Two-digit Multiplication Research Articles

How Distracting Events Influence Young and Older adults’ Arithmetic Performance?

ABSTRACT In this study, I investigated the role of distraction on arithmetic performance and whether this role changes with aging during adulthood. Young and older adults were asked to verify one-digit addition problems (Expt. 1) or to estimate the results of two-digit multiplication problems (Expt. 2). In both experiments, true and false simple problems (Expt. 1) or easier and harder complex problems (Expt. 2) were displayed superimposed or not on irrelevant, emotionally neutral pictures (e.g. mushrooms). In both simple and complex arithmetic, young and older adults obtained poorer arithmetic performance under distraction relative to no-distraction conditions. Most interesting, deleterious effects of irrelevant stimuli on arithmetic performance were larger in older than in young adults. Moreover, magnitude of distraction effects increased with longer solution latencies in young (but not in older) adults while solving complex arithmetic problems. These findings have important implications for furthering our understanding of the role of distraction on cognitive performance in general, and arithmetic performance in particular, as well as age-related differences in this role.

Fast FPGA-Based Multipliers by Constant for Digital Signal Processing Systems

Traditionally, the usual multipliers are used to multiply signals by a constant, but multiplication by a constant can be considered as a special operation requiring the development of specialized multipliers. Different methods are being developed to accelerate multiplications. A large list of methods implement multiplication on a group of bits. The most known one is Booth’s algorithm, which implements two-digit multiplication. We propose a modification of the algorithm for the multiplication by three digits at the same time. This solution reduces the number of partial products and accelerates the operation of the multiplier. The paper presents the results of a comparative analysis of the characteristics of Booth’s algorithm and the proposed algorithm. Additionally, a comparison with built-in FPGA multipliers is illustrated.

Influence of emotional stimuli on metacognition: A study in arithmetic.

We investigate the role of negative emotional stimuli on direct and indirect metacognition, and document age-related differences in this role during adulthood. Participants were presented with negative or neutral pictures while asked to select which of two available strategies was the better strategy to find approximate estimates of two-digit multiplication problems. Following each strategy selection, participants provided either a direct (confidence judgment; Expt. 1) or an indirect (opt-out judgment; Expt. 2) evaluation of their strategy choice. Negative emotional stimuli decreased metacognitive accuracy for arithmetic strategy selection, but only when indirect metacognitive measures were collected. No differences were found when direct metacognitive judgments were requested. The effects of emotional stimuli on indirect metacognition and lack of effects on direct metacognition were found in both young and older adults. These findings have important implications for our understanding of the mechanisms underlying the effects of emotion on metacognition in young and older adults.

Estimation Strategy Selection Is Modulated by Snapshot Emotional Priming, but Not Math Anxiety.

The present study explored the role of snapshot emotional priming and math anxiety in estimation strategy selection. Participants were asked to complete a two-digit multiplication estimation task (e.g., 34 × 67) under explicit (Experiment 1) and implicit (Experiment 2) snapshot emotional priming conditions by freely choosing to use DU (down-up, e.g., doing 30 × 70 = 2100 for 34 × 67) or UD (up-down, e.g., doing 40 × 60 = 2400 for 34 × 67) strategies to arrive as close as possible to the correct answer. In Experiment 1, individuals’ estimation performance was positively influenced by explicit happy priming (shorter RT (reaction time)), while not affected by explicit fear priming. In Experiment 2, individuals’ estimation ACC (accuracy) when using the UD strategy was negatively affected by both implicit happy and fear priming, but their RT when using DU and UD strategies was positively impacted by implicit happy priming. In both experiments, the correlations between math anxiety and estimation performance (ACC, RT, and strategy selection adaptivity) was not significant. The present study suggests that fear priming was not always detrimental to individuals’ estimation performance, and happy priming did not always universally improve individuals’ estimation performance. Additionally, estimation strategy selection was not influenced by math anxiety.

Neural Correlates of Numerical Estimation: The Role of Strategy Use.

