Working memory, strategy execution, and strategy selection in mental arithmetic

A total of 72 participants estimated products of complex multiplications of two-digit operands (e.g., 63 × 78), using two strategies that differed in complexity. The simple strategy involved rounding both operands down to the closest decades (e.g., 60 × 70), whereas the complex strategy required rounding both operands up to the closest decades (e.g., 70 × 80). Participants accomplished this estimation task in two conditions: a no-load condition and a working-memory load condition in which executive components of working memory were taxed. The choice/no-choice method was used to obtain unbiased strategy execution and strategy selection data. Results showed that loading working-memory resources led participants to poorer strategy execution. Additionally, participants selected the simple strategy more often under working-memory load. We discuss the implications of the results to further our understanding of variations in strategy selection and execution, as well as our understanding of the impact of working-memory load on arithmetic performance and other cognitive domains.     

Effects of strategy sequences and response–stimulus intervals on children’s strategy selection and strategy execution: A study in computational estimation

The present study investigates how children’s better strategy selection and strategy execution on a given problem are influenced by which strategy was used on the immediately preceding problem and by the duration between their answer to the previous problem and current problem display. These goals are pursued in the context of an arithmetic problem solving task. Third and fifth graders were asked to select the better strategy to find estimates to two-digit addition problems like 36 + 78. On each problem, children could choose rounding-down (i.e., rounding both operands down to the closest smaller decades, like doing 40 + 60 to solve 42 + 67) or rounding-up strategies (i.e., rounding both operands up to the closest larger decades, like doing 50 + 70 to solve 42 + 67). Children were tested under a short RSI condition (i.e., the next problem was displayed 900 ms after participants’ answer) or under a long RSI condition (i.e., the next problem was displayed 1,900 ms after participants’ answer). Results showed that both strategy selection (e.g., children selected the better strategy more often under long RSI condition and after selecting the poorer strategy on the immediately preceding problem) and strategy execution (e.g., children executed strategy more efficiently under long RSI condition and were slower when switching strategy over two consecutive problems) were influenced by RSI and which strategy was used on the immediately preceding problem. Moreover, data showed age-related changes in effects of RSI and strategy sequence on mean percent better strategy selection and on strategy performance. The present findings have important theoretical and empirical implications for our understanding of general and specific processes involved in strategy selection, strategy execution, and strategic development.     

Effects of execution duration on within-item strategy switching in young and older adults

This study aimed at determining whether previously found age-related differences in within-item strategy switching is modulated by duration of engagement in initial strategy execution. In a computational estimation task, young and older adults had to find estimates to arithmetic problems like 37×64 while either rounding down (i.e., 30×60) or rounding up (i.e., 40×70) both operands to the closest decades. Participants were asked to execute a cued strategy for different durations (i.e., 1, 2, or 3 s), before deciding whether the cued strategy was the best strategy and to switch to the best strategy if the cued strategy was not the best. The main findings revealed that (1) young and older adults were able to switch strategies, especially when they started to execute poorer strategy on a given item; (2) older adults switched strategy less often than young adults, and (3) both young and older participants switched strategy less frequently for long than for short durations. These findings suggest that to be able to switch strategy within item, participants should not be too much engaged in execution of the initially selected strategy, and that duration of engagement in initial strategy execution does not modulate age-related differences in within-item strategy execution.     

Children's strategies in computational estimation

We investigated strategies used to estimate answers to addition problems. Two hundred and sixteen participants (72 adults, 72 sixth graders, and 72 fourth graders) had to provide estimates of three-by-three digit addition problems (e.g., 249 + 743). The choice/no-choice method was used to obtain unbiased estimates of the performance characteristics of strategies. Results showed that (a) at all ages, the most common strategy was to round both operands down to the closest smaller decades, (b) strategy use and execution were influenced by participants' age, problem features, and relative strategy performance, and (c) age-related changes in computational estimation include changes in relative strategy use and execution, as well as in the relative influences of problem and strategy characteristics on strategy choices. Implications of these findings for understanding age-related differences in strategic aspects of computational estimation performance are discussed.     

