The role of short-term working memory in mental arithmetic

A series of experiments explored the role of information storage in working memory in performing mental arithmetic. Experiment 1 assessed the strategies people report for solving auditorily presented multidigit problems such as 325 + 46. As expected, all subjects reported breaking down the problems into a series of elementary stages, though there were considerable individual differences with regard to the order of their execution. Strategies of this type necessitate both the temporary storage of information and the mobilization of long-term knowledge. Experiments 2 and 3 examined the effects of delaying the output of individual partial results on calculation accuracy and showed that interim information is forgotten if it is not utilized immediately. Experiment 4 showed that forgetting the initial information is also a source of error and suggested that forgetting increases as a function of the number of calculation stages intervening between initial presentation and subsequent utilization of information. Two simple quantitative models were derived from a general task analysis, one of which assumed a decay process in working storage and the other no decay. The decay model gave a reasonable fit to data from Experiments 2–4, and in doing so it coped appreciably well with the effects of a large variety of task variables (e.g., carrying, the provision of written notes, calculation strategy, output order). The decay model is a tractable analysis of a complex task, and it is suggested that similar analyses may prove fruitful for other problem-solving activities which involve the use of working memory.     

Dissociating mental transformations and visuo-spatial storage in working memory: evidence from representational neglect

A study is reported of visuo-spatial working memory in two individuals suffering from a cognitive deficit known as unilateral spatial neglect, and seven healthy control participants. Both patients have difficulties reporting details on the left side of imaged representations, and one has an additional difficulty with perceptual input to the left of his body midline. All participants were asked to report the location and identity of objects presented in novel 2 x 2 arrays that were either present throughout or were described orally by the experimenter, with no visual input. On half of the trials, the report was to be made from the opposite perspective, requiring 180 degree mental rotation of the mentally represented array. The patients show an impaired ability to report details from the presented or the imagined left, but had no difficulty with mental rotation. Results point to a clear separation between the processes of perception and those of visuo-spatial working memory. Results also suggest that the patients might be suffering from damage to the system used for holding visuo-spatial representations rather than a difficulty with attending to elements of that representation.     

Separate roles for executive and phonological components of working memory in mental arithmetic

A dual-task methodology was used to investigate the roles played by executive and phonological aspects of working memory in mental arithmetic. Experiment 1 showed that suppression of articulation impaired the ability to add a pair of briefly presented three-digit numbers. Suppression had no effect when the need to store temporarily was minimized by making the numbers visible throughout calculation. Experiment 2 showed that disrupting executive processes by requiring concurrent performance of a Trails task impaired the ability to add numbers that remained permanently visible. Performance on the Trails task deteriorated as the number of carry operations in the addition increased. Experiment 3 showed that this decline in Trails performance was not simply due to the extra time taken by carrying. These and other features of the results suggest that the carrying component of mental arithmetic places substantial demands on executive processes, whereas the need to retain problem information is met by the phonological loop. The results are consistent with an interpretation of executive processes according to which there is a limit on the capacity to inhibit strongly primed routine operations.     

Representation and working memory in early arithmetic

Working memory has been implicated in the early acquisition of arithmetic skill, but the relations among different components of working memory, performance on different types of arithmetic problems, and development have not been explored. Preschool and Grade 1 children completed measures of phonological, visual-spatial, and central executive working memory, as well as nonverbal and verbal arithmetic problems, some of which included irrelevant information. For preschool children, accuracy was higher on nonverbal problems than on verbal problems, and the best and only unique predictor of performance on the standard nonverbal problems was visual-spatial working memory. This finding is consistent with the view that most preschoolers use a mental model for arithmetic that requires visual-spatial working memory. For Grade 1 children, performance was equivalent on nonverbal and verbal problems, and phonological working memory was the best predictor of performance on standard verbal problems. For both age groups, problems with added irrelevant information were substantially more difficult than standard problems, and in some cases measures of the central executive predicted performance. Assessing performance on different components of working memory in conjunction with different types of arithmetic problems provided new insights into the developing relations between working memory and how children do arithmetic.     