Introduction: Computation estimation is the ability to provide an approximate answer to a complex arithmetic problem without calculating it exactly. Despite its importance in daily life, the neuronal network underlying computation estimation is largely unknown. Methods: We looked at the neuronal correlates of two computational estimation strategies: approximated calculation and sense of magnitude (SOM)–intuitive representation of magnitude, without calculation. During an fMRI scan, thirty-one college students judged whether the result of a two-digit multiplication problem was larger or smaller than a given reference number. In two different blocks, they were asked to use a specific strategy (AC or SOM). Results: The two strategies activated brain regions related to calculation, numerical cognition, decision-making, and working memory. AC more than SOM elicited activations in multiple, domain-specific brain regions in the parietal lobule, including the left SMG (BA 40), the bilateral superior parietal lobule (BA 7), and the right inferior parietal lobule (BA 7). The activation level of the IFG was positively correlated to individual accuracy, indicating that the IFG has an essential role in both strategies. Conclusions: These finding suggest that the analogic code of magnitude is more involved in the AC than the SOM strategy.

Age-related differences in how negative emotions influence arithmetic performance

ABSTRACT In this study, we investigated the role of negative emotions on arithmetic and whether this role changes with aging during adulthood. Young and older adults were asked to verify one-digit addition problems (Experiment 1) and to estimate the results of two-digit multiplication problems (Experiment 2). In both experiments, easier and harder problems were displayed superimposed on emotionally neutral (e.g. mushrooms) or emotionally negative (e.g. a corpse) pictures. In both simple and complex arithmetic, young and older adults obtained poorer arithmetic performance under negative emotion conditions, especially while solving harder problems. Most interesting, deleterious effects of negative emotions on arithmetic performance were larger in young than in older adults. These findings have important implications for further our understanding of the role of negative emotions in the domain of arithmetic and age-related differences in this role.

Implicit happy and fear experience contributes to computational estimation strategy execution: Behavioral and neurophysiological evidence

Previous study have explored the influence of explicit emotion priming on computational estimation strategy execution, but the corresponding influence of implicit emotion priming still remains unknown. The present study aimed to solve this problem. Participants were asked to complete two-digit multiplication computational estimation task, under different implicit emotion priming conditions (gender judgment task). In the two-digit multiplication computational estimation task, the computational estimation question was presented in the middle of the screen, meanwhile, two alternative answers were presented side by side at the bottom of the screen, participants were required to select which answer is correct, by using the down-up strategy (e.g., doing 30 × 50 = 1500 for 34 × 46). Behavioral results showed that the response speed under implicit happy and fear (vs. neutral and angry) priming conditions was quicker, and the accuracy under different priming conditions showed no significant difference. The ERP results showed that the influence of implicit emotion priming on computational estimation strategy execution consisted of two phases: in the first phase, the N1 amplitudes elicited by completing the multiplication computational estimation task were smaller under implicit fear (vs. angry) priming condition; in the second phase, the corresponding P2 amplitudes under implicit happy (vs. fear) priming condition were smaller. The present study indicated that implicit happy and fear experience contributed to complete computational estimation tasks, suggesting that implicit negative emotional (e.g., fear) experience was not always detrimental to computational estimation strategy execution.

Emotional valence modulates arithmetic strategy execution in priming paradigm: an event-related potential study.

Combined with the prime paradigm, the present study aimed to explore the influence of emotion (anger, fear, happiness, and neutral) on performing multiplication estimation. Participants were asked to complete a two-digit multiplication estimation task using the down-up strategy (e.g., doing 20 × 80 = 1600 for 24 × 79). Behavioral results showed that the reaction time for completing multiplication estimation tasks under happy conditions was shorter than for those under anger and fear, and it was shorter under neutral than under fearful conditions. The ERP results showed that about 100ms after multiplication estimation task onset, multiplication estimation execution in the context of happiness (vs. neutral) elicited smaller P1 amplitudes; about 170ms after multiplication estimation task onset, the N170 amplitudes elicited by multiplication estimation strategy execution under different emotional priming conditions showed no significant differences. The above results showed that the impact of emotion priming demonstrates the occurrence of a dynamic process when participants use a specified strategy to complete the multiplication estimation task. The present study revealed that emotional valence modulated arithmetic strategy execution, suggesting the role of different emotions should be fully considered in similar study.

