Adults' strategies for simple addition and multiplication: verbal self-reports and the operand recognition paradigm

Accurate measurement of cognitive strategies is important in diverse areas of psychological research. Strategy self-reports are a common measure, but C. Thevenot, M. Fanget, and M. Fayol (2007) proposed a more objective method to distinguish different strategies in the context of mental arithmetic. In their operand recognition paradigm, speed of recognition memory for problem operands after solving a problem indexes strategy (e.g., direct memory retrieval vs. a procedural strategy). Here, in 2 experiments, operand recognition time was the same following simple addition or multiplication, but, consistent with a wide variety of previous research, strategy reports indicated much greater use of procedures (e.g., counting) for addition than multiplication. Operation, problem size (e.g., 2 + 3 vs. 8 + 9), and operand format (digits vs. words) had interactive effects on reported procedure use that were not reflected in recognition performance. Regression analyses suggested that recognition time was influenced at least as much by the relative difficulty of the preceding problem as by the strategy used. The findings indicate that the operand recognition paradigm is not a reliable substitute for strategy reports and highlight the potential impact of difficulty-related carryover effects in sequential cognitive tasks.     

Retrieval or nonretrieval strategies in mental arithmetic? An operand recognition paradigm

According to LeFevre, Sadesky, and Bisanz, averaging solution latencies in order to study individuals' arithmetic strategies can result in misleading conclusions. Therefore, in addition to classical chronometric data, they collected verbal reports and challenged the assumption that adults rely systematically on retrieval of arithmetic facts from memory to solve simple addition problems. However, Kirk and Ashcraft questioned the validity of such a methodology and concluded that a more appropriate method has to be found. Thus, we developed an operand recognition paradigm that does not rely on verbal reports or on solution latencies. In accordance with LeFevre et al., we show in a first experiment that adults resort to nonretrieval strategies to solve addition problems involving medium numbers. However, in a second experiment, we show that high-skilled individuals can solve the same problems using a retrieval strategy. The benefits of our paradigm to the study of arithmetic strategies are discussed.     

认知控制的层级性：来自任务切换的脑电证据*

认知控制的主要研究范式之一是任务切换。以往研究发现切换代价受到认知控制层级性的调节, 但鲜有研究探索这一调节过程的动态神经机制。本研究通过嵌套的线索-任务切换范式考察不同层级任务切换代价的差异及其神经机制。在实验中, 要求被试完成高低两种层级任务, 低层级任务要求被试判断数字大小(或奇偶); 高层级任务则须先加工数字的某一语义特征(如当前数字是否是偶数), 然后进行大小判断。行为结果表明, 高层级任务切换代价显著大于低层级任务切换代价。线索锁时的脑电结果表明, 层级效应最早出现于P2成分, 切换效应(切换与重复之差)在CNV成分上受到任务层级的调控, 反映了在任务目标重构阶段给予高层级任务更多的选择性注意以及更高的主动性控制。目标锁时的脑电结果表明, 在N2及慢波(SP)成分上, 高层级任务切换与重复的波幅差异相比低层级任务显著更大, 反映了在抑制旧任务集与重构新反应集的过程中增强的反应性控制。这些结果为任务设置重构论和认知控制的层级性提供了新的证据。     

Simple addition and multiplication: No comparison

Some models of memory for arithmetic facts (e.g., 5+2=7, 6×7=42) assume that only the max-left order is stored in memory (e.g., 5+2=7 is stored but not 2+5=7). These models further assume an initial comparison of the two operands so that either operand order (5+2 or 2+5) can be mapped to the common internal representation. We sought evidence of number comparison in simple addition and multiplication by manipulating size congruity. In number comparison tasks, performance costs occur when the physical and numerical size of numerals are incongruent (8 3) relative to when they are congruent (8 3). Sixty-four volunteers completed a number comparison task, an addition task, and a multiplication task with both size congruent and size incongruent stimuli. The comparison task demonstrated that our stimuli were capable of producing robust size congruity and split effects. In the addition and multiplication task, however, we were unable to detect any of the RT signatures of comparison or reordering processes despite ample statistical power: Specifically, there was no evidence of size congruity, split, or order effects in either the addition or multiplication data. We conclude that our participants did not routinely engage a comparison operation and did not consistently reorder the operands to a preferred orientation.     

