Why 2×2=5 looks so wrong: On the odd-even rule in product verification

Subjects judged whether the proposed product for two multipliers was true or false. Each equation could be judged as plausible or implausible because a product must be even if any of its multipliers is even; otherwise, it must be odd. A proposed product that violates the required odd-even status of the product—that is, deviates from the correct product, whether odd or even, by an odd value (e.g., splits of ±l, ±3, ±5)—can be rejected as false before normal processing is completed (i.e., before the correct product is retrieved and compared with the proposed product). Subjects were indeed faster and more accurate in rejecting a split of +1 or +3 than a split of +2 or +4, and this effect increased as the number of even multipliers in the pair increased. Subjects did not use the odd-even rule when either multiplier was 0 or 1 (Experiment 2), perhaps because other rules are available to bypass normal processing in those cases. A similar odd-even rule is used in sum verification (Krueger &Hallford), and use of the odd-even rules may help to explain why oddness and evenness are such salient features of numbers as abstract concepts (Shepard, Kilpatric, & Cunningham).     

Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification

The odd-even status of a sum depends on the odd-even status of its addends. A sum must be odd if an odd number of its addends are odd; else it must be even. A proposed sum that violates the required odd-even status of the sum—that is, deviates from the correct sum, whether odd or even, by an odd value (e.g., splits of ±1, ±3, ±5)—can be rejected immediately as false. Subjects in the present study did indeed use the odd-even rule in sum verification, because they were as fast and accurate in rejecting a split of ±1 as one of ±2, and a split of ±3 as one of ±4, even though a larger split generally is easier to reject (symbolic distance effect), and splits of ±3 and ±4 were rejected faster and more accurately than those of ±1 and ±2. Performance on separate odd-even tasks indicated that the odd-even properties of numbers and sums are readily available for use by adults, and that persons who do well on such tasks are especially likely to use the odd-even rule in sum verification.     

The odd-even effect in multiplication: Parity rule or familiarity with even numbers?

This study questions the evidence that a parity rule is used during the verification of multiplication. Previous studies reported that products are rejected faster when they violate the expected parity, which was attributed to the use of a rule (Krueger, 1986; Lemaire & Fayol, 1995). This experiment tested an alternative explanation of this effect: the familiarity hypothesis. Fifty subjects participated in a verification task with contrasting types of problems (even x even, odd x odd, mixed). Some aspects of our results constitute evidence against the use of the parity rule: False even answers were rejected slowly, even when the two operands were odd. We suggest that the odd-even effect in verification of multiplication could not be due to the use of the parity rule, but rather to a familiarity with even numbers (three quarters of products are indeed even).     

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.     

When is an odd number not odd? Influence of task rule on the MARC effect for numeric classification

When classifying numbers as odd or even with left-right keypresses, performance is better with the mapping even-right/odd-left than with the opposite mapping. This linguistic markedness association of response codes (MARC) effect has been attributed to compatibility between the linguistic markedness of stimulus and response codes. In 2 experiments participants made keypresses to the Arabic numerals or number words 3, 4, 8, and 9 using the odd-even parity rule or a multiple-of-3 rule, which yield the same keypress response for each stimulus. For both stimulus modes, the MARC effect was obtained with the odd-even rule, but tended to reverse with the multiple-of-3 rule. The reversal was complete for the right response, but task rule had little influence on the left response. The results are consistent with the view that the MARC effect and its reversal are caused by correspondence of the stimulus code designated as positive by the task rule with the positive-polarity right response code.     

Individual differences in conflict-monitoring: testing means and covariance hypothesis about the Simon and the Eriksen Flanker task

Conflict and context slow-down have been proposed as indicators of a conflict-monitoring system that initiates cognitive control to resolve conflicts in information processing. We investigated individual differences in conflict-monitoring and their associations with working memory (WM) and impulsivity. A total of 150 adults completed a Simon and an Eriksen flanker task, together with measures of WM and impulsivity. On both tasks, responses were slower and less accurate on incompatible than on compatible trials (conflict effect), and the conflict effect was larger when the preceding trial was compatible than when it was incompatible (context effect). Stimulus repetition did not explain the context effect. Individual differences could be attributed to three separable factors for each task: general speeded performance, conflict effect, and context effect. Evidence for across-task generality of these factors was sparse. Associations of these factors with impulsivity were weak at best. WM was correlated with general speed, and also with some but not all factors reflecting conflict-related processes.     

