Problem Size Effect Research Articles

Brain Mechanisms of Arithmetic: A Crucial Role for Ventral Temporal Cortex.

Elementary arithmetic requires a complex interplay between several brain regions. The classical view, arising from fMRI, is that the intraparietal sulcus (IPS) and the superior parietal lobe (SPL) are the main hubs for arithmetic calculations. However, recent studies using intracranial electroencephalography have discovered a specific site, within the posterior inferior temporal cortex (pITG), that activates during visual perception of numerals, with widespread adjacent responses when numerals are used in calculation. Here, we reexamined the contribution of the IPS, SPL, and pITG to arithmetic by recording intracranial electroencephalography signals while participants solved addition problems. Behavioral results showed a classical problem size effect: RTs increased with the size of the operands. We then examined how high-frequency broadband (HFB) activity is modulated by problem size. As expected from previous fMRI findings, we showed that the total HFB activity in IPS and SPL sites increased with problem size. More surprisingly, pITG sites showed an initial burst of HFB activity that decreased as the operands got larger, yet with a constant integral over the whole trial, thus making these signals invisible to slow fMRI. Although parietal sites appear to have a more sustained function in arithmetic computations, the pITG may have a role of early identification of the problem difficulty, beyond merely digit recognition. Our results ask for a reevaluation of the current models of numerical cognition and reveal that the ventral temporal cortex contains regions specifically engaged in mathematical processing.

3G1-K3 繰り下がりのあるひき算における問題のサイズ効果(科学教育各論(1))

3G1-K3 繰り下がりのあるひき算における問題のサイズ効果(科学教育各論(1))

Impact of ageing on problem size and proactive interference in arithmetic facts solving.

Arithmetic facts (AFs) are required when solving problems such as "3 × 4" and refer to calculations for which the correct answer is retrieved from memory. Currently, two important effects that modulate the performance in AFs have been highlighted: the problem size effect and the proactive interference effect. The aim of this study is to investigate possible age-related changes of the problem size effect and the proactive interference effect in AF solving. To this end, the performance of young and older adults was compared in a multiplication production task. Furthermore, an independent measure of proactive interference was assessed to further define the architecture underlying this effect in multiplication solving. The results indicate that both young and older adults were sensitive to the effects of interference and of the problem size. That is, both interference and problem size affected performance negatively: the time needed to solve a multiplication problem increases as the level of interference and the size of the problem increase. Regarding the effect of ageing, the problem size effect remains constant with age, indicating a preserved AF network in older adults. Interestingly, sensitivity to proactive interference in multiplication solving was less pronounced in older than in younger adults suggesting that part of the proactive interference has been overcome with age.

Interference and problem size effect in multiplication fact solving: Individual differences in brain activations and arithmetic performance

In the development of math ability, a large variability of performance in solving simple arithmetic problems is observed and has not found a compelling explanation yet. One robust effect in simple multiplication facts is the problem size effect, indicating better performance for small problems compared to large ones. Recently, behavioral studies brought to light another effect in multiplication facts, the interference effect. That is, high interfering problems (receiving more proactive interference from previously learned problems) are more difficult to retrieve than low interfering problems (in terms of physical feature overlap, namely the digits, De Visscher and Noël, 2014). At the behavioral level, the sensitivity to the interference effect is shown to explain individual differences in the performance of solving multiplications in children as well as in adults. The aim of the present study was to investigate the individual differences in multiplication ability in relation to the neural interference effect and the neural problem size effect.To that end, we used a paradigm developed by De Visscher, Berens, et al. (2015) that contrasts the interference effect and the problem size effect in a multiplication verification task, during functional magnetic resonance imaging (fMRI) acquisition. Forty-two healthy adults, who showed high variability in an arithmetic fluency test, participated in our fMRI study. In order to control for the general reasoning level, the IQ was taken into account in the individual differences analyses.Our findings revealed a neural interference effect linked to individual differences in multiplication in the left inferior frontal gyrus, while controlling for the IQ. This interference effect in the left inferior frontal gyrus showed a negative relation with individual differences in arithmetic fluency, indicating a higher interference effect for low performers compared to high performers. This region is suggested in the literature to be involved in resolution of proactive interference. Besides, no correlation between the neural problem size effect and multiplication performance was found. This study supports the idea that the interference due to similarities/overlap of physical traits (the digits) is crucial in memorizing arithmetic facts and in determining individual differences in arithmetic.