Using dual-process theory and analogical transfer to explain facilitation on a hypothetico-deductive reasoning task

Using the analogical transfer paradigm, the present study investigated the competing explanations of Girotto and Legrenzi (Psychological Research 51: 129–135, 1993) and Griggs, Platt, Newstead, and Jackson (Thinking and Reasoning 4: 1–14, 1998) for facilitation on the SARS version of the THOG problem, a hypothetico-deductive reasoning task. Girotto and Legrenzi argue that facilitation is based on logical analysis of the task [System 2 reasoning in Evans’s (Trends in Cognitive Sciences 7: 454–459, 2003) dual-process account of reasoning] while Griggs et al. maintain that facilitation is due to an attentional heuristic produced by the wording of the problem (System 1 reasoning). If Girotto and Legrenzi are correct, then System 2 reasoning, which is volitional and responsible for deductive reasoning, should be elicited, and participants should comprehend the solution principle of the THOG task and exhibit analogical transfer. However, if Griggs et al. are correct, then System 1 reasoning, which is responsible for heuristic problem solving strategies such as an attentional heuristic, should occur, and participants should not abstract the solution principle and transfer should not occur. Significant facilitation (68 and 82% correct) was only observed for the two SARS source problems, but significant analogical transfer did not occur. This lack of transfer suggests that System 1 reasoning was responsible for the facilitation observed in the SARS problem, supporting Griggs et al.’s attentional heuristic explanation. The present results also underscore the explanatory value of using analogical transfer rather than facilitation as the criterion for problem understanding.     

Telling stories: the perils and promise of using verbal reports to study math strategies

The problem size effect in adult arithmetic performance is generally attributed to direct retrieval processes operating on a network representation in long-term memory. J. LeFevre and her colleagues (J. LeFevre, J. Bisanz, et al., 1996; J. LeFevre, G. S. Sadesky, & J. Bisanz, 1996) challenged this explanation using verbal report evidence that adults also use time consuming nonretrieval strategies to solve simple addition and multiplication. The authors replicated J. LeFevre and colleagues' methods, but added instructional biasing and silent control conditions to test these methods. Both reaction time and report results suggest that LeFevre and colleagues' conclusions about nonretrieval frequency may have been influenced by instructions that revealed the experimental hypothesis and affected participants' strategy reports. Obtaining evidence about adult strategy use in simple arithmetic will require understanding instructional demand and appropriate report methodology.     

Task switching: effects of practice on switch and mixing costs

In the task-switching paradigm, mixing costs indicate the performance costs to mix two different tasks, while switch costs indicate the performance costs to switch between two sequentially presented tasks. Applying tasks with bivalent stimuli and responses, many studies demonstrated substantial mixing and switch costs and a reduction of these costs as a result of practice. The present study investigates whether extensive practice of a task-switching situation including tasks with univalent stimuli eliminates these costs. Participants practiced switching between a visual and an auditory task. These tasks were chosen because they had shown eliminated performance costs in a comparable dual-task practice study (Schumacher et al. Psychol Sci 12:101–108, 2001). Participants either performed the tasks with univalent responses (i.e., visual-manual and auditory-verbal stimulus–response mappings) or bivalent responses (i.e., visual-manual and auditory-manual stimulus–response mappings). Both valence conditions revealed substantial mixing and switch costs at the beginning of practice, yet, mixing costs were largely eliminated after eight practice sessions while switch costs were still existent.     

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。     

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.     

Mental representations of arithmetic facts: Evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150–300 ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000 ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000 ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

Is there semantic interference in delayed naming?