Can an old dog learn (and want to experience) new tricks? Cognitive training increases openness to experience in older adults

The present study investigated whether an intervention aimed to increase cognitive ability in older adults also changes the personality trait of openness to experience. Older adults completed a 16-week program in inductive reasoning training supplemented by weekly crossword and Sudoku puzzles. Changes in openness to experience were modeled across four assessments over 30 weeks using latent growth curve models. Results indicate that participants in the intervention condition increased in the trait of openness compared with a waitlist control group. The study is one of the first to demonstrate that personality traits can change through nonpsychopharmocological interventions.     

The cuing and priming of cognitive operations

The effect on performance of advance information about the specific cognitive operations to be performed on a stimulus was investigated in two experiments using cues (information useful and necessary for performance) and primes (information useful but not necessary for performance). In the first experiment, a cue presented prior to a digit stimulus indicated whether the digit was to be classified as odd or even, or low (less than 6) or high (greater than 5). Results showed that performance improved with increasing time between cue and digit and with practice. A stroop-like asymmetric interference of the low-high operation on the odd-even operation was also observed. In the second experiment, a prime that matched the cue, mismatched it, or was neutral was presented before the cue. Results showed facilitatory and inhibitory priming effects, as well as a distance effect based on the position of a digit relative to the boundary between 5 and 6. The results of both experiments were discussed in terms of a model based on relative processing speeds of the two relevant properties of the digits.     

Aging, working memory, and discrimination learning

Older adults easily learn probabilistic relationships between cues and outcomes when the predictive event is the occurrence of a cue, but have greater difficulty when the predictive event is the nonoccurrence of a cue (Mutter & Pliske, 1996; Mutter & Plumlee, 2004; Mutter & Williams, 2004). This study explored whether this age-related deficit occurs in a simpler learning context and whether it might be related to working memory (WM) decline. We gave younger and older adults simultaneous discrimination tasks that allowed us to compare their ability to learn deterministic relationships when either the occurrence (feature positive; FP) or the nonoccurrence (feature negative; FN) of a distinctive feature predicted reinforcement. We also included a group of younger adults who performed the discrimination tasks under a concurrent WM load. Both age and WM load had a detrimental effect on initial FP and FN discrimination; however, these effects persisted only in FN discrimination after additional learning experience. Learning predictive relationships requires inductive reasoning processes that apparently do not operate as efficiently in individuals with reduced WM capacity. The impact of WM decline may ultimately be greater for negative cue-outcome relationships because learning these relationships requires more difficult inductive reasoning processes, which place greater demands on WM.     

Magnitude processing and complex calculation is negatively impacted by mathematics anxiety while retrieval‐based simple calculation is not

Mathematics anxiety (MA) refers to the experience of negative affect when engaging in mathematical activity. According to Ashcraft and Kirk (2001), MA selectively affects calculation with high working memory (WM) demand. On the other hand, Maloney, Ansari, and Fugelsang (2011) claim that MA affects all mathematical activities, including even the most basic ones such as magnitude comparison. The two theories make opposing predictions on the negative effect of MA on magnitude processing and simple calculation that make minimal demands on WM. We propose that MA has a selective impact on mathematical problem solving that likely involves processing of magnitude representations. Based on our hypothesis, MA will impinge upon magnitude processing even though it makes minimal demand on WM, but will spare retrieval‐based, simple calculation, because it does not require magnitude processing. Our hypothesis can reconcile opposing predictions on the negative effect of MA on magnitude processing and simple calculation. In the present study, we observed a negative relationship between MA and performance on magnitude comparison and calculation with high but not low WM demand. These results demonstrate that MA has an impact on a wide range of mathematical performance, which depends on one's sense of magnitude, but spares over‐practiced, retrieval‐based calculation.     

Aging,working memory,and discrimination learning

Older adults easily learn probabilistic relationships between cues and outcomes when the predictive event is the occurrence of a cue, but have greater difficulty when the predictive event is the nonoccurrence of a cue (Mutter & Pliske, 1996; Mutter & Plumlee, 2004; Mutter & Williams, 2004). This study explored whether this age-related deficit occurs in a simpler learning context and whether it might be related to working memory (WM) decline. We gave younger and older adults simultaneous discrimination tasks that allowed us to compare their ability to learn deterministic relationships when either the occurrence (feature positive; FP) or the nonoccurrence (feature negative; FN) of a distinctive feature predicted reinforcement. We also included a group of younger adults who performed the discrimination tasks under a concurrent WM load. Both age and WM load had a detrimental effect on initial FP and FN discrimination; however, these effects persisted only in FN discrimination after additional learning experience. Learning predictive relationships requires inductive reasoning processes that apparently do not operate as efficiently in individuals with reduced WM capacity. The impact of WM decline may ultimately be greater for negative cue–outcome relationships because learning these relationships requires more difficult inductive reasoning processes, which place greater demands on WM.     

