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.     

Mental rotation: cross-task training and generalization

It is well established that performance on standard mental rotation tasks improves with training (Peters et al., 1995), but thus far there is little consensus regarding the degree of transfer to other tasks which also involve mental rotation. In Experiment 1, we assessed the effect of mental rotation training on participants' Mental Rotation Test (MRT) scores. Twenty-eight participants were randomly assigned to one of three groups: a "One-Day Training," "Spaced Training," or "No Training" group. Participants who received training achieved higher scores on the MRT, an advantage that was still evident after 1 week. Distribution of training did not affect performance. Experiment 2 assessed generalization of mental rotation training to a more complex mental rotation task, laparoscopic surgery. Laparoscopic surgical skills were assessed using Fundamentals of Laparoscopic Surgery (FLS) tasks. Thirty-four participants were randomly assigned to a "Full Mental Rotation Training, MRT and FLS," "MRT and FLS," or "FLS-only" group. MRT results from Experiment 1 were replicated and mental rotation training was found to elicit higher scores on the MRT. Further, mental rotation training was found to generalize to certain laparoscopic surgical tasks. Participants who obtained mental rotation training performed significantly better on mental-rotation dependent surgical tasks than participants who did not receive training. Therefore, surgical training programs can use simple computer or paper-based mental rotation training instead of more expensive materials to enhance certain aspects of surgical performance of trainees.     

The impact of finger counting habits on arithmetic in adults and children

Here, we explored the impact of finger counting habits on arithmetic in both adults and children. Two groups of participants were examined, those that begin counting with their left hand (left-starters) and those that begin counting with their right hand (right-starters). For the adults, performance on an addition task in which participants added 2 two-digit numbers was compared. The results revealed that left-starters were slower than right-starters when adding and they had lower forward and backward digit-span scores. The children (aged 5–12) showed similar results on a single-digit timed addition task—right-starters outperformed left-starters. However, the children did not reveal differences in working memory or verbal and non-verbal intelligence as a function of finger counting habit. We argue that the motor act of finger counting influences how number is represented and suggest that left-starters may have a more bilateral representation that accounts for the slower processing.     

Improving arithmetic performance with number sense training: An investigation of underlying mechanism

A nonverbal primitive number sense allows approximate estimation and mental manipulations on numerical quantities without the use of numerical symbols. In a recent randomized controlled intervention study in adults, we demonstrated that repeated training on a non-symbolic approximate arithmetic task resulted in improved exact symbolic arithmetic performance, suggesting a causal relationship between the primitive number sense and arithmetic competence. Here, we investigate the potential mechanisms underlying this causal relationship. We constructed multiple training conditions designed to isolate distinct cognitive components of the approximate arithmetic task. We then assessed the effectiveness of these training conditions in improving exact symbolic arithmetic in adults. We found that training on approximate arithmetic, but not on numerical comparison, numerical matching, or visuo-spatial short-term memory, improves symbolic arithmetic performance. In addition, a second experiment revealed that our approximate arithmetic task does not require verbal encoding of number, ruling out an alternative explanation that participants use exact symbolic strategies during approximate arithmetic training. Based on these results, we propose that nonverbal numerical quantity manipulation is one key factor that drives the link between the primitive number sense and symbolic arithmetic competence. Future work should investigate whether training young children on approximate arithmetic tasks even before they solidify their symbolic number understanding is fruitful for improving readiness for math education.     

Children’s strategies in complex arithmetic

Strategies used to solve two-digit addition problems (e.g., 27 + 48, Experiment 1) and two-digit subtraction problems (e.g., 73 – 59, Experiment 2) were investigated in adults and in children from Grades 3, 5, and 7. Participants were tested in choice and no-choice conditions. Results showed that (a) participants used the full decomposition strategy more often than the partial decomposition strategy to solve addition problems but used both strategies equally often to solve subtraction problems; (b) strategy use and execution were influenced by participants’ age, problem features, relative strategy performance, and whether the problems were displayed horizontally or vertically; and (c) age-related changes in complex arithmetic concern relative strategy use and execution as well as the relative influences of problem characteristics, strategy characteristics, and problem presentation on strategy choices and strategy performance. Implications of these findings for understanding age-related changes in strategic aspects of complex arithmetic performance are discussed.     