The contribution of working memory resources in the verification of simple mental arithmetic sums

The present study replicated the investigations of Lemaire, Abdi, and Fayol with some modifications: the random time interval generation (RIG) task was used and the stimuli were created in another way. The results provide additional evidence for the crucial role of the central executive in the speed of solving both true and false sums and for the role of the phonological loop in solving false sums. However, the findings concerning the role of this slave system in solving true sums were different. Possible explanations and limitations of these results are discussed. Received: 16 April 1998 / Accepted: 20 July 1998     

Visuospatial and verbal memory in mental arithmetic

Working memory allows complex information to be remembered and manipulated over short periods of time. Correlations between working memory and mathematics achievement have been shown across the lifespan. However, only a few studies have examined the potentially distinct contributions of domain-specific visuospatial and verbal working memory resources in mental arithmetic computation. Here we aimed to fill this gap in a series of six experiments pairing addition and subtraction tasks with verbal and visuospatial working memory and interference tasks. In general, we found higher levels of interference between mental arithmetic and visuospatial working memory tasks than between mental arithmetic and verbal working memory tasks. Additionally, we found that interference that matched the working memory domain of the task (e.g., verbal task with verbal interference) lowered working memory performance more than mismatched interference (verbal task with visuospatial interference). Findings suggest that mental arithmetic relies on domain-specific working memory resources.     

Effect of working memory training on working memory,arithmetic and following instructions

Mathematical ability is dependent on specific mathematical training but also associated with a range of cognitive factors, including working memory (WM) capacity. Previous studies have shown that WM training leads to improvement in non-trained WM tasks, but the results regarding transfer to mathematics are inconclusive. In the present study, 176 children with WM deficits, aged 7–15 years performed 5 weeks of WM training. During the training period, they were assessed five times with a test of complex WM (the Odd One Out), a test of remembering and following instructions and a test of arithmetic. The improvements were compared to the performance of a control group of 304 typically developing children aged 7–15 years who performed the same transfer tasks at the same time intervals, but without training. The training group improved significantly more than the control group on all three transfer tests (all p < 0.0001), after correction for baseline performance, age and sex. The effect size for mathematics was small and the effect sizes for the WM tasks were moderate to large. The transfer increased linearly with the amount of training time and correlated with the amount of improvement on the trained tasks. These results confirm previous findings of training-induced improvements in non-trained WM tasks including the ability to follow instructions, but extend previous findings by showing improvements also for arithmetic. This is encouraging regarding the potential role of cognitive training for education, but it is desirable to find paradigms that would enhance the effect of the training on mathematics. One of the future challenges for studying training effects is combining large sample sizes with high quality and compliance, to detect relevant but smaller effects of cognitive training.     

Goal-directed access to mental objects in working memory: The role of task-specific feature retrieval

In the present study, we examined the hypothesis of task-specific access to mental objects from verbal working memory. It is currently assumed that a mental object is brought into the focus of attention in working memory by a process of object selection, which provides this object for any upcoming mental operation (Oberauer, 2002). We argue that this view must be extended, since the selection of information for processing is always guided by current intentions and task goals. In our experiments, it was required that two kinds of comparison tasks be executed on digits selected from a set of three digits held in working memory. The tasks differed in regard to the object features the comparison was based on. Access to a new mental object (object switch) took consistently longer on the semantic comparison task than on the recognition task. This difference is not attributable to object selection difficulty and cannot be fully accounted for by task difficulty or differences in rehearsal processes. The results support our assumptions that (1) mental objects are selected for a given specific task and, so, are accessed with their specific task-relevant object features; (2) verbal mental objects outside the focus of attention are usually not maintained at a full feature level but are refreshed phonologically by subvocal rehearsal; and (3) if more than phonological information is required, access to mental objects involves feature retrieval processes in addition to object selection.     

The role of working memory in the carry operation of mental arithmetic: number and value of the carry

Two experiments were conducted to investigate the role of phonological and executive working-memory components in the carry operation in mental arithmetic. We manipulated the number of carry operations, as previous research had done, but also the value that had to be carried. Results of these experiments show that in addition to the number of carry operations, the value of the carry is also an important variable determining the difficulty of arithmetical sums. Furthermore, both variables (number and value) interacted with each other in such a way that the combination of multiple carries and values of carries larger than one resulted in more difficult problems irrespective of the presence of a working-memory load. The findings with respect to working-memory load suggest that mainly the central executive is important in handling the number of carry operations as well as the value that has to be carried. The implications of the present findings for our views on mental arithmetic and its reliance on working memory are discussed.     