When less is more and when more is more: The mediating roles of capacity and speed in brain-behavior efficiency

An enduring enterprise of experimental psychology has been to account for individual differences in human performance. Recent advances in neuroimaging have permitted testing of hypotheses regarding the neural bases of individual differences but this burgeoning literature has been characterized by inconsistent results. We argue that careful design and analysis of neuroimaging studies is required to separate individual differences in processing capacity from individual differences in processing speed to account for these differences in the literature. We utilized task designs which permitted separation of processing capacity influences on brain-behavior relationships from those related to processing speed. In one set of studies, participants performed verbal delayed-recognition tasks during blocked and event-related fMRI scanning. The results indicated that those participants with greater working memory (WM) capacity showed greater prefrontal cortical activity, strategically capitalized on the additional processing time available in the delay period, and evinced faster WM-retrieval rates than low-capacity participants. In another study, participants performed a digit-symbol substitution task (DSST) designed to minimize WM storage capacity requirements and maximize processing speed requirements during fMRI scanning. In some prefrontal cortical (PFC) brain regions, participants with faster processing speed showed less PFC activity than slower performers while in other PFC and parietal regions they showed greater activity. Regional-causality analysis indicated that PFC exerted more influence over other brain regions for slower than for faster individuals. These results support a model of neural efficiency in which individuals differ in the extent of direct processing links between neural nodes. One benefit of direct processing links may be a surplus of resources that maximize available capacity permitting fast and accurate performance.     

不同注意条件下的空间－数字反应编码联合效应

考察在无线索、内源性线索与外源性线索时不同符号数字在注意与非注意条件下的空间－数字的反应编码联合效应（Spatial Numerical Association of Response Codes,简称SNARC效应）。采用1到9的中文与阿拉伯数字为材料,以判断数字奇偶为任务。实验结果表明：⑴无线索时注意条件下阿拉伯和中文数字都出现了SNARC效应,而非注意条件下则都没有出现,并且受影响的主要是较大的数字（8、9）;⑵外源性线索和内源性线索时,我们得到一个逐渐递减的SNARC效应,受影响的也主要是较大的数字（8、9）。在内源性线索的注意条件阿拉伯和中文数都有SNARC效应,而在非注意条件只有阿拉伯数有SNARC效应;在外源性线索的注意条件只有阿拉伯数有SNARC效应,而在非注意条件阿拉伯和中文都没有SNARC效应,说明外源性注意的影响比内源性注意更大,中文数字所受的影响比阿拉伯数字更大。     

The odd-even effect in addition: an analysis per problem type

The odd-even effect is a well documented finding in the literature on mental arithmetic, at least for multiplication. It implies that false answers with the same parity as the correct answer are rejected more slowly than false answers with a different parity. For addition, this effect is not so well documented. The study by Krueger and Hallford (1984) is the only one that investigated odd-even effects for addition. However, they did not investigate odd-even effects per problem, even though there are indications that problem type can modulate odd-even effects for multiplication (Lemaire & Reder, 1999). Therefore, we wanted to get more insight into odd-even effects for addition by investigating odd-even effects per problem type. Our results extended the findings of Krueger and Hallford. First of all, we found an interaction between split and problem type. The most important and new result of present study, however, was a strong parity effect for E + E problems. We discuss our results in terms of two alternative explanations for odd-even effects, namely use of a parity rule on the one hand and familiarity with even outcomes on the other.     

Working memory binding of visual object features in older adults

Accurate mental representation of visual stimuli requires retaining not only the individual features but also the correct relationship between them. This associative process of binding is mediated by working memory (WM) mechanisms. The present study re-examined reports of WM-related binding deficits with aging. In Experiment 1, 31 older and 31 younger adults completed a visual change detection task with feature–location relations presented either simultaneously or sequentially; the paradigm was also designed specifically to minimize the impact of lengthy retention intervals, elaborative rehearsal, and processing demands of multi-stimulus probes. In Experiment 2, 38 older and 42 younger adults completed a modified task containing both feature–location relations and feature–feature conjunctions. In Experiment 1 although feature–location binding was more difficult with sequential compared with simultaneous presentation, the effect was independent of age. In Experiment 2 while older adults were overall slower and less accurate than young adults, there were no age-specific deficits in WM binding. Overall, after controlling for methodological factors, there was no evidence of an age-related visual WM binding deficit for surface or location features. However, unlike younger adults, older adults appeared less able to restrict processing of irrelevant features, consistent with reported declines with age in strategic capacities of WM.     