Do multiplication and division strategies rely on executive and phonological working memory resources?

The role of executive and phonological working memory resources in simple arithmetic was investigated in two experiments. Participants had to solve simple multiplication problems (e.g., 4 x 8; Experiment 1) or simple division problems (e.g., 42 / 7; Experiment 2) under no-load, phonological-load, and executive-load conditions. The choice/no-choice method was used to investigate strategy execution and strategy selection independently. Results for strategy execution showed that executive working memory resources were involved in direct memory retrieval of both multiplication and division facts. Executive working memory resources were also involved in the use of nonretrieval strategies. Phonological working memory resources, on the other hand, tended to be involved in nonretrieval strategies only. Results for strategy selection showed no effects of working memory load. Finally, correlation analyses showed that both strategy execution and strategy selection correlated with individual-difference variables, such as gender, math anxiety, associative strength, calculator use, arithmetic skill, and math experience.     

Age-related changes in children's executive functions and strategy selection: A study in computational estimation

Third, fifth, and seventh graders selected the best strategy (rounding up or rounding down) for estimating answers to two-digit addition problems. Executive function measures were collected for each individual. Data showed that (a) children's skill at both strategy selection and execution improved with age and (b) increased efficiency in executive functions contributed significantly to age-related improvement in children's strategy selection skill. These findings have implications for understanding of age-related differences in computational estimation, strategy selection processes, and mechanisms of strategic development in children.     

Information acquisition strategies and the cognitive structure of arithmetic

In three experiments, we examined the sensitivity of information acquisition strategies to the underlying cognitive structure of arithmetic tasks. Previous work has shown that individuals solve arithmetic problems more quickly when they consider operators before operands. The operators establish the goal, and the operands are then assimilated into that goal. In the present experiments, participants performed arithmetic tasks in which they controlled the display of operators and operands, using separate keystrokes. The participants chose to see the operator before the operands in most conditions. This ordering was more likely when task constraints made it easier to achieve, when feedback emphasized execution time, and in a more complex multiple-step task. These results extend previous research on strategy selection and information acquisition, supporting the idea that coordination between the environment and ongoing cognition is an important feature of cognitive skill.     

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.     

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.     

Instruction and load effects on high-skill and low-skill individuals: A study in the domain of mental arithmetic

What happens when people are asked to respond as quickly or as accurately as possible? This study tested the effects of speed/accuracy instructions and working-memory load on people's strategy efficiency and strategy selection. Adult participants solved simple addition problems (Experiment 1) and simple multiplication problems (Experiment 2) under load and no-load conditions and provided trial-by-trial strategy reports. High-skill participants were more efficient than low-skill participants, but the underlying causes of these skill-related effects differed across experiments. In the addition experiment, high-skill participants responded adaptively to the changing situations by changing their strategy choices, which resulted in smaller effects on their actual performance. Low-skill participants in contrast, did not change their strategy choices as adaptively, which resulted in less efficient performance—and especially so under load conditions. In the multiplication experiment, high-skill and low-skill participants differed in strategy efficiencies rather than in strategy choices. In the discussion, the results are further interpreted and future adaptations for the adaptive strategy choice model (ASCM; Siegler & Jenkins, 1989) are suggested.     

Approximate quantification in young,healthy older adults’, and Alzheimer patients

Forty young adults, 40 healthy older adults, and 39 probable AD patients were asked to estimate small (e.g., 25) and large (e.g., 60) collections of dots in a choice condition and in two no-choice conditions. Participants could choose between benchmark and anchoring strategies on each collection of dots in the choice condition and were required to use either benchmark or anchoring on all configurations in the no-choice conditions (one per strategy). The benchmark strategy involves visual estimation processes whereas the anchoring strategy involves both enumeration and estimation processes. Results showed that strategy use was influenced by collection, participant, and strategy characteristics. Age-related and dementia-related differences were found in both strategy use and strategy execution. The findings have implications for our understanding of aging effects in approximate quantification, strategic variations in Alzheimer’s patients, and sources of cognitive decline during early stages of Alzheimer’s disease.     