The role of working memory in the carry operation of mental arithmetic: Number and value of the carry

Two experiments were conducted to investigate the role of phonological and executive working-memory components in the carry operation in mental arithmetic. We manipulated the number of carry operations, as previous research had done, but also the value that had to be carried. Results of these experiments show that in addition to the number of carry operations, the value of the carry is also an important variable determining the difficulty of arithmetical sums. Furthermore, both variables (number and value) interacted with each other in such a way that the combination of multiple carries and values of carries larger than one resulted in more difficult problems irrespective of the presence of a working-memory load. The findings with respect to working-memory load suggest that mainly the central executive is important in handling the number of carry operations as well as the value that has to be carried. The implications of the present findings for our views on mental arithmetic and its reliance on working memory are discussed.     

Counting on working memory in arithmetic problem solving

Mental calculation is an important everyday skill involving access to well-learned procedures, problem solving, and working memory. Although there is an active literature on acquiring concepts and procedures for mental arithmetic, relatively little is known about the role of working memory in this task. This paper reports two experiments in which dual-task methodology is used to study the role of components of working memory in mental addition. In Experiment 1, mental addition of auditorily presented two-digit numbers was significantly disrupted by concurrent random letter generation and, to a lesser extent, by concurrent articulatory suppression, but was unimpaired by concurrent hand movement or by presentation of irrelevant pictures. Although the number of errors increased with two of the dual tasks, the incorrect responses tended to be quite close to the correct answer. In Experiment 2, the numbers for addition were presented visually. Here again, random generation produced the largest disruption of mental arithmetic performance, while a smaller amount of disruption was observed for articulatory suppression, hand movement, and unattended auditorily presented two-digit numbers. The overall levels of performance were better and the absolute size of the disruptive effects shown with visual presentation was very small compared with those found for auditory presentation. This pattern of results is consistent with a role for a central executive component of working memory in performing the calculations required for mental addition and in producing approximately correct answers. Visuospatial resources in working memory may also be involved in approximations. The data support the view that the subvocal rehearsal component of working memory provides a means of maintaining accuracy in mental arithmetic, and this matches a similar conclusion derived from previous work on counting. The general implications for the role of working memory in arithmetic problem solving will be discussed.     

High-performers use the phonological loop less to process mental arithmetic during working memory tasks

This study investigated the effects of three working memory components—the central executive, phonological loop, and visuospatial sketchpad—on performance differences in complex mental arithmetic between individuals. Using the dual-task method, we examined how performance during two-digit addition was affected by load on the central executive (random tapping condition), phonological loop (articulatory suppression condition), and visuospatial sketchpad (spatial tapping condition) compared to that under no load (control condition) in high- and low-performers of complex mental arithmetic in Experiment 1. Low-performers showed an increase in errors under the random tapping and articulatory suppression conditions, whereas high-performers showed an increase of errors only under the random tapping condition. In Experiment 2, we conducted similar experiments on only the high-performers but used a shorter presentation time of each number. We found the same pattern for performing complex mental arithmetic as seen in Experiment 1. These results indicate that high-performers might reduce their dependence on the phonological loop, because the central executive enables them to choose a strategy in which they use less working memory capacity.     

White matter hyperintensities and working memory: an explorative study

White matter hyperintensities (WMH) are commonly observed in elderly people and may have the most profound effect on executive functions, including working memory. Surprisingly, the Digit Span backward, a frequently employed working memory task, reveals no association with WMH. In the present study, it was investigated whether more detailed analyses of WMH variables and study sample selection are important when establishing a possible relationship between the Digit Span backward and WMH. To accomplish this, the Digit Span backward and additional working memory tests, WMH subscores, and cardiovascular risk factors were examined. The results revealed that performance on the Digit Span backward test is unrelated to WMH, whereas a relationship between other working memory tests and WMH was confirmed. Furthermore, a division between several white matter regions seems important; hyperintensities in the frontal deep white matter regions were the strongest predictor of working memory performance.     

Phonological and visual working memory in mental addition

The goal of the present research was to examine the role of working memory in mental arithmetic. Adults (n = 96) solved multidigit arithmetic problems (e.g., 52 + 3; 3 + 52) alone and in combination with either a phonological memory load (i.e., nonwords, such as gup) or a visual memory load (i.e., random pattern of asterisks). The participants solved problems presented in a vertical format significantly faster than problems presented in a horizontal format. They also solved double digit first problems (e.g., 52 + 3) more quickly than the reverse (e.g., 3 + 52), but only when the problems were presented horizontally. Performance was worse in the phonological load condition than in the visual load condition for the participants who solved problems presented horizontally, whereas performance was worse in the visual load condition than in the phonological load condition when problems were presented vertically. The present research provides evidence that both phonological and visual aspects of working memory are involved in mental arithmetic but that the role of each working memory component will depend on such factors as presentation format.     