Combining of Concrete and Abstract Instruction for Teaching Strategies for Solving Two-digit Multiplication Problems

Combining of Concrete and Abstract Instruction for Teaching Strategies for Solving Two-digit Multiplication Problems

PENGGUNAAN METODE LATTICE DALAM MENGATASI RENDAHNYA KEMAMPUAN BERHITUNG OPERASI PERKALIAN

One of the problems often faced by students is the low ability to calculate multiplication operations. One of the causes of this condition is still a lack of understanding of the two-digit multiplication concept and the procedure for working on it. In line with this, the existence of this study is expected to overcome the low ability to calculate multiplication operations of students by applying the lattice method. The aim of this study is describe of using the lattice method in overcoming the low ability to calculate multiplication operations. This research is a qualitative study. The stages in this study are divided into several treatments, where each treatment students get treatment from the introduction of the lattice method to use the advanced lattice method. The results of this study show that students are very interested in the new methods they know. They were able to complete independently the multiplication operation after going through four treatments that had been given. At the end of the treatment the average student achieved was 84.30 with a percentage of completeness reaching 96.56% and his numeracy ability increased by 0.81 (high category).

Using Brief Experimental Analysis to Identify the Right Math Intervention at the Right Time

Brief experimental analysis (BEA) is a methodology of rapidly implementing interventions and observing the effect each has on student performance. Extensive research exists demonstrating the utility of BEA in identifying effective reading interventions for students, but comparatively little research exists regarding BEA and mathematics. The current study utilized BEA procedures to identify an intervention targeting skill- or performance-based deficits that would be effective for remediating 4 middle school students’ two-digit by two-digit multiplication skills. Each student had a clearly differentiated intervention identified by BEA as being most effective. Findings from the current study provide evidence for the utility of BEA in matching deficits with mathematics interventions and illustrate their sensitivity to changes in student performance.

Aging and Sequential Strategy Interference: A Magnetoencephalography Study in Arithmetic Problem Solving.

This study investigated age-related changes in the neural bases of sequential strategy interference. Sequential strategy interference refers to decreased strategy interference (i.e., poorer performance when the cued strategy is not the best) after executing a poorer strategy relative to after a better strategy. Young and older adults performed a computational estimation task (e.g., providing approximate products to two-digit multiplication problems, like 38 × 74) and were matched on behavioral sequential strategy interference effects. Analyses of magnetoencephalography (MEG) data revealed differences between young and older adults in brain activities underlying sequential strategy interference. More specifically, relative to young adults, older adults showed additional recruitments in frontal, temporal, and parietal regions. Also, age-related differences were found in the temporal dynamics of brain activations, with modulations occurring both earlier and later in older than young adults. These results suggest that highly functioning older adults rely on additional mechanisms to process sequential strategy interference as efficiently as young adults. Our findings inform mechanisms by which highly functioning older adults obtain as good performance as young adults, and suggest that these older adults may compensate deleterious effects of aging to efficiently execute arithmetic strategies.

Do Exact Calculation and Computation Estimation Reflect the Same Skills? Developmental and Individual Differences Perspectives.

Groups of children in 4th, 5th, and 6th grades and college students performed exact calculation and computation estimation tasks with two-digit multiplication problems. In the former they calculated the exact answer for each problem, and in the latter they estimated whether the result of each problem was larger or smaller than a given reference number. The analyses of speed and accuracy both showed different developmental patterns of the two tasks. While the accuracy of exact calculation increased with age in childhood, the accuracy of the estimation task reached its maximum level already in 4th grade and did not change with age. The reaction time of the exact calculation task was longer than that of the estimation task. The reaction time for both tasks remained constant in childhood and decreased in adulthood, with the improvement in speed larger for the exact calculation task. Similarly, within group variability in accuracy was larger in the exact calculation task than in the computation estimation task. Finally, low correlation was found between the accuracy of the two tasks. Together, these findings suggest that exact calculation and computation estimation reflect at least in part different skills.

