Bidirectional links in the network of multiplication facts

In three experiments, we tested the hypothesis that activation of multiplication operand nodes (e.g., 3 and 8) can occur through presentation of their product (e.g., 24). In Experiments 1 and 2 we found activation of the operands when the product was presented as a cue in a number-matching task. In Experiment 3, activation also occurred in a parity-matching task, where the product (24) was not relevant to the parity matching on its operands (3 and 8). We concluded that bidirectional links exist among the operands and their product for multiplication problems and these links can be activated in a purely stimulus-driven manner. We suggest this may constitute the basis for the solution of simple divisions by mediation through the complementary multiplication facts.     

Production,verification, and priming of multiplication facts

In the arithmetic-verification procedure, subjects are presented with a simple equation (e.g., 4 × 8 = 24) and must decide quickly whether it is true or false. The prevailing model of arithmetic verification holds that the presented answer (e.g., 24) has no direct effect on the speed and accuracy of retrieving an answer to the problem. It follows that models of the retrieval stage based on verification are also valid models of retrieval in the production task, in which subjects simply retrieve and state the answer to a given problem. Results of two experiments using singledigit multiplication problems challenge these assumptions. It is argued that the presented answer in verification functions as a priming stimulus and that on “true” verification trials the effects of priming are sufficient to distort estimates of problem difficulty and to mask important evidence about the nature of the retrieval process. It is also argued that the priming of false answers that have associative links to a presented problem induces interference that disrupts both speed and accuracy of retrieval. The results raise questions about the interpretation of verification data and offer support for a network-interference theory of the mental processes underlying simple multiplication.     

The representation of multiplication and division facts in memory

Recently, using a training paradigm, Campbell and Agnew (2009) observed cross-operation response time savings with nonidentical elements (e.g., practice 3 + 2, test 5 - 2) for addition and subtraction, showing that a single memory representation underlies addition and subtraction performance. Evidence for cross-operation savings between multiplication and division have been described frequently (e.g., Campbell, Fuchs-Lacelle, & Phenix, 2006) but they have always been attributed to a mediation strategy (reformulating a division problem as a multiplication problem, e.g., Campbell et al., 2006). Campbell and Agnew (2009) therefore concluded that there exists a fundamental difference between addition and subtraction on the one hand and multiplication and division on the other hand. However, our results suggest that retrieval savings between inverse multiplication and division problems can be observed. Even for small problems (solved by direct retrieval) practicing a division problem facilitated the corresponding multiplication problem and vice versa. These findings indicate that shared memory representations underlie multiplication and division retrieval. Hence, memory and learning processes do not seem to differ fundamentally between addition-subtraction and multiplication-division.     

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.     

Learning multiplication facts: a study of children taught by discovery methods in England

The development of multiplication skills was examined in a group of children ages 8 to 12 years who were taught by discovery methods. Strategies used by the children included direct retrieval, retrieval + calculation, and counting-in-series. Repeated addition was not observed. Retrieval was the fastest and least error-prone strategy; counting-in-series was the slowest and most error prone. Children ages 8 and 9 years used mainly mixed strategies. Children ages 10 to 12 years used mainly retrieval or retrieval and calculation for low operands, but reverted to back-up strategies for high operands based on the strategies available for low operands. There was a general shift away from less effective strategies across ages 8 to 12 years but, by the end of the primary school (age 11 years), relatively few children used the most effective strategy of retrieval for all operands. The development of effective strategies was related to nonverbal reasoning ability and to working memory capacity. The results are considered with reference to experiential and pedagogical models of multiplication.     

Effects of problem format on division and multiplication performance: division facts are mediated via multiplication-based representations

In 2 experiments participants solved division problems presented in multiplication-based formats (e.g., 8 x _ = 72) more quickly than division problems presented in division-based formats (e.g., 72 / 8 = _). In contrast, participants solved multiplication problems presented in a division-based format (e.g., _ / 8 = 9) slowly and made many errors. In both experiments, the advantage for multiplication-based formats on division problems was found only for large problems (i.e., those with products or dividends greater than 25). These findings provide support for the view that large single-digit division facts are mediated via multiplication-based representations and that multiplication is the primary mode of representation for both division and multiplication facts.     

The representation of multiplication facts: developmental changes in the problem size, five, and tie effects

In this study, we investigated the development of basic effects that have been found in single-digit multiplication arithmetic: the problem size, five, and tie effects. Participants (9-, 10-, and 11-year-olds and adults) performed a production task on simple multiplication. The procedure replicated study [Canadian Journal of Psychology, Vol. 39, pp. 338-366], but the results show that the gradual decrease of the problem size effect ends in sixth grade. We report analyses on raw latencies and state trace analyses that take into account reaction time scaling as a function of age. The results show that 11-year-olds do not differ significantly from adults on any of the three effects. Before 11 years of age, interesting developmental changes occur.     