Brain Mechanisms of Arithmetic: A Crucial Role for Ventral Temporal Cortex

Mental arithmetic is central to our society, yet the roles of different brain regions involved in calculation remain poorly understood. To tackle this issue, we recorded electrocorticography signals from subjects solving addition problems. Behavioral results showed a classical problem-size effect: RTs increased with the size of the operands. We then examined how high-frequency broadband (HFB) activity is modulated by problem size. Sites around the intraparietal sulculs (IPS) showed increased total activity with problem size, compatible with a role in arithmetic decision making. More surprisingly, sites within the posterior inferior temporal gyrus (pITG) showed an initial burst of HFB activity that decreased with problem size, yet with a constant integral, suggesting that pITG appears to represent the amount of semantic evidence available for a calculation. Our results ask for a re-evaluation of current models of arithmetic and demonstrate that the ventral temporal cortex contain regions involved in symbolic forms of reasoning, such as mental calculation.

How do individual differences in children's domain specific and domain general abilities relate to brain activity within the intraparietal sulcus during arithmetic? An fMRI study.

Previous research has demonstrated that children recruit the intraparietal sulcus (IPS) during arithmetic, which has largely been attributed to domain-specific processes such as quantity manipulations. However, the IPS has also been found to be important for domain-general abilities, such as visuo-spatial working memory (VSWM). Based on the current literature it is unclear whether individual differences in domain-specific skills, domain-general skills, or a combination of the two, are related to the recruitment of the IPS during arithmetic. This study examines how individual differences in both domain general and domain specific competencies relate to brain activity in the IPS during arithmetic, and whether the relationships are related to how brain activity is measured. In a sample of 44 school-aged children, we found that VSWM was only weakly related to a neural index of arithmetic complexity (neural problem size effect), whereas symbolic number processing skills (symbolic comparison and ordering) were related to overall arithmetic activity (both small and large problems). By simultaneously examining multiple domain-general and domain specific measures, we were also able to determine that symbolic skills were a stronger predictor of brain activity within the IPS than domain general skills such as VSWM and domain specific skills such as non-symbolic number processing. Together, these findings highlight that neural problem size effect may reflect different cognitive processes than brain activity across both small and large arithmetic problems, and that symbolic number processing skills are a critical predictor of variability in IPS activity during arithmetic. Hum Brain Mapp 38:3941-3956, 2017. © 2017 Wiley Periodicals, Inc.

Editorial: Abstract Mathematical Cognition.

Editorial: Abstract Mathematical Cognition.

Monkeys display classic signatures of human symbolic arithmetic.

Non-human primates compare quantities in a crude manner, by approximating their values. Less is known about the mental transformations that non-humans can perform over approximate quantities, such as arithmetic transformations. There is evidence that human symbolic arithmetic has a deep psychological connection with the primitive, approximate forms of quantification of non-human animals. Here, we ask whether the subtle performance signatures that humans exhibit during symbolic arithmetic also bear a connection to primitive arithmetic. Specifically, we examined the problem size effect, the tie effect, and the practice effect-effects which are commonly observed in children's math performance in school. We show that, like humans, monkeys exhibited the problem size and tie effects, indicating commonalities in arithmetic algorithms with humans. Unlike humans, however, monkeys did not exhibit a practice effect. Together, these findings provide new evidence for a cognitive relation between non-symbolic and symbolic arithmetic.

Age-related differences in sequential modulations of problem-size and rule-violation effects during arithmetic problem verification tasks.