Dissociating familiarity from recollection in human recognition memory: Different rates of forgetting over short retention intervals

Two functionally distinct forms of recognition memory have been identified in human and nonhuman species—the ability to recollect qualitative information about previous events, and the ability to differentiate between familiar and novel stimuli. Separate dual-process theories have been developed in the animal and human literatures to account for these findings. However, it is not clear whether these theories describe the same two underlying memory processes. On the basis of animal studies of medial temporal lobe function, familiarity is expected to exhibit disproportionately fast forgetting compared with recollection over short retention intervals. We tested this prediction in healthy human subjects by examining recognition forgetting rates across a range of 8–32 intervening items and found significant forgetting in the accuracy of familiarity-based discriminations and no evidence of forgetting in the accuracy of recollection-based discriminations. In agreement with the results from nonhuman species, the present results indicate that item familiarity decreases more rapidly than recollection over short retention intervals.     

Dissociating the neural correlates of item and context memory: an ERP study of face recognition.

We investigated the neural correlates of item and context retrieval using event-related potentials (ERPs). Participants studied unfamiliar faces with happy or neutral expressions, and at test, they decided whether test faces were studied in the same or in a different expression, or were new. The parietal ERP effect, which is hypothesized to indirectly reflect medial-temporal lobe (MTL) function, was sensitive to item retrieval, whereas the frontal ERP effect, which is thought to reflect prefrontal cortex (PFC) function, was sensitive to context retrieval. Converging with lesion, functional neuroimaging (PET and fMRI), and ERP evidence, these results support the notion that item retrieval is primarily associated with MTL function whereas context retrieval is primarily associated by PFC function.     

Asymmetrical access to color and location in visual working memory

Models of visual working memory (VWM) have benefitted greatly from the use of the delayed-matching paradigm. However, in this task, the ability to recall a probed feature is confounded with the ability to maintain the proper binding between the feature that is to be reported and the feature (typically location) that is used to cue a particular item for report. Given that location is typically used as a cue-feature, we used the delayed-estimation paradigm to compare memory for location to memory for color, rotating which feature was used as a cue and which was reported. Our results revealed several novel findings: 1) the likelihood of reporting a probed object’s feature was superior when reporting location with a color cue than when reporting color with a location cue; 2) location report errors were composed entirely of swap errors, with little to no random location reports; and 3) both colour and location reports greatly benefitted from the presence of nonprobed items at test. This last finding suggests that it is uncertainty over the bindings between locations and colors at memory retrieval that drive swap errors, not at encoding. We interpret our findings as consistent with a representational architecture that nests remembered object features within remembered locations.     

The role of phonological and executive working memory resources in simple arithmetic strategies

The current study investigated the role of the central executive and the phonological loop in arithmetic strategies to solve simple addition problems (Experiment 1) and simple subtraction problems (Experiment 2). The choice/no-choice method was used to investigate strategy execution and strategy selection independently. The central executive was involved in both retrieval and procedural strategies, but played a larger role in the latter than in the former. Active phonological processes played a role in procedural strategies only. Passive phonological resources, finally, were only needed when counting was used to solve subtraction problems. No effects of working memory load on strategy selection were observed.     

Working memory and mental arithmetic: A case for dual central executive resources

The current study was designed to examine the possible existence of two limited-capacity pools of central executive resources: one each for verbal and visuospatial processing. Ninety-one college students (M age = 19.0, SD = 2.2) were administered a verbal working memory task that involved updating numbers in 2-, 3-, and 4-load conditions. The task was administered in both single task (no-interference condition) and dual-task (verbal interference and visuospatial interference conditions) formats. Findings indicated main effects for both memory load and type of interference, as well as, a load × interference interaction. Verbal interference led to a steeper increase in errors on the primary verbal working memory task; whereas, there was a smaller increase in errors across load in both the non-verbal and no-interference conditions. The effect of verbal interference and the lack of a spatial interference effect on a primary task that utilized verbal working memory resources, suggests that the processing of verbal and spatial stimuli in a dual-task paradigm requires separate central executive resources.