Cognitive compensations for blindness in children: an investigation using odour naming

Historically, blindness has been associated with compensation for the loss of vision by the other senses. However, research to date has focused on perceptual compensations, largely ignoring possible cognitive compensations. We explored the notion that cognitive skills of blind children may facilitate performance in apparently perceptual tasks, by investigating the cognitive factors related to naming a familiar odour. Eighty-three children participated in olfactory and cognitive tasks (thirty-two early-blind, five late-blind, fourteen low-vision, and thirty-two sighted). In the olfactory tasks, the early-blind children performed significantly better than the sighted children on the odour-naming task but not on the odour-sensitivity task. From the cognitive tasks, scores on a nonvisualisable word-pairs task and a sound-word-pairs task were significantly higher for early-blind children and were highly correlated with odour-naming score. The early-blind children outperformed the sighted controls on a task of directed attention. The groups did not differ on memory for a story or for visualisable word pairs. The results suggest that blind children enjoy an advantage in tasks that assess nonvisual memory for paired associates and directed attention, and that superiority on these tasks facilitates performance in the odour-naming task. Other data suggest that sighted children rely on visualisation as a strategy to aid their performance on the cognitive tasks, and are disadvantaged when these strategies cannot be utilised.     

Interpretive biases in chronic insomnia: an investigation using a priming paradigm

Disorder-congruent interpretations of ambiguous stimuli characterize several psychological disorders and have been implicated in their maintenance. Models of insomnia have highlighted the importance of cognitive processes, but the possibility that biased interpretations are important has been minimally investigated. Hence, a priming methodology was employed to investigate the presence of an interpretive bias in insomnia. A sample of 78 participants, differing in the presence of a diagnosis of insomnia, severity of sleep disturbance, and sleepiness, was required to read ambiguous sentences and make a lexical decision about target words that followed. Sleepiness at the time of the experiment was associated with the likelihood with which participants made insomnia and threat consistent interpretations of ambiguous sentences. The results suggest that there is a general bias towards threatening interpretations when individuals are sleepy and suggests that cognitive accounts of insomnia require revision to include a role for interpretative bias when people are sleepy. Future research is required to investigate whether this interpretive bias plays a causal role in the maintenance of insomnia.     

A software system for the collection of retrospective protocols prompted by eye fixations

A software system is capable of recording eye position, reducing these data to a sequence of eye fixations, and immediately replaying this fixation sequence superimposed over the original stimulus. The replay can be controlled by the subject to produce various speeds, to pause, or to back up to earlier fixations. The purpose of the replay software is to enable the eye fixations to prompt a retrospective verbal protocol. Such a prompted protocol should reduce the forgetting and fabrication problems that are often associated with retrospective protocols. In a preliminary experimental test, the prompted protocol method yielded longer protocols and contained more words than those generated by either the concurrent or retrospective protocol methods.     

Attentional biases in social anxiety: an investigation using the inattentional blindness paradigm

One line of research has examined attentional bias as a potential maintenance factor in social anxiety using cognitive experiment paradigms. The present study sought to examine the utility of the inattentional blindness (IB) paradigm for assessing attentional bias in social anxiety. Unlike other existing paradigms such as the emotional Stroop or dot-probe tasks, the IB paradigm has the advantage of eliminating the individual's expectation and intention to search for social cues, which would reduce strategic or effortful responses. Two independent experiments were conducted using college students scoring high or low on the Fear of Negative Evaluation Scale. In both Experiments 1 and 2, participants were randomized to one of three IB experiment conditions, in which a positive face, a negative face, or a neutral item was unexpectedly presented, in the presence/absence of a bogus-speech threat. The overall pattern of our data suggests the presence of hypervigilant attentional processing in social anxiety. The IB paradigm appears to be a useful addition to existing experiment paradigms for investigating attentional bias in social anxiety and perhaps other psychopathology.     

The structure of short-term memory scanning: an investigation using response time distribution models

A classic question in cognitive psychology concerns the nature of memory search in short-term recognition. Despite its long history of investigation, however, there is still no consensus on whether memory search takes place serially or in parallel or is based on global access. In the present investigation, we formalize a variety of models designed to account for detailed response time distribution data in the classic Sternberg (Science 153: 652-654, 1966) memory-scanning task. The models vary in their mental architectures (serial exhaustive, parallel self-terminating, and global access). Furthermore, the component processes within the architectures that make match/mismatch decisions are formalized as linear ballistic accumulators (LBAs). In fast presentation rate conditions, the parallel and global access models provide far better accounts of the data than does the serial model. LBA drift rates are found to depend almost solely on the lag between study items and test probes, whereas response thresholds change with memory set size. Under slow presentation rate conditions, even simple versions of the serial-exhaustive model provide accounts of the data that are as good as those of the parallel and global access models. We provide alternative interpretations of the results in our General Discussion.     

