retrieval from organized memory representations was proposed (M. H. Ashcraft, Developmental Review, 1982, 2, 213–236). In contrast to this view A. J. Baroody (Developmental Review, 1983, 3, 225–230) proposes that most mental arithmetic performance depends on procedural knowledge such as rules, heuristics, and principles. While Baroody's idea is both intriguing and potentially important, its exposition is quite vague and speculative. Without concrete suggestions as to the nature of the proposed rules and heuristics, especially for routine problems like 4 + 3 and 8 × 5, Baroody's proposal appears to be pertinent only to special cases like N + 0 and N + 1. Lacking this sort of elaboration, the alternative does not provide a useful or compelling explanation of the existing Chronometric results, and seems, at best, to be premature.     

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.     

Children's cognitive maps of large-scale spaces: Effects of exploration,direction, and repeated experience

Kindergarteners and third graders (mean ages 5–10 and 8–9 years) repeatedly encountered a model town and then constructed the town from memory. In Experiment 1, the effect of different types of exploration on the development of a cognitive map was assessed. Children who were directed to walk within the town placed buildings more accurately than children who had walked along the town's perimeter. Children who walked within the town and were directed to the spatial relationships among buildings had the highest placement accuracy. Third graders were more accurate than kindergarteners across the three types of exploration. In Experiment 2, children were permitted to explore the town alone for an unlimited amount of time. Under these conditions, third graders still placed buildings more accurately than kindergarteners. A comparison of Experiments 1 and 2 indicated that children developed more accurate cognitive maps when motor activity and attention were directed by the experimenter as opposed to being directed by the child. Accuracy improved with repeated walks and constructions in both experiments, and the results of Experiment 3 suggested that constructing facilitated the development of the cognitive map as much as walking. It was concluded that third graders' cognitive maps were more accurate than kindergarteners' maps due to differences in the speed of acquisition and storage of spatial information.     

The effects of specific and categorical orienting on children's incidental and intentional memory for pictures and words

This study examines the effects of differences in the encoding of specific and categorical information on second graders' (7 years; 7 months), fifth graders' (10; 6), and college adults' cued recall for cue-target picture and word pairs. The cues at retrieval were either same-modal (P-P and W-W) or cross-modal (P-W and W-P) as the cues presented in acquisition, and acquisition encoding was either incidental or intentional and constrained by orienting questions or unconstrained. The most important results were that both picture and word recall varied with the encoding of both specific and categorical information and that children differed from adults in the encoding of both kinds of information in both incidental and intentional encoding conditions. In addition, both children and adults showed congruency effects for pictures and words that were greater for specific than for categorical orienting questions. The results suggest that differences in the encoding of both specific and categorical attribute information contribute to developmental recall differences independently of encoding intent and stimulus modality.     

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.     

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.     

The relationship between phonological processing and arithmetic in children with learning disabilities

Phonological processing skills have not only been shown to be important for reading skills, but also for arithmetic skills. Specifically, previous research in typically developing children has suggested that phonological processing skills may be more closely related to arithmetic problems that are solved through fact retrieval (e.g., remembering the solution from memory) than procedural computation (e.g., counting). However, the relationship between phonological processing and arithmetic in children with learning disabilities (LDs) has not been investigated. Yet, understanding these relationships in children with LDs is especially important because it can help elucidate the cognitive underpinnings of math difficulties, explain why reading and math disabilities frequently co-occur, and provide information on which cognitive skills to target for interventions. In 63 children with LDs, we examined the relationship between different phonological processing skills (phonemic awareness, phonological memory, and rapid serial naming) and arithmetic. We distinguished between arithmetic problems that tend to be solved with fact retrieval versus procedural computation to determine whether phonological processing skills are differentially related to these two arithmetic processes. We found that phonemic awareness, but not phonological memory or rapid serial naming, was related to arithmetic fact retrieval. We also found no association between any phonological processing skills and procedural computation. These results converge with prior research in typically developing children and suggest that phonemic awareness is also related to arithmetic fact retrieval in children with LD. These results raise the possibility that phonemic awareness training might improve both reading and arithmetic fact retrieval skills.