Resisting emotional interference: Brain regions facilitating working memory performance during negative distraction

Survival-relevant information has privileged access to our awareness even during active cognitive engagement. Previous work has demonstrated that during working memory (WM) negative emotional distraction disrupts activation in the lateral prefrontal regions while also engaging the amygdala. Here, using slow eventrelated fMRI, we replicate and extend previous work examining the effect of negative emotional distraction on WM: (1) We demonstrate that prefrontal regions showed activation differences between correct and incorrect trials during negative, but not neutral, distraction. Specifically, frontopolar prefrontal cortex showed more deactivation for incorrect trials faced with negative distraction, whereas ventrolateral prefrontal regions showed less activation; (2) individual differences in amygdala activity predicted WM performance during negative as well as neutral distraction, such that lower activity predicted better performance; and (3) amygdala showed negative correlations with prefrontal and parietal cortical regions during resting state. However, during negative distraction, amygdala signals were more negatively correlated with prefrontal cortical regions than was found for resting state and neutral distraction. These results provide further evidence for an inverse relationship between dorsal prefrontal cortical regions and the amygdala when processing aversive stimuli competes with ongoing cognitive operations, and further support the importance of the prefrontal cortex in resisting emotional interference. Supplemental materials associated with this article may be downloaded from http://cabn.psychonomic-journals .org/content/supplemental.     

The effect of saccades on number processing

Recent research has shown that saccadic eye movements interfere with dorsal-stream tasks such as judgments of object orientation, but not with ventral-stream tasks such as object recognition. Because saccade programming and execution also rely on the dorsal stream, it has been hypothesized that cognitive saccadic suppression occurs as a result of dual-task interference within the dorsal stream. Judging whether one number is larger or smaller than another (magnitude comparison) is a dorsal-stream task that relies especially on the right parietal cortex. In contrast, judging whether a number is odd or even (parity judgment) does not involve the dorsal stream. In the present study, one group of subjects judged whether two-digit numbers were greater than or less than 65, whereas another group judged whether two-digit numbers were odd or even. Subjects in both groups made these judgments while making no, short, or long saccades. Saccade distance had no effect on parity judgments, but reaction times to make magnitude comparison judgments increased with saccade distance when the eyes moved from right to left. Because the right parietal cortex is instrumental in generating leftward saccades, these results provide further evidence for the hypothesis that cognitive suppression during saccades occurs as a result of dual-task interference within the dorsal stream.     

Visual working memory is disrupted by covert verbal retrieval

If working memory (WM) depends on a central resource—as is posited in some theories, but not in others—it should be possible to observe interference between tasks that share few features with each other. We investigated whether interference with WM for visual arrays would occur, even if the interfering task required neither visual processing nor overt responding. In an auditory-verbal interfering task, a response was to be made if a word was recognized as having come from one of two prelearned lists, but not from the other list. As compared with nonretrieval control conditions, even covert verbal memory retrieval disrupted the storage of visual items held in WM. A second experiment ruled out verbal recoding of the visual arrays as the basis of interference. The results indicate that visual WM and verbal long-term retrieval share a central resource (e.g., attention).     

内源性注意和外源性注意对中小学生SNARC效应的影响

采用内源性注意和外源性注意实验范式,以判断数字奇偶为任务,材料为1～9的阿拉伯数字,考察内源性和外源性线索不同注意条件下中小学生的数字加工SNARC效应。结果发现:(1)内源性线索和外源性线索条件下,中小学生在数字加工任务中所表现出来的SNARC效应随着年龄增长逐渐增大;(2)内源性线索和外源性线索注意条件下,各年级学生均表现出显著的SNARC效应;(3)内源性线索非注意条件下,中学生能够表现出显著的SNARC效应,但小学生的SNARC效应不明显;外源性线索非注意条件下,高中生能够表现出显著的SNARC效应,但小学生和初中生的SNARC效应不明显,说明外源性注意对中小学生数字加工SNARC效应的影响比内源性注意大,且对SNARC效应的影响主要表现在较大的数字8和9上。     

The influence of an external symbol system on number parity representation, or What’s odd about 6?

The aim of this study was to investigate whether language-specific properties influence mental number processing. German Sign Language (DGS) numbers differ from those in spoken German not only in terms of modality but also in their basic language structure. A group of 20 congenitally deaf German signers participated in a number parity (odd/even) judgment task with DGS and printed German number words. The results indicated that two-handed DGS number signs are processed in a decomposed way. This language-specific effect also generalized to another linguistic number notation, German number words, but not to Arabic digit notation. These differences are discussed with respect to two possible routes to number parity.