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。     

Attentional resources in timing: Interference effects in concurrent temporal and nontemporal working memory tasks

Three experiments examined interference effects in concurrent temporal and nontemporal tasks. The timing task in each experiment required subjects to generate a series of 2- or 5-sec temporal productions. The nontemporal tasks were pursuit rotor tracking (Experiment 1), visual search (Experiment 2), and mental arithmetic (Experiment 3). Each nontemporal task had two levels of difficulty. All tasks were performed under both single- and dual-task conditions. A simple attentional allocation model predicts bidirectional interference between concurrent tasks. The main results showed the classic interference effect in timing. That is, the concurrent nontemporal tasks caused temporal productions to become longer (longer productions represent a shortening of perceived time) and/or more variable than did timing-only conditions. In general, the difficult version of each nontemporal task disrupted timing more than the easier version. The timing data also exhibited a serial lengthening effect, in which temporal productions became longer across trials. Nontemporal task performance showed a mixed pattern. Tracking and visual search were essentially unaffected by the addition of a timing task, whereas mental arithmetic was disrupted by concurrent timing. These results call for a modification of the attentional allocation model to incorporate the idea of specialized processing resources. Two major theoretical frameworks—multiple resource theory and the working memory model—are critically evaluated with respect to the resource demands of timing and temporal/ nontemporal dual-task performance.     

Anatomically ordered tapping interferes more with one-digit addition than two-digit addition: a dual-task fMRI study

Fingers are used as canonical representations for numbers across cultures. In previous imaging studies, it was shown that arithmetic processing activates neural resources that are known to participate in finger movements. Additionally, in one dual-task study, it was shown that anatomically ordered finger tapping disrupts addition and subtraction more than multiplication, possibly due to a long-lasting effect of early finger counting experiences on the neural correlates and organization of addition and subtraction processes. How arithmetic task difficulty and tapping complexity affect the concurrent performance is still unclear. If early finger counting experiences have bearing on the neural correlates of arithmetic in adults, then one would expect anatomically and non-anatomically ordered tapping to have different interference effects, given that finger counting is usually anatomically ordered. To unravel these issues, we studied how (1) arithmetic task difficulty and (2) the complexity of the finger tapping sequence (anatomical vs. non-anatomical ordering) affect concurrent performance and use of key neural circuits using a mixed block/event-related dual-task fMRI design with adult participants. The results suggest that complexity of the tapping sequence modulates interference on addition, and that one-digit addition (fact retrieval), compared to two-digit addition (calculation), is more affected from anatomically ordered tapping. The region-of-interest analysis showed higher left angular gyrus BOLD response for one-digit compared to two-digit addition, and in no-tapping conditions than dual tapping conditions. The results support a specific association between addition fact retrieval and anatomically ordered finger movements in adults, possibly due to finger counting strategies that deploy anatomically ordered finger movements early in the development.     

Typical and atypical development of basic numerical skills in elementary school

Deficits in basic numerical processing have been identified as a central and potentially causal problem in developmental dyscalculia; however, so far not much is known about the typical and atypical development of such skills. This study assessed basic number skills cross-sectionally in 262 typically developing and 51 dyscalculic children in Grades 2, 3, and 4. Findings indicate that the efficiency of number processing improves over time and that dyscalculic children are generally less efficient than children with typical development. For children with typical arithmetic development, robust effects of numerical distance, size congruity, and compatibility of ten and unit digits in two-digit numbers could be identified as early as the end of Grade 2. Only the distance effect for comparing symbolic representations of numerosities changed developmentally. Dyscalculic children did not show a size congruity effect but showed a more marked compatibility effect for two-digit numbers. We did not find strong evidence that dyscalculic children process numbers qualitatively differently from children with typical arithmetic development.     

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。     

Is adding 48 + 25 and 45 + 28 the same? How addend compatibility influences the strategy execution in mental addition

A recent study revealed that adults frequently start to add two two-digit numbers from the larger one, suggesting that addend magnitudes are compared at an early stage of processing. However, several studies showed that symbolic number comparison involves compatibility effects: Such numerical comparison is easier when the larger number also contains the larger unit (48_25) than in the opposite, incompatible case (45_28). In this context, whether the compatibility between tens and units across operands affects the execution of arithmetic-solving strategies remains an open question. In this study, we used two kinds of verbal protocols to assess how addend compatibility influences the implementation of magnitude-based strategies. We observed that participants started their computations from the larger operand more frequently when solving compatible additions than they did when solving incompatible ones. The presence of a compatibility effect extends the view that multidigit number processing is componential rather than holistic, even in an arithmetic task that did not explicitly require a number magnitude comparison. Further, the findings corroborate the notion that number magnitude is used in mental calculation and influences the way calculation strategies are implemented.     

