Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect

Knops, André; Viarouge, Arnaud; Dehaene, Stanislas

doi:10.3758/app.71.4.803

Cited by 149 publications

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“…OM has initially been observed with non-symbolic operations where participants estimated the number of visual objects (dots) that would result from adding two sets of dots or subtracting one set from the other (McCrink et al, 2007). However, OM has also been found in symbolic notation (Knops, Dehaene, Berteletti, & Zorzi, 2014;Knops, Viarouge, & Dehaene, 2009) which has been interpreted as evidence for a common underlying mechanism. OM has also been observed with paradigms that require translating the cognitively generated numerical estimate into a position on a labeled line (Pinhas & Fischer, 2008), or actively producing the outcome via a dot generating manual device (Lindemann & Tira, 2011).…”

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“…14 × 3 = 42) an OM effect was observed (Katz & Knops, 2014). However, for the corresponding non-symbolic problems where the quantities were presented as dot patterns a regular OM effect was observed.One crucial difference between the procedure adopted by Katz and Knops (2014) and previous studies (Knops, Viarouge, et al, 2009) was that the correct solution was presented as one of five response alternatives, potentially encouraging the engagement in exact calculation and direct retrieval rather than approximating the outcome. This may have reduced the opportunity to detect any systematic biases due to attentional shifts that accompany approximate quantity manipulations.…”

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A role for attentional reorienting during approximate multiplication and division

2017

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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.Keywords: operational momentum, approximate calculation, spatial attention, mental number line, reorienting, mental arithmetic Journal of Numerical Cognition, 2017, Vol. 3(2), 246-269, doi:10.5964/jnc.v3i2.62 Received: 2016-06-13. Accepted: 2017-11-17. Published (VoR): 2017 Handling Editors: Silke Goebel, University of York, York, United Kingdom; Hans-Christoph Nuerk, Universität Tübingen, Tübingen, Germany *Corresponding author at: Unter den Linden 6, 10099 Berlin, Germany. E-mail: andre. knops@hu-berlin.de This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 International License, CC BY 4.0 (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Numerical cognition has long been thought to represent a prime example of an abstract propositional symbolic system with no obvious reference to the outer world. For example, number names such as 'one', 'two' and so on do not bear any obvious association to the referenced cardinal meaning. Recent evidence, however, implies that numbers and mental arithmetic bear numerous associations with space and time (Fischer & Knops, 2014;H...

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A role for attentional reorienting during approximate multiplication and division

2017

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“…Mc-Crink [9] firstly have shown, when performing approximate additions and subtractions, participants favour larger numbers for addition and smaller numbers for subtraction. Knops et al, [10] add information to this issue by demonstrating that during nonsymbolic addition, the subjects preferentially selected numbers at the upper right location, whereas during subtraction, they were biased toward the upper left location. The same group of researchers [11] further expanded this evidence by reporting a pattern of eye movement shifts to the left or to the right, during execution of arithmetic operations in association to posterior parietal cortex activation.…”

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Understanding economical outcomes with the mental number line

Vicario

2011

Nat Prec

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With this study we provide evidence that the cognitive processes involved in addition/subtraction, mapped along the mental number line, seem to mediate our understanding of trading verbs.When left-to-right culture participants read "loss" verbs, cognitive activation moves "leftward" as in arithmetical subtraction, while reading "gain" verbs activates a mental rightward space as in addition.We test this hypothesis by asking to a group of participants to use their left and right hand in judging (as correct of not) the syntactic form of several verbs meaning financial outcomes. Results show that processing "gain verbs" was associated with shorter latencies when responding with the right hand similarly when performing an addition task, while processing "loss verbs" was associated with shorter latencies when responding with the left, similarly when performing a subtraction task.

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“…In many studies, participants respond along a particular spatial axis-for example, by pressing left-and right-side response keys (e.g., Dehaene et al, 1993;Pinhas & Fischer, 2008) or by making leftward and rightward head or arm movements (e.g., Loetscher, Schwarz, Schubiger, & Brugger, 2008;Marghetis et al, 2014;Wiemers, Bekkering, & Lindemann, 2014;Winter & Matlock, 2013). In other studies, the numerical stimuli themselves are