Sequential Difficulty Effects in Cognitive and Sensorimotor Tasks: Insights from Arithmetic and Fitts Tasks

Abstract The present study tested sequential difficulty effects (SDEs) in arithmetic problem solving and Fitts’ aiming task for the same individuals. SDEs refer to poorer performance on current items following harder items relative to after easier items. Young and older adults accomplished an arithmetic task (i.e., finding the approximate products to two-digit multiplication problems) and a Fitts aiming task (i.e., performing rapid pointing movements to reach the finish areas). Current items were preceded by two easy or difficult items (i.e., in the repeated-precursor condition) or only one easy or difficult item (i.e., in the unrepeated-precursor condition). Participants’ performance revealed SDEs in both the arithmetic and the aiming tasks only when the difficulty level of the precursor items was repeated. Data also revealed comparable SDEs in both age groups during the arithmetic task but SDEs only in older adults while participants accomplished the aiming task. These findings have a number of implications for our understanding of mechanisms underlying SDEs and age-related differences in SDEs, as they suggest that SDEs involve both domain-general and domain-specific mechanisms that are differentially influenced by aging.

Aging and List-Wide Modulations of Strategy Execution:A Study in Arithmetic

ABSTRACTBackground/Study Context: This study aimed at further our understanding of the cognitive processes involved during strategy execution, and how the processes involved change with age. More specifically, the main goal was to investigate whether poorer-strategy effects (i.e., poorer performance when a cued strategy is not the best) and sequential modulations of poorer-strategy effects (i.e., decreased poorer-strategy effects on current problems following poorer-strategy problems compared with after better-strategy problems) are influenced by proportions of poorer-strategy problems.Methods: We used a computational estimation task (i.e., providing approximate products to two-digit multiplication problems such as 38 × 74) with problems sets including 75%, 50%, or 25% of poorer-strategy problems (i.e., participants have to estimate products with another strategy than the better strategy). The remaining problems were cued with the better strategy. Age-related differences were also investigated.Results: We found that proportions of poorer-strategy problems influenced sequential modulations of poorer-strategy effects. Indeed, sequential modulations of poorer-strategy effects were larger when proportions of poorer-strategy problems were equal than unequal. Moreover, proportion effects were different for young and older adults, as older adults benefited more from low proportions of poorer-strategy problems compared with young adults.Conclusion: These findings have important implications regarding cognitive control mechanisms underlying both list-wide and trial-to-trial modulations of strategy execution, and how these processes change during aging.

The Sources of Sequential Modulations of Control Processes in Arithmetic Strategies: A Magnetoencephalography Study.

In a wide variety of cognitive domains, performance is determined by the selection and execution of cognitive strategies to solve problems. We used magnetoencephalography to identify the brain regions involved and specify the time course of dynamic modulations of executive control processes during strategy execution. Participants performed a computational estimation task in which they were instructed to execute a poorer or better strategy to estimate results of two-digit multiplication problems. When participants were asked to execute the poorer strategy, two distinct sets of brain activations were identified, depending on whether the poorer strategy (engaging the left inferior frontal junction) or the better strategy (engaging ACC) had been executed on the immediately preceding items. Our findings also revealed the time course of activations in regions involved in sequential modulations of cognitive control processes during arithmetic strategy execution. These findings point at processes of proactive preparation on items after poorer strategy items and dynamics of reactive adjustments after better strategy items.

Increased arithmetic complexity is associated with domain-general but not domain-specific magnitude processing in children: A simultaneous fNIRS-EEG study.

The investigation of the neural underpinnings of increased arithmetic complexity in children is essential for developing educational and therapeutic approaches and might provide novel measures to assess the effects of interventions. Although a few studies in adults and children have revealed the activation of bilateral brain regions during more complex calculations, little is known about children. We investigated 24 children undergoing one-digit and two-digit multiplication tasks while simultaneously recording functional near-infrared spectroscopy (fNIRS) and electroencephalography (EEG) data. FNIRS data indicated that one-digit multiplication was associated with brain activity in the left superior parietal lobule (SPL) and intraparietal sulcus (IPS) extending to the left motor area, and two-digit multiplication was associated with activity in bilateral SPL, IPS, middle frontal gyrus (MFG), left inferior parietal lobule (IPL), and motor areas. Oscillatory EEG data indicated theta increase and alpha decrease in parieto-occipital sites for both one-digit and two-digit multiplication. The contrast of two-digit versus one-digit multiplication yielded greater activity in right MFG and greater theta increase in frontocentral sites. Activation in frontal areas and theta band data jointly indicate additional domain-general cognitive control and working memory demands for heightened arithmetic complexity in children. The similarity in parietal activation between conditions suggests that children rely on domain-specific magnitude processing not only for two-digit but-in contrast to adults-also for one-digit multiplication problem solving. We conclude that in children, increased arithmetic complexity tested in an ecologically valid setting is associated with domain-general processes but not with alteration of domain-specific magnitude processing.