Young and older adults were asked to verify true (e.g., 5 × 61 = 305) and false (5 × 61 = 315) arithmetic problems. Half the problems were small (e.g., 5 × 17 = 85) and half were large problems (e.g., 5 × 93 = 465). Half the false problems respected the five rule (i.e., the product of an operand multiplied by five ends with either 5 or 0), and half violated this rule (e.g., 21 × 5 = 115 vs. 21 × 5 = 113). Both young and older adults showed problem-size effects (i.e., they verified small problems more quickly than large problems) and five-rule violation effects (i.e., they verified problem violating five rule more quickly than problems respecting five rule). Moreover, we found sequential modulations of these problem-size and five-rule effects. Problem-size effects were larger on current problems following large problems than after small problems, and five-rule violation effects were larger after problems violating the five rule than after no-rule violation problems. Finally, sequential modulations of problem-size effects were larger in older adults than in young adults, and there were no age-related differences in sequential modulations of five-rule violation effects. These findings speak to the determiners of arithmetic performance, as to how well arithmetic calculation and non-calculation strategies are executed and selected on current problems depends on strategies used with preceding problems.

Learning multiplication: An integrated analysis of the multiplication ability of primary school children and the difficulty of single digit and multidigit multiplication problems

In this study the mental multiplication ability of primary school children and the difficulty structure of all single digit and 469 multidigit multiplication problems, each solved tens of thousands of times in a web-based practice program, were investigated.Child analyses indicated three groups: single digit problem solvers, multidigit problem solvers, and high performers. Within-grade ability differences were very large.In the item analyses, previously identified effects in single digit multiplication, such as the problem size effect and the tie effect, were replicated in one integrated analysis. Data from two tasks were contrasted: one in which children used predominantly computational strategies, and one in which they are expected to rely mostly on retrieval. In both tasks we found most support for the computational efficiency model. Finally, exploratory analyses on the difficulty of multidigit problems suggest that children rely on the base 10 structure of our number system.

Evaluation of linear relaxations in Ad Network optimization for online marketing

Ad Networks connect advertisers to websites that want to host advertisements. When users request websites, the Ad Network decides which ad to send so as to maximize the number of ads that are clicked by users. Due to the difficulty in solving such maximization problems, there are solutions in the literature that are based on linear programming (LP) relaxations. We contribute with a formulation for the Ad Network optimization problem, where it is cast as a Markov decision process (MDP). We analyze theoretically the relative performance of MDP and LP solutions. We report on an empirical evaluation of solutions for LP relaxations, in which we analyze the effect of problem size in performance loss compared to the MDP solution. We show that in some configurations, the LP relaxations incur in approximately 58 % revenue loss when compared to MDP solutions. However, such relative loss decreases as problem size increases. We also propose new heuristics to improve the use of solutions achieved by LP relaxation. We show that solutions obtained by LP relaxations are suitable for Ad Networks, as the performance loss introduced by such solutions are small in large problems observed in practice.

The interference effect in arithmetic fact solving: An fMRI study

Some multiplication facts share common digits with other, previously learned facts, and as a result, different problems are associated with different levels of interference. The detrimental effect of interference in arithmetic facts knowledge has been recently highlighted in behavioral studies, in children as well as in adults, both in typical and atypical development. The present study investigated the brain regions involved in the interference effect when solving multiplication problems. Twenty healthy adults carried out a multiplication task in an MRI scanner. The event-related design comprised problems whose interference level and problem size were manipulated in a 2×2 factorial design. After each trial, individuals were requested to indicate whether they solved the trial by retrieving the answer from long-term memory. This allowed us to examine which brain areas were sensitive to the interference effect and problem size effect as well as the retrieval strategy. The results highlighted two specific regions: the left angular gyrus was more activated for low interfering than for high interfering problems, and the right intraparietal sulcus was more activated for large problems than for small problems. In both regions, brain activity was not modulated by the other effect. These results suggest that the left angular gyrus is sensitive to the level of interference of the multiplication problems, whereas previously this region was thought to be more activated by small problems or by retrieval strategy. Here, in a design manipulating interference and problem size, while controlling for retrieval strategy, we showed that it rather reflects an automatic mapping between the problem and the answer stored in long-term memory. The right intraparietal sulcus was modulated by the problem size effect, which supports the idea that the problem size effect comes from the higher overlap between magnitude of the answers of large problems compared to small ones. Importantly, neither effects can be reduced to a strategy effect since they were present when analyzing only retrieval trials.