The semantic interference effect in the picture-word interference task is interpreted as an index of lexical competition in prominent speech production models. Janssen, Schirm, Mahon, and Caramazza (2008) challenged this interpretation on the basis of experiments with a novel version of this task, which introduced a task-switching component. Participants either named the picture or read the word, depending on the word's color. Janssen et al. reported semantic interference in picture naming, regardless of whether the word appeared simultaneously with the picture (immediate naming) or 1,000 ms after the picture (delayed naming). Because picture name retrieval is completed in less than 1,000 ms, the finding in delayed naming was taken as evidence against the lexical competition account. In 3 sets of experiments conducted in German and English, we tested for semantic effects in Janssen et al.'s task-switching version and in the standard picture-word interference task. Using identical materials, we obtained sizeable interference effects in the standard task (Experiments 2, 4, and 6) but no effects in the task-switching version (Experiments 1, 3, and 5). When the word reading trials of the task-switching version were replaced with no-go trials (Experiment 7), semantic interference reemerged in immediate naming but was still absent in delayed naming. The experiments question the reliability of Janssen et al.'s critical finding and suggest that theoretical inferences about the origin of semantic effects in the standard picture-word interference task based on results from the task-switching version used by Janssen et al. are difficult to draw.     

Extensive practice does not eliminate human switch costs

Numerous human task-switching studies have shown that decision making that follows a task switch is slower and less accurate than that which follows a task repetition. Stoet and Snyder (2003a) tested humans and rhesus monkeys on a task-switching paradigm, but found, surprisingly, no switch costs in the monkeys. We hypothesized that the exceptional monkey behavior may have been due to the more extensive practice the monkeys received in comparison with human subjects. In the present study, we tested the hypothesis that extensive practice can abolish switch costs in humans. Four human subjects each performed 23,000 trials in a task-switching paradigm. We found that this amount of practice does not abolish switch costs.     

Task switching training effects are mediated by working-memory management

Task switching is an important executive function, and finding ways to improve it has become a major goal of contemporary scientists. Karbach and Kray (2009) found that training in the Alternating-Runs Task-Switching (AR-TS) paradigm (in which the task changed every second trial) reduced the costs of switching in untrained tasks, as well as led to far transfer to interference control ability and fluid intelligence. However, AR-TS is known to involve working memory updating (WMU). Therefore, we hypothesized that AR-TS training involves WMU and not task-switching proper. Participants were trained using Karbach and Kray's protocol. Results indicate a highly specific transfer pattern in which participants showed near transfer to switching cost in the AR-TS paradigm, but did not significantly improve in another version of the task switching paradigm in which the tasks were randomly ordered or a version in which the task changed every 3rd trial. The results suggest that what has been trained is not a broad task-switching ability but rather a specific skill related to the unique WMU requirements of the training paradigm.     

Mental representations of arithmetic facts: evidence from eye movement recordings supports the preferred operand-order-specific representation hypothesis

There are three main hypotheses about mental representations of arithmetic facts: the independent representation hypothesis, the operand-order-free single-representation hypothesis, and the operand-order-specific single-representation hypothesis. The current study used electrical recordings of eye movements to examine the organization of arithmetic facts in long-term memory. Subjects were presented single-digit addition and multiplication problems and were asked to report the solutions. Analyses of the horizontal electrooculograph (HEOG) showed an operand order effect for multiplication in the time windows 150-300?ms (larger negative potentials for smaller operand first problems than for larger operand first ones). The operand order effect was reversed in the time windows from 400 to 1,000?ms (i.e., larger operand first problems had larger negative potentials than smaller operand first problems). For addition, larger operand first problems had larger negative potentials than smaller operand first in the series of time windows from 300 to 1,000?ms, but the effect was smaller than that for multiplication. These results confirmed the dissociated representation of addition and multiplication facts and were consistent with the prediction of the preferred operand-order-specific representation hypothesis.     

Anticipation of cognitive demand during decision-making

An abundance of evidence indicates that action selection is guided, at least in certain contexts, by anticipation of action outcomes. In one particularly clear demonstration of this principle, Bechara and colleagues, studying a gambling task, observed phasic skin conductance responses just prior to actions associated with a relatively high risk of monetary loss (Bechara et al. in J Neurosci 19:5473–5481, 1999; Bechara et al. in Science 275:1293–1295, 1997; Bechara et al. in Cereb Cortex 6:215–225, 1996). In the present work, we tested for the same effect in a paradigm where choices resulted not in differential monetary outcomes, but in differential requirements for subsequent mental effort. In two experiments, we observed an anticipatory skin conductance response prior to actions resulting in a high level of cognitive demand. This finding indicates that requirements for effortful cognitive control are anticipated during action selection. We argue, based on convergent evidence, that such anticipation may not only trigger preparation; it may also play a direct role in effort-based decision-making.     