When combined spatial polarities activated through spatio-temporal asynchrony lead to better mathematical reasoning for addition

Several recent studies have supported the existence of a link between spatial processing and some aspects of mathematical reasoning, including mental arithmetic. Some of these studies suggested that people are more accurate when performing arithmetic operations for which the operands appeared in the lower-left and upper-right spaces than in the upper-left and lower-right spaces. However, this cross-over Horizontality × Verticality interaction effect on arithmetic accuracy was only apparent for multiplication, not for addition. In these studies, the authors used a spatio-temporal synchronous operand presentation in which all the operands appeared simultaneously in the same part of space along the horizontal and vertical dimensions. In the present paper, we report studies designed to investigate whether these results can be generalized to mental arithmetic tasks using a spatio-temporal asynchronous operand presentation. We present three studies in which participants had to solve addition (Study 1a), subtraction (Study 1b), and multiplication (Study 2) in which the operands appeared successively at different locations along the horizontal and vertical dimensions. We found that the cross-over Horizontality × Verticality interaction effect on arithmetic accuracy emerged for addition but not for subtraction and multiplication. These results are consistent with our predictions derived from the spatial polarity correspondence account and suggest interesting directions for the study of the link between spatial processing and mental arithmetic performances.     

Strategies for written additions in adults

Studies about strategies used by adults to solve multi-digit written additions are very scarce. However, as advocated here, the specificity and characteristics of written calculations are of undeniable interest. The originality of our approach lies in part in the presentation of two-digit addition problems on a graphics tablet, which allowed us to precisely follow and analyse individuals’ solving process. Not only classic solution times and accuracy measures were recorded but also initiation times and starting positions of the calculations. Our results show that adults largely prefer the fixed columnar strategy taught at school rather than more flexible mental strategies. Moreover, the columnar strategy is executed faster and as accurately as other strategies, which suggests that individuals’ choice is usually well adapted. This result contradicts past educational intuitions that the use of rigid algorithms might be detrimental to performance. We also demonstrate that a minority of adults can modulate their strategy choice as a function of the characteristics of the problems. Tie problems and additions without carry were indeed solved less frequently through the columnar strategy than non-tie problems and additions with a carry. We conclude that the working memory demand of the arithmetic operation influences strategy selection in written problem-solving.     

Brain injury and movement recall: Preselection, active-passive and interference effects

Stroke patients with unilateral lesions were compared with age-controls and students on their ability to reproduce a terminal location established kinesthetically by a previous movement. Conditions for the criterion movement differed over active/passive and preselected/constrained (experiment 1) and whether the retention interval between the criterion and recall movements involved mental rehearsal of the criterion movement or yes/no responding to a mental arithmetic task (experiment 2). Whereas students showed more accurate recall with little effect of criterion movement condition, patient groups showed a preselection effect, but only with active movements. A preferred hand advantage observed for the patient controls did not occur with stroke patients, and prevention of mental rehearsal during the retention interval disrupted recall more for the stroke patients. These findings are interpreted in terms of hemisphere-specific coding strategies whose relative use depends on the attentional demands of the task.     

Differentiating two- from three-dimensional mental rotation training effects

Block videogame training has consistently demonstrated transfer effects to mental rotation tasks, yet how variations in training influence performance with different stimuli remains unclear. In this study, participants took mental rotation assessments before and after a 3-week training programme based on 2D or 3D block videogames. Assessments varied in terms of dimensionality (2D or 3D) and stimulus type (polygon or body). Increases in videogame scores throughout training were correlated with mental rotation improvements. In particular, 2D training led to improvements in 2D tasks, whereas 3D training led to improvements in both 2D and 3D tasks. This effect did not depend on stimulus type, demonstrating that training can transfer to different stimuli of identical dimensionality. Interestingly, traditional gender differences in 3D mental rotation tasks vanished after 3D videogame training, highlighting the malleability of mental rotation ability given adequate training. These findings emphasize the influence of dimensionality in transfer effects and offer promising perspectives to reduce differences in mental rotation via designed training programmes.     

