scholarly journals A role for attentional reorienting during approximate multiplication and division

2017 ◽  
Vol 3 (2) ◽  
pp. 246-269 ◽  
Author(s):  
Curren Katz ◽  
Hannes Hoesterey ◽  
André Knops
Keyword(s):  
Cognitive Processes ◽  
Incorrect Response ◽  
Number Representation ◽  
Momentum Effect ◽  
Division Problems ◽  
Operational Momentum ◽  
Response Alternatives ◽  
Attentional Shifts ◽  
Fact Retrieval ◽  
Subsequent Target

When asked to estimate the outcome of arithmetic problems, participants overestimate for addition problems and underestimate for subtraction problems, both in symbolic and non-symbolic format. This bias is referred to as operational momentum effect (OM). The attentional shifts account holds that during computation of the outcome participants are propelled too far along a spatial number representation. OM was observed in non-symbolic multiplication and division while being absent in symbolic multiplication and division. Here, we investigate whether (a) the absence of the OM in symbolic multiplication and division was due to the presentation of the correct outcome amongst the response alternatives, putatively triggering verbally mediated fact retrieval, and whether (b) OM is correlated with attentional parameters, as stipulated by the attentional account. Participants were presented with symbolic and non-symbolic multiplication and division problems. Among seven incorrect response alternatives participants selected the most plausible result. Participants were also presented with a Posner task, with valid (70%), invalid (15%) and neutral (15%) cues pointing to the position at which a subsequent target would appear. While no OM was observed in symbolic format, non-symbolic problems were subject to OM. The non-symbolic OM was positively correlated with reorienting after invalid cues. These results provide further evidence for a functional association between spatial attention and approximate arithmetic, as stipulated by the attentional shifts account of OM. They also suggest that the cognitive processes underlying multiplication and division are less prone to spatial biases compared to addition and subtraction, further underlining the involvement of differential cognitive processes.

scholarly journals Operational momentum effect in children with and without developmental dyscalculia

2013 ◽  
Vol 4 ◽  
Author(s):  
Karin Kucian ◽  
Fabienne Plangger ◽  
Ruth O'Gorman ◽  
Michael von Aster
Keyword(s):  
Momentum Effect ◽  
Developmental Dyscalculia ◽  
Operational Momentum

scholarly journals Fact retrieval or compacted procedures in arithmetic – a neurophysiological investigation of two hypotheses

2020 ◽  
Author(s):  
Roland H. Grabner ◽  
Clemens Brunner ◽  
Valerie Lorenz ◽  
Stephan E. Vogel ◽  
Bert De Smedt
Keyword(s):  
Problem Solving ◽  
Cognitive Processes ◽  
High Sensitivity ◽  
List Type ◽  
Single Digit ◽  
Frequency Bands ◽  
Eeg Data ◽  
Neurophysiological Test ◽  
Problem Solving Strategies ◽  
Fact Retrieval

ABSTRACTThere is broad consensus that adults solve single-digit multiplication problems almost exclusively by fact retrieval (i.e., retrieval of the solution from an arithmetic fact network). In contrast, there has been a long-standing debate on the cognitive processes involved in solving single-digit addition problems. This debate has evolved around two theoretical accounts. The fact-retrieval account postulates that these are solved through fact retrieval, just like multiplications, whereas the compacted-procedure account proposes that solving very small additions (i.e., problems with operands between 1 and 4) involves highly automatized and unconscious compacted procedures. In the present electroencephalography (EEG) study, we put these two accounts to the test by comparing neurophysiological correlates of solving very small additions and multiplications. A sample of 40 adults worked on an arithmetic production task involving all (non-tie) single-digit additions and multiplications. Afterwards, participants completed trial-by-trial strategy self-reports. In our EEG analyses, we focused on induced activity (event-related synchronization/desynchronization, ERS/ERD) in three frequency bands (theta, lower alpha, upper alpha). Across all frequency bands, we found higher evidential strength for similar rather than different neurophysiological processes accompanying the solution of very small addition and multiplication problems. This was also true when n + 1 and n × 1 problems were excluded from the analyses. In two additional analyses, we showed that ERS/ERD can differentiate between self-reported problem-solving strategies (retrieval vs. procedure) and even between n + 1 and n + m problems in very small additions, demonstrating its high sensitivity to cognitive processes in arithmetic. The present findings clearly support the fact-retrieval account, suggesting that both very small additions and multiplications are solved through fact retrieval.HIGHLIGHTSNeurophysiological test of fact retrieval and compacted procedures accountInduced EEG data are sensitive to cognitive processes in arithmetic problem solvingBoth very small additions and multiplications are solved through fact retrieval

scholarly journals The Developmental Trajectory of the Operational Momentum Effect

2018 ◽  
Vol 9 ◽  
Author(s):  
Pedro Pinheiro-Chagas ◽  
Daniele Didino ◽  
Vitor G. Haase ◽  
Guilherme Wood ◽  
André Knops
Keyword(s):  
Developmental Trajectory ◽  
Momentum Effect ◽  
Operational Momentum

scholarly journals Addition goes where the big numbers are: evidence for a reversed operational momentum effect

2014 ◽  
Vol 22 (4) ◽  
pp. 993-1000 ◽  
Author(s):  
Michal Pinhas ◽  
Samuel Shaki ◽  
Martin H. Fischer
Keyword(s):  
Momentum Effect ◽  
Operational Momentum

scholarly journals Perceived Motion and Operational Momentum: How Speed, Distance, and Time Influence Two-Digit Arithmetic