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。     

Interactive effects of numerical surface form and operand parity in cognitive arithmetic

In Experiment 1, adults (n = 48) performed simple addition, multiplication, and parity (i.e., odd-even) comparisons on pairs of Arabic digits or English number words. For addition and comparison, but not multiplication, response time increased with the number of odd operands. For addition, but not comparison, this parity effect was greater for words than for digits. In Experiment 2, adults (n = 50) solved simple addition problems in digit and word format and reported their strategies (i.e., retrieval or procedures). Procedural strategies were used more for odd than even addends and much more for word than digit problems. The results indicate that problem encoding and answer retrieval processes for cognitive arithmetic are interactive rather than strictly additive stages.     

Automatic activation of multiplication facts: Evidence from the nodes adjacent to the product

Adult observers are widely assumed to be equipped with a specific memory store containing arithmetic facts. The present study was aimed at exploring the possibility of obtaining an automatic activation of multiplication facts by using the number-matching paradigm (LeFevre, Bisanz, & Mrkonjic, 1988), in which mental arithmetic is task irrelevant. In particular, we were interested in exploring whether the nodes that precede or follow the product node in the multiplication table can also be automatically activated as a consequence of the mere presentation of two numbers. In Experiments 1 and 2, we showed that participants were slower in responding “no” to probes that were numbers adjacent to the product in the table related to the first operand of the initial pair than to probes that were unrelated to the initial pair. In Experiments 3 and 4, we showed a similar pattern for probes that were numbers adjacent to the product in the table related to the second operand of the initial pair. Experiments 5 and 6 ruled out alternative accounts and confirmed the results of the previous experiments. Taken together the present findings suggest that multiplication facts are stored in a highly related network in which activation spreads from the product node to adjacent nodes.     

Does the asymmetry effect inflate the temporal expansion of odd stimuli?

Tse et al. (Percept Psychophys 66:1171–1189, 2004) reported that participants tend to overestimate the duration of an oddball stimulus. The size of this effect was much larger than the one reported by Ulrich et al. (Psychol Res 70:77–87, 2006). More crucially, the effect in the study of Tse et al. already emerged at short standard durations, arguing against the arousal account proposed by Ulrich et al. This study investigated whether the oddball effect reported by Tse et al. was inflated by an asymmetry effect, that is, by an asymmetrical distribution of physical comparison durations around the duration of the standard. Experiment 1 demonstrated that an asymmetry effect could mimic an oddball effect. Therefore, we conducted Experiment 2 to replicate the results by Tse et al. employing not only their original procedure but also an adaptive procedure that rather avoids an asymmetry effect. Both psychophysical procedures in this experiment revealed an oddball effect, which, however, was of smaller size than the one reported by Tse et al. Furthermore, this effect emerged only at longer standard durations, which is in agreement with the arousal account as the underlying mechanism of this robust temporal illusion.     

Complexity factors in visuo-spatial working memory

Three experiments are presented that use a technique of selective interference--irrelevant pictures--to develop our understanding of visuo-spatial working memory. Visual noise fields are used as the irrelevant pictures. Using two related measures of simple visual complexity, the experiments demonstrate that the greater the complexity the greater the degree of interference, even within a paradigm where subjects are instructed to look at but otherwise ignore the irrelevant pictures. Both the number of dots and the density of the dots comprising the visual noise fields affect the degree of interference in a concurrent memory task. In addition, increasing the size of the field increases the amount of interference. It is argued that the results give insight into the properties of visual working memory and contribute to its theoretical development. For example, it is argued that the store is directly accessible by externally presented interference and that particular aspects of the noise displays cause interference with visual memory.