Inhibition and shifting capacities mediate adults' age-related differences in strategy selection and repertoire

Young and older adults differ in how many strategies they use to accomplish cognitive tasks. They also differ in how often they select the best strategy on each problem. Two experiments were run to determine whether two executive functions—inhibition and shifting capacities—mediate age-related differences in strategy repertoire and in strategy selection. Both experiments were run in arithmetic problem solving. In Experiment 1, young and older adults’ strategy repertoire was assessed on a problem-by-problem basis while participants solved two-digit addition problems (e.g., 42 + 76). In Experiment 2, young and older participants had to select the best strategy on each problem to find estimates of two-digit multiplication problems (e.g., 43 × 72). In both experiments, individuals’ inhibition and shifting capacities were assessed with the Trail Making Test and the Stroop Test. The main results showed that (a) older adults used a smaller strategy repertoire (Expt. 1) and selected the best strategy on each problem less frequently (Expt. 2) than young adults, (b) inhibition and shifting capacities mediated age-related differences in strategy repertoire and strategy selection, and unique age effects were no longer significant in strategy repertoire but were still significant in strategy selection after statistical control of inhibition and shifting capacities. We discuss important implications of these findings to further our understanding of strategic variations during cognitive aging.     

Strategy sequential difficulty effects vary with working-memory and response–stimulus-intervals: A study in arithmetic

Strategy sequential difficulty effects are the findings that when participants execute strategies, performance is worse after a difficult strategy than after an easy strategy (Uittenhove & Lemaire, 2012). Strategy sequential difficulty effects are hypothesized to result from decreased working-memory resources following difficult strategy execution. In the present study we found a correlation between individuals' working memory and strategy sequential difficulty effects in arithmetic, supporting a working-memory account of these effects. Furthermore, we varied response–stimulus intervals, and we found decreased strategy sequential difficulty effects with increasing response–stimulus intervals. Implications of these findings for further understanding of strategic variations in human cognition are discussed.     

Strategy combination during execution of memory strategies in young and older adults

The present study investigated whether people can combine two memory strategies to encode pairs of words more efficiently than with a single strategy, and age-related differences in such strategy combination. Young and older adults were asked to encode pairs of words (e.g., satellite-tunnel). For each item, participants were told to use either the interactive-imagery strategy (e.g., mentally visualising the two words and making them interact), the sentence-generation strategy (i.e., generate a sentence linking the two words), or with strategy combination (i.e., generating a sentence while mentally visualising it). Participants obtained better recall performance on items encoded with strategy combination than on items encoded with interactive-imagery or sentence-generation strategies. Moreover, we found age-related decline in such strategy combination. These findings have important implications to further our understanding of execution of memory strategies, and suggest that strategy combination occurs in a variety of cognitive domains.     

What affects strategy selection in arithmetic? The example of parity and five effects on product verification

The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people's strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.     

Young and older adults' strategies in complex arithmetic

Young and older adults solved complex addition problems such as 49 + 56. Verbal protocols, solution times, and percentage errors documented strategy repertoire and strategy selection in young and older participants and age-related differences in complex arithmetic performance. Both young and older adults used a set of 9 strategies to solve complex addition problems, although many older adults used a smaller strategy repertoire. The data also showed age-related differences in strategy execution and strategy selection. We discuss general implications of the present data to further our understanding of complex arithmetic and the role of strategic variations during aging.     

Effects of problem size,operation, and working-memory span on simple-arithmetic strategies: differences between children and adults?