Research Highlights

• Relationships between phonological processing and various arithmetic skills were investigated in children with learning disabilities (LDs) for the first time.
• We found phonemic awareness was related to arithmetic involving fact retrieval, but not to arithmetic involving procedural computation in LDs.
• The results suggest that phonemic awareness is not only important to skilled reading, but also to some aspects of arithmetic.
• These results raise the question of whether intervention in phonemic awareness might improve arithmetic fact retrieval skills.

     

Representation and retrieval of arithmetical facts: Developmental difficulties

One form of developmental difficulty with arithmetic affects the storage or retrieval of arithmetical facts, such as tables, which are required to implement arithmetical computations (Temple, 1991, 1994). Such difficulties may arise because of impairment in a specialized system for the storage of arithmetical facts or as a result of causally linked impairment in another cognitive domain. This study explored issues concerning the representation and retrieval of arithmetical facts in children with number fact disorders (NF) and in normal children, in particular the status of hypothesized linked impairments: short-term memory (STM) spans, counting skills, speed of speech, and speed of number fact and lexical retrieval. There was no evidence that NF children had weak STM spans on any span measure or that STM spans related to arithmetical fact skills. There was also no evidence that NF children had weak counting abilities or free counting speeds. The NF children were slower in speeded counting, which also correlated with number fact skill. The significance or not of this is discussed. The NF children were also slower than controls in speed of speech and on some measures of speed of access. However, the absence of correlation with number fact skill, the absence of generality across tasks, and the possibility that delayed speeds in fact retrieval reflect the use of alternative strategies, together suggest that the increased speeds are not causally linked to number fact skill. The results are consistent with modular accounts, in which there is a specialized system for the storage and retrieval of arithmetical facts.     

心算困难儿童的语音复述

心算中语音复述和计算加工交替进行,其中任何一方面落后都会导致心算困难。使用视觉心算成绩和听觉心算成绩相减的范式,研究计算能力正常但心算水平落后的儿童的语音复述水平。结果发现,不同年级心算困难儿童呈现不同的困难模式,2、4年级心算困难儿童语音复述水平正常,但由于语音工作记忆容量的落后,导致缺少足够的注意资源进行计算加工。3年级心算困难儿童在语音工作记忆容量和语音复述上都存在落后,导致心算困难,呈现出双重困难模式。     

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.     

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.     

运算动量效应的理论解释及其发展性预测因素

了解运算偏差的形成与发展对探索算数运算系统的内在机制具有重要意义, 早期的算数运算能力是儿童理解和进行复杂数学运算的基础。运算动量偏差是指个体在进行基本数学运算时倾向于高估加法运算结果而低估减法运算结果的一种运算偏差, 主要包括三种理论解释, 即注意转移假说、启发式解释和压缩解释。鉴于运算动量效应在成年群体中相对稳定却在不同发展阶段儿童中存在不一致的证据, 数学能力的提高与空间注意的成熟可结合不同的理论解释来阐明儿童发展过程中运算动量效应的变化趋势。未来可以进一步整合多种研究任务以揭示运算动量效应的发展轨迹, 考察数量表征系统与运算动量效应间的关联, 探究运算动量效应在不同运算符号中的稳定性, 探讨不同因素共同作用对运算动量效应的影响, 并设计有关数学能力的干预措施以减少运算动量效应这一运算偏差。     

Fragile X syndrome: Neural network models of sequencing and memory

A comparative framework of memory processes in males with fragile X syndrome (FXS) and typically developing (TYP) mental age-match children is presented. Results indicate a divergence in sequencing skills, such that males with FXS recall sequences similarly to TYP children around five and a half years of age, but the males with FXS recall significantly worse when compared to TYP children around seven and a half years of age. Performance on one working memory measure, an n-back Card Task, is modeled with a neural network. To date, no network models explicate the sequencing and memory processes in those with FXS. Noise was added to various levels (weight matrices) in the FXS model and outputs approximated human FXS performance. Three models were compared: (1) FXS; (2) younger mental age-TYP matches; (3) older reading level-TYP matches. Modeling can help to reify conceptualizations of deficits and to guide in the creation of more valid, science-based remediations. The FXS model suggested that the levels of phonological representation and sequencing in memory were candidates for targeted therapies in males with FXS.     

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.