Lying and executive control: an experimental investigation using ego depletion and goal neglect

This study investigated whether lying requires executive control using a reaction-time based lie test. We hypothesized that (1) goal neglect induced by a long response-stimulus interval (RSI; 5-8s) would make lying harder relative to a short RSI (.2 s) that promoted attentional focus, and (2) participants whose executive control resources were depleted by an initial executive control task would experience more difficulty to lie than control participants who performed a task that required little executive control. Across two experiments, the ego depletion manipulation did not reliably affect lying. Both experiments revealed that the cognitive cost associated with lying was larger for the long compared to the short RSI. This finding supports the idea that lying requires more executive control than truth telling. The manipulation of RSI may provide a simple, yet effective means to improve lie detection accuracy.     

Internal versus external attention in social anxiety: an investigation using a novel paradigm

Several cognitive models propose that social anxiety is associated with increased self-focused attention. Indirect evidence for this hypothesis has been provided by questionnaire studies, and by cognitive psychology paradigms that have demonstrated reduced processing of external information during feared social-evaluative situations. However, no studies have simultaneously measured on-line attention to internal and external events. A probe detection task that aimed to measure the balance of attention between internal and external stimuli was developed. High and low socially anxious individuals were instructed to detect two probes. The external probe was superimposed on pictures of faces (happy, neutral, angry) or household objects that were presented on a VDU. The 'internal' probe was a pulse to the finger which participants were led to believe represented significant changes in their physiology. Compared to low speech anxious individuals, high speech anxious individuals showed an internal attentional bias, that was specific to conditions of social-evaluative threat.     

Dissociating expectancy of shock and changes in skin conductance: an investigation of the Perruchet effect using an electrodermal paradigm

Is human Pavlovian conditioning driven by a unitary, propositional system (as claimed by Mitchell, De Houwer, & Lovibond, 2009) or by dual systems; one under conscious control, symbolic in nature, and requiring effort to deploy, and the other utilizing associative processes and automatic in its operation (McLaren, Green, & Mackintosh, 1994)? Past research has suggested that for electrodermal conditioning to occur in humans, conscious awareness of the contingencies is necessary to produce conditioned responding (e.g., Hinchy, Lovibond, & Ter-Horst, 1995), as predicted by single process theories that attribute the conditioned response (CR) to conscious expectancy of the shock. In this article, the authors examined the Perruchet effect (Perruchet, 1985), using an electrodermal paradigm to determine whether there is any role for associative processes in human electrodermal conditioning. The authors attempted to replicate the basic effect, whereby expectancy of an unconditioned stimulus (US) increases over a run of nonreinforced trials while the CR to the conditional stimulus (CS) declines, and the complementary pattern in which expectancy decreases over a run of reinforced trials while the CR to the CS grows in strength. In line with these patterns, the change in skin conductance response (our CR) as a function of US run length was found to follow a linear trend opposite to that of conscious expectancy of shock with respect to US run length. This dissociation supports a dual-processing system account of human Pavlovian conditioning, with conscious, controlled processes governing expectancy (and subject to the gambler's fallacy), whereas automatic, associative processes determine at least some of the strength of the CR to the CS.     

Gray's three-arousal model: an empirical investigation

We evaluated the validity of Gray's and Fowles' three-arousal model in two studies of criminal offenders using a continuous motor task involving rewards and punishments. Consistent with predictions for the behavioral approach/activation system (BAS), offenders displayed significant (p<0.00001) increases in response time and heart rate (HR) from a no-incentive practice phase to a reward-only (experiment 1) and active-avoidance (experiment 2) phase. Trait impulsivity was correlated with the response time index of BAS activation in experiment 1 but not experiment 2. Consistent with predictions for the behavioral inhibition system (BIS), offenders showed a significant increase in number of skin conductance responses (SCRs) (p<0.05) from a reward-only to a mixed-incentive phase in experiment 1 and a significant increase in SC amplitude to punishment cues in both experiments. Consistent with predictions for the dynamics of the model, participants showed significant slowing of response time (p<0.0001) from reward-only (experiment 1) or active avoidance (experiment 2) to mixed-incentive phases despite showing an initial tendency toward response facilitation to the onset of the punishment cue signifying the beginning of the mixed-incentive phases. Participants also showed significant (p<0.002) decreases in HR between these phases in both studies, but this effect was only evident on trial 1. The BIS-influenced response time and HR indices were significantly (p<0.05) correlated with anxiety in experiment 1, but unexpectedly, anxiety was not correlated with SC indices of the BIS in either study. Although much of the data support the validity of the Gray/Fowles model, particular findings suggest that further refinement of this theory is indicated.