2021 ◽  
Vol 12 ◽  
Author(s):  
Maciej Haman ◽  
Hubert Młodzianowski ◽  
Michał Gołȩbiowski
Keyword(s):  
Spatial Attention ◽  
Mental Arithmetic ◽  
Arithmetic Operation ◽  
Internal Representation ◽  
Representational Momentum ◽  
Number Line ◽  
Mental Number Line ◽  
Moving Object ◽  
Momentum Effect ◽  
Operational Momentum

Operational momentum was originally defined as a bias toward underestimating outcomes of subtraction and overestimating outcomes of addition. It was suggested that these estimation biases are due to leftward attentional shift along the mental number-line (spatially organized internal representation of number) in subtraction and rightward shift in addition. This assumes the use of “recycled” mechanisms of spatial attention, including “representational momentum” – a tendency to overestimate future position of a moving object, which compensates for the moving object’s shift during preparation of a reaction. We tested a strong version of this assumption directly, priming two-digit addition and subtraction problems with leftward and rightward motion of varied velocity, as velocity of the tracked object was found to be a factor in determining representational momentum effect size. Operands were subsequently moving across the computer screen, and the participants’ task was to validate an outcome proposed at the end of the event, which was either too low, correct, or too high. We found improved accuracy in detecting too-high outcomes of addition, as well as complex patterns of interactions involving arithmetic operation, outcome option, speed, and direction of motion, in the analysis of reaction times. These results significantly extend previous evidence for the involvement of spatial attention in mental arithmetic, showing movement of the external attention focus as a factor directing internal attention in processing numerical information. As a whole, however, the results are incompatible with expectations derived from the strong analogy between operational and representational momenta. We suggest that the full model may be more complex than simply “moving attention along the mental number-line” as a direct counterpart of attention directed at a moving object.

scholarly journals Dividing attention increases operational momentum

2017 ◽  
Vol 3 (2) ◽  
pp. 230-245 ◽  
Author(s):  
Koleen McCrink ◽  
Timothy Hubbard
Keyword(s):  
Divided Attention ◽  
Baseline Condition ◽  
Operational Momentum ◽  
Actual Outcome ◽  
Attentional Shifts ◽  
Spatial Judgment ◽  
Probe Array

When adding or subtracting two quantities, adults often compute an estimated outcome that is larger or smaller, respectively, than the actual outcome, a bias referred to as “operational momentum”. The effects of attention on operational momentum were investigated. Participants viewed a display in which two arrays of objects were added, or one array was subtracted from another array, and judged whether a subsequent outcome (probe) array contained the correct or incorrect number of objects. In a baseline condition, only the arrays to be added or subtracted were viewed. In divided attention conditions, participants simultaneously viewed a sequence of colors or shapes, and judged which color (a non-spatial judgment) or shape (a spatial judgment) was repeated. Operational momentum occurred in all conditions, but was higher in divided attention conditions than in the baseline condition, primarily for addition problems. This pattern suggests that dividing attention, rather than decreasing operational momentum by decreasing attentional shifts, actually increased operational momentum. These results are consistent with a heightened use of arithmetic heuristics under conditions of divided attention.

Operational Momentum in Numerosity Production Judgments of Multi-Digit Number Problems

2011 ◽  
Vol 219 (1) ◽  
pp. 50-57 ◽  
Author(s):  
Oliver Lindemann ◽  
Michael D. Tira
Keyword(s):  
Empirical Support ◽  
Number Processing ◽  
Production Task ◽  
Momentum Effect ◽  
Digit Number ◽  
Judgment Bias ◽  
Operational Momentum ◽  
Calculation Task ◽  
Addition And Subtraction ◽  
Random Dot Patterns

The current study demonstrates a numerosity production task and investigates approximate mental calculations with two-digit numbers. Participants were required to produce random dot patterns to indicate the size of two-digit numbers and the results of addition and subtraction problems. The stimuli in the calculation task consisted of problems requiring a carry operation (e.g., 24 + 18) or no-carry problems (e.g., 24 + 53) or zero problems (e.g., 24 + 0). Our analysis revealed that the outcomes of additions were estimated to be larger than the outcomes of subtractions. Interestingly, this judgment bias was present for no-carry and zero problems but not for carry problems. Taken together, the presented data provide empirical support for the presence of an operational momentum effect (OM effect) in multi-digit number arithmetic. These findings and the dissociation of the OM effect for carry and no-carry problems are discussed in the context of recent models on multi-digit number processing.

Eye Movements and Cognitive Strategies

1981 ◽  
Vol 53 (2) ◽  
pp. 623-632 ◽  
Author(s):  
Cris W. Johnston ◽  
Francis J. Pirozzolo
Keyword(s):  
Eye Movements ◽  
Cognitive Processes ◽  
Visual Information ◽  
Cognitive Strategies ◽  
Incorrect Response ◽  
Reflection Method ◽  
Oculomotor Behavior ◽  
Test Items ◽  
Initial Fixation ◽  
Scan Pattern

Eye movements were recorded using an infra-red reflection method from two female subjects while they took the Peabody Picture Vocabulary Test. The purpose of the study was to investigate the manner in which oculomotor behavior may characterize an individual's verbal-cognitive ability, and to study processing and evaluating visual information. Correct responses on the test were best associated with relatively high fixation density, i.e., frequency, for the chosen item compared to alternative selections. When the chosen item was an incorrect response die most predictive measure was that the chosen item received the longest duration of fixation. Less useful measures studied were mean duration of fixation and total time spent looking at each alternative (gaze time). Upon exposure of the test items, the initial fixation was on the left and the initial direction of eye movement was clockwise. Based on a sequential “scan pattern” analysis of location, frequency, and duration of fixation, other evidence of psycho-oculomotor strategies was not observed. It is suggested that a trade-off may exist between the various parameters of oculomotor behavior and that perhaps by some unique combination and analysis of selected measures it would be possible to further elucidate how eye movements reflect cognitive processes.

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