Involvement of working memory in mental multiplication in Chinese elementary students

ABSTRACTThe authors' aim was to examine the relation between two-digit mental multiplication and working memory. In Study 1, involving 30 fifth-grade students, we used digit span backward as an abbreviated measure of working memory. In Study 2, involving 41 fourth-grade students, working memory comprised measures of phonological loop, visuospatial sketchpad, and central executive (i.e., updating) based on A. Miyake et al. (2000). In Study 1, working memory measured through the use of digit span backward explained unique variance in response time. In Study 2, participants' accuracy and response time were most susceptible to phonological loop influence. The findings support the argument that multiplication facts are stored in a verbal form and the retrieval of multiplication facts relies more on verbal modality. Unique features of Chinese mathematics instruction of mental multiplication were discussed. The findings suggested that the involvement of different subcomponents of working memory in mental arithmetic may be subject to instructional, contextual, and linguistic factors.

The obligatory activation of practiced complex multiplication facts and what it tells us about models of arithmetic processing

Three experiments were conducted in which adults practiced complex multiplication problems (e.g., 4 x 17). In Experiments 1 and 2, after practice participants completed a number-matching task in which two digits (cues) were followed by a single digit (probe) and had to determine whether the probe matched either of the cues. In simple arithmetic (e.g., 4 x 3), when the probe is the product of the cues (12), participants are slower/more error prone when determining whether there is a match. Results of Experiment 1 extended this effect to complex multiplication. In Experiment 2, participants practiced problems with the larger operand first (e.g., 17 x 4) or with the smaller operand first (e.g., 4 x 17). The number-matching interference effect from Experiment 1 was replicated, and was equal across the two groups whether cues were presented in their practiced or non-practiced order. Experiment 3 was conducted to determine if two additional simple multiplication effects, consistency and relatedness, could be documented for complex multiplication. After practice, in a verification task (4 x 13 = 56?) it was found that when presented answers shared a digit with the decade digit of the correct answer (consistency) or were a correct answer to another practiced problem (relatedness), participants rejected answers more slowly and/or less accurately. Together, findings from the three experiments support arithmetic models that posit that commuted pairs are not represented in long-term memory independently and that posit representations of two-digit multiplication answers are decomposed into decades and units during arithmetic processing.

Relationships Between Strategy Switching and Strategy Switch Costs in Young and Older Adults: A Study in Arithmetic Problem Solving

Background/Study Context: This study investigated age-related differences in within-item strategy switching (i.e., revising initial strategy choices to select a better strategy while solving a given problem) and in strategy switch costs (i.e., longer latencies when participants switch strategies than when they do not switch strategy during strategy execution).Methods: In a computational estimation task, participants had to give approximate products to two-digit multiplication problems (e.g., 41 × 67) while rounding up (i.e., do 50 × 70 for 41 × 67) or rounding down (i.e., do 40 × 60 for 41 × 67) operands to their nearest decades. After executing a cued strategy during 1000 ms, participants had the possibility to switch to another strategy (or repeat the same strategy) in a selection condition. In an execution condition, participants were forced to repeat the same strategy or to switch to another strategy.Results: It was found that (1) older adults were less able than young adults to switch strategy after starting to execute a cued strategy (36.1% vs. 45.8%); (2) older adults showed larger switch costs than young adults (422 vs. 223 ms); and (3) strategy switches and strategy switch costs correlated in older adults but not in young adults.Conclusion: These findings have important implications for our understanding of the mechanisms underlying within-item strategy switching and aging effects on these mechanisms as well as, more generally, of strategic variations during cognitive aging.