Optimising sampling designs for the maximum coverage problem of plume detection

The location of sensors to detect outbreaks of hazardous plumes in the atmosphere can be improved by considering possible paths of such plumes. Atmospheric dispersion models can provide simulations of such paths under realistic weather conditions. Numeric simulation always goes along with discretisation, and if a plume is detected or not can be regarded as a binary question. Thus optimising the locations of a fixed number of sensors can be regarded as finite; in fact it is a variety of the classical maximum coverage problem. We present an algorithm to completely solve this problem. It is based on complete deep tree search that finds all globally optimal sensor configurations, but the effort is reduced by skipping non-promising configurations and deleting redundant data. To determine the effort and to learn about optimal sensor configurations, the algorithm was tested on several scenarios based on plume simulations or on random data. This was completed by some theoretical results on the effect of problem size and plume detection probability on the effort. Finally we used the determined optimal configurations to evaluate two well known heuristics: greedy search and a genetic algorithm.

When problem size matters: differential effects of brain stimulation on arithmetic problem solving and neural oscillations.

The problem size effect is a well-established finding in arithmetic problem solving and is characterized by worse performance in problems with larger compared to smaller operand size. Solving small and large arithmetic problems has also been shown to involve different cognitive processes and distinct electroencephalography (EEG) oscillations over the left posterior parietal cortex (LPPC). In this study, we aimed to provide further evidence for these dissociations by using transcranial direct current stimulation (tDCS). Participants underwent anodal (30min, 1.5 mA, LPPC) and sham tDCS. After the stimulation, we recorded their neural activity using EEG while the participants solved small and large arithmetic problems. We found that the tDCS effects on performance and oscillatory activity critically depended on the problem size. While anodal tDCS improved response latencies in large arithmetic problems, it decreased solution rates in small arithmetic problems. Likewise, the lower-alpha desynchronization in large problems increased, whereas the theta synchronization in small problems decreased. These findings reveal that the LPPC is differentially involved in solving small and large arithmetic problems and demonstrate that the effects of brain stimulation strikingly differ depending on the involved neuro-cognitive processes.

Efficiency of Arithmetic Procedures Modulates the Problem-Size Effect in Subtraction.

The aim of this study was to examine whether differences in strategy selection and/or strategy efficiency can explain the modulation of the problem-size effect by arithmetic skill. More specifically, we wondered whether arithmetic skill increases the use of retrieval strategy in large problems, and/or enhances the efficiency of either retrieval or procedural strategies. The performance of highly-skilled (HS) and less highly-skilled (LS) individuals on a subtraction verification task was analyzed according to problem size and to the strategy reported on a trial-by-trial basis after each problem. The problem size effect was larger for LS individuals than for their HS peers, both in response time and in hit rate. Nevertheless, groups did not differ regarding the strategy reported for each subtraction size. As expected, problems in which retrieval strategy was reported were solved more quickly and more accurately than problems solved by procedural strategies. Responses using retrieval strategy were equally fast in the two groups, but HS individuals performed better than LS when using procedural strategies. The results therefore suggest that the differences in behavioral measures between groups might specifically be due to differences in the efficiency of procedural strategies.

How number line estimation skills relate to neural activations in single digit subtraction problems

The Number Line (NL) task requires judging the relative numerical magnitude of a number and estimating its value spatially on a continuous line. Children's skill on this task has been shown to correlate with and predict future mathematical competence. Neurofunctionally, this task has been shown to rely on brain regions involved in numerical processing. However, there is no direct evidence that performance on the NL task is related to brain areas recruited during arithmetical processing and that these areas are domain-specific to numerical processing. In this study, we test whether 8- to 14-year-old's behavioral performance on the NL task is related to fMRI activation during small and large single-digit subtraction problems. Domain-specific areas for numerical processing were independently localized through a numerosity judgment task. Results show a direct relation between NL estimation performance and the amount of the activation in key areas for arithmetical processing. Better NL estimators showed a larger problem size effect than poorer NL estimators in numerical magnitude (i.e., intraparietal sulcus) and visuospatial areas (i.e., posterior superior parietal lobules), marked by less activation for small problems. In addition, the direction of the activation with problem size within the IPS was associated with differences in accuracies for small subtraction problems. This study is the first to show that performance in the NL task, i.e. estimating the spatial position of a number on an interval, correlates with brain activity observed during single-digit subtraction problem in regions thought to be involved in numerical magnitude and spatial processes.