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。     

MENTAL ARITHMETIC IN AFRICA AND AMERICA: STRATEGIES,PRINCIPLES, AND EXPLANATIONS*

Mental arithmetic abilities were studied among unschooled African adults and American college students. A set of problems tested the use of the four basic arithmetic operations. Performance was analyzed for: strategies, implicit arithmetic principles, and explicit explanations of the principles. Both groups showed accurate mental arithmetic strategies related to the base ten structure of their native counting systems. The American students' mental strategies additionally made use of algorithms based on written place values. Several principles were implicit in the calculation strategies of both groups, and both used these principles to short-cut calculations. Though the African subjects did not, the American subjects did describe these abstract principles in explicit common language or algebraic expressions. Relationships between schooling and cognition are discussed.     

Temporal tuning in the acquisition of cognitive skill

Thetemporal tuning hypothesis suggests that individuals adjust the timing of cognitive performances to achieve temporal coordination of mental processes and the data on which they operate, and that this adjustment becomes more precise with practice. Participants in two experiments performed self-paced multiple-step arithmetic tasks in which the information needed for each step was briefly displayed at the participants’ request. Timing constraints were manipulated by varying between subjects the delay between requests and displays of information. In Experiment 1, both operators and operands appeared step by step, and participants achieved a modest degree of temporal adjustment that did not change with practice. In Experiment 2, participants could preview operators while operands appeared step by step. In that experiment, participants achieved more precise temporal adjustment, and the amount of adjustment increased with practice. These results demonstrate the phenomenon of temporal tuning in symbolic cognitive skills and suggest some constraints on the ability to anticipate the time course of one’s mental processes.     

Holistic or compositional representation of two-digit numbers? Evidence from the distance, magnitude, and SNARC effects in a number-matching task

Whether two-digit numbers are represented holistically (each digit pair processed as one number) or compositionally (each digit pair processed separately as a decade digit and a unit digit) remains unresolved. Two experiments were conducted to examine the distance, magnitude, and SNARC effects in a number-matching task involving two-digit numbers. Forty undergraduates were asked to judge whether two two-digit numbers (presented serially in Experiment 1 and simultaneously in Experiment 2) were the same or not. Results showed that, when numbers were presented serially, unit digits did not make unique contributions to the magnitude and distance effects, supporting the holistic model. When numbers were presented simultaneously, unit digits made unique contributions, supporting the compositional model. The SNARC (Spatial-Numerical Association of Response Codes) effect was evident for the whole numbers and the decade digits, but not for the unit digits in both experiments, which indicates that two-digit numbers are represented on one mental number line. Taken together, these results suggested that the representation of two-digit numbers is on a single mental number line, but it depends on the stage of processing whether they are processed holistically or compositionally.     

Conceptual Knowledge,Procedural Knowledge,and Metacognition in Routine and Nonroutine Problem Solving

When, how, and why students use conceptual knowledge during math problem solving is not well understood. We propose that when solving routine problems, students are more likely to recruit conceptual knowledge if their procedural knowledge is weak than if it is strong, and that in this context, metacognitive processes, specifically feelings of doubt, mediate interactions between procedural and conceptual knowledge. To test these hypotheses, in two studies (Ns = 64 and 138), university students solved fraction and decimal arithmetic problems while thinking aloud; verbal protocols and written work were coded for overt uses of conceptual knowledge and displays of doubt. Consistent with the hypotheses, use of conceptual knowledge during calculation was not significantly positively associated with accuracy, but was positively associated with displays of doubt, which were negatively associated with accuracy. In Study 1, participants also explained solutions to rational arithmetic problems; using conceptual knowledge in this context was positively correlated with calculation accuracy, but only among participants who did not use conceptual knowledge during calculation, suggesting that the correlation did not reflect “online” effects of using conceptual knowledge. In Study 2, participants also completed a nonroutine problem-solving task; displays of doubt on this task were positively associated with accuracy, suggesting that metacognitive processes play different roles when solving routine and nonroutine problems. We discuss implications of the results regarding interactions between procedural knowledge, conceptual knowledge, and metacognitive processes in math problem solving.