Adult's simple-arithmetic strategy use depends on problem-related characteristics, such as problem size and operation, and on individual-difference variables, such as working-memory span. The current study investigates (a) whether the effects of problem size, operation, and working-memory span on children's simple-arithmetic strategy use are equal to those observed in adults, and (b) how these effects emerge and change across age. To this end, simple-arithmetic performance measures and a working-memory span measure were obtained from 8-year-old, 10-year-old, and 12-year-old children. Results showed that the problem-size effect in children results from the same strategic performance differences as in adults (i.e., size-related differences in strategy selection, retrieval efficiency, and procedural efficiency). Operation-related effects in children were equal to those observed in adults as well, with more frequent retrieval use on multiplication, more efficient strategy execution in addition, and more pronounced changes in multiplication. Finally, the advantage of having a large working-memory span was also present in children. The differences and similarities across children's and adult's strategic performance and the relevance of arithmetic models are discussed.     

Examining the influence of action on spatial working memory: The importance of selection

We report three experiments that examine the influence of pointing-to relative to passively viewing an array of objects that participants are attempting to memorize. Recently, Chum, Bekkering, Dodd, and Pratt (2007) provided evidence that pointing to objects enhanced memory relative to passively viewing objects when pointing instruction was manipulated within trial (e.g., point to one array but passively view the other). We replicate this result but also demonstrate that when pointing instruction is blocked (e.g., participants point to or passively view all items in an array as opposed to pointing to some while passively viewing others), pointing to an array of objects actually decreases memory relative to passively viewing that array. Moreover, when pointing is manipulated within trial, the influence of action on working-memory performance appears to be attributable to an enhancement of processing of the pointed-to items as well as a subsequent inhibition of the passively viewed array. These results demonstrate that while action can enhance working memory under conditions where a subset of items is actively selected for additional processing, when selection is not a requirement (e.g., either point to everything or passively view everything), action decreases working-memory performance. Thus, the relationship between action and spatial working memory is complex and context dependent. These results are also discussed as they relate to other similar phenomena (e.g., retrieval-induced forgetting, Corsi Blocks test) in which selection during processing may be critical, and collectively these results provide important insight into spatial working memory and the factors that influence it.     

Sequential modulations of poorer-strategy effects during strategy execution: An event-related potential study in arithmetic

When participants accomplish cognitive tasks, they obtain poorer performance if asked to execute a poorer strategy than a better strategy on a given problem. These poorer-strategy effects are smaller following execution of a poorer strategy relative to following a better strategy. To investigate ERP correlates of sequential modulations of poorer-strategy effects, we asked participants (n = 20) to accomplish a computational estimation task (i.e., provide approximate products to two-digit multiplication problems like 38 × 74). For each problem, they were cued to execute a better versus a poorer strategy. We found event-related potentials signatures of sequential modulations of poorer-strategy effects in two crucial windows (i.e., between 200 and 550 ms and between 850 and 1250 ms) associated with executive control mechanisms and allowing conflict monitoring between the better and the cued strategy. These results have important implications on theories of strategies as they suggest that sequential modulations of poorer-strategy effects involve earlier as well as later mechanisms of cognitive control during strategy execution.     

Sequential difficulty effects during execution of memory strategies in young and older adults

This study aimed at uncovering factors influencing execution of memory strategies and at furthering our understanding of ageing effects on memory performance. To achieve this end, we investigated strategy sequential difficulty (SSD) effects recently demonstrated by Uittenhove and Lemaire in the domain of problem solving. We found that both young and older participants correctly recalled more words using a sentence-construction strategy when this strategy followed an easier strategy (i.e., repetition strategy) or a harder strategy (i.e., mental-image strategy). These SSD effects were of equal magnitude in young and older adults, correlated significantly with Stroop performance in both young and older adults and correlated with N-back performance only in young adults. These findings have important implications for furthering our understanding of memory strategy execution and age-related variations in memory performance, as well for understanding mechanisms underlying SSD effects.