The arithmetic problem size effect in children: an event-related potential study.

This study used for the first time event-related potentials (ERPs) to examine the well-known arithmetic problem size effect in children. The electrophysiological correlates of this problem size effect have been well documented in adults, but such information in children is lacking. In the present study, 22 typically developing 12-year-olds were asked to solve single-digit addition problems of small (sum ≤ 10) and large problem size (sum > 10) and to speak the solution into a voice key while ERPs were recorded. Children displayed similar early and late components compared to previous adult studies on the problem size effect. There was no effect of problem size on the early components P1, N1, and P2. The peak amplitude of the N2 component showed more negative potentials on left and right anterior electrodes for large additions compared to small additions, which might reflect differences in attentional and working memory resources between large and small problems. The mean amplitude of the late positivity component which follows the N2, was significantly larger for large than for small additions at right parieto-occipital electrodes, in line with previous adult data. The ERPs of the problem size effect during arithmetic might be a useful neural marker for future studies on fact retrieval impairments in children with mathematical difficulties.

Dynamically structured holographic memory

Abstract We describe the DSHM (Dynamically Structured Holographic Memory) model of human memory, which uses high dimensional vectors to represent items in memory. The complexity and intelligence of human behavior can be attributed, in part, to our ability to utilize vast knowledge acquired over a lifetime of experience with our environment. Thus models of memory, particularly models that can scale up to lifetime learning, are critical to modeling human intelligence. DHSM is based on the BEAGLE model of language acquisition ( Jones & Mewhort, 2007 ) and extends this type of model to general memory phenomena. We demonstrate that DHSM can model a wide variety of human memory effects. Specifically, we model the fan effect, the problem size effect (from math cognition), dynamic game playing (detecting sequential dependencies from memories of past moves), and time delay learning (using an instance based approach). This work suggests that DSHM is suitable as a basis for learning both over the short-term and over the lifetime of the agent, and as a basis for both procedural and declarative memory. We argue that cognition needs to be understood at both the symbolic and sub-symbolic levels, and demonstrate that DSHM intrinsically operates at both of these levels of description. In order to situate DSHM in a familiar context, we discuss the relationship between DHSM and ACT-R.

Effect of problem size on the performance of constructive algorithms for flow shop scheduling problems with sequence dependent setup times

This paper focus on the scheduling problem of a flow shop operating in a sequence dependent setup time (SDST) environment. Two constructive algorithms of contrasting characteristics are analysed for the performance with respect to change in problem size; the first algorithm is processing time-based and the second algorithm is setup time-based. The problem size is characterised by the variables namely, number of jobs and number of machines. An extensive performance analysis of the two constructive algorithms has been carried out using 960 SDST flow shop benchmark problem instances. The graphical analysis of the results reveals the correlation between the relative performance of the algorithms and problem size. The study shows that the performance of the setup time-based algorithm increases with increase in number of jobs and decreases with increase in number of machines. The coefficient of variation analysis is used to investigate the performance variation of the algorithm with change in number of machines. The analysis reveals that as the number machines increases, the coefficient of variation of the summed setup time matrix decreases which causes the change in performance of the setup time-based algorithm.

On the problem-size effect in small additions: Can we really discard any counting-based account?

The problem-size effect in simple additions, that is the increase in response times (RTs) and error rates with the size of the operands, is one of the most robust effects in cognitive arithmetic. Current accounts focus on factors that could affect speed of retrieval of the answers from long-term memory such as the occurrence of interference in a memory network or the strength of memory traces that would differ from problem to problem. The present study analyses chronometric data from a sample of 91 adults solving very small additions (operands from 1 to 4) that are generally considered as being solved by retrieval. The results reveal a monotonic linear increase in RTs with the magnitude of both operands. This pattern is at odds with the retrieval-based accounts of the problem-size effect and challenges the well-established view that small additions are solved through retrieval of the answer from long-term memory. Our results are more compatible with the hypothesis that even very small additions are solved using compacted fast procedures that scroll an ordered representation such as a number line or a verbal number sequence. This interpretation is corroborated by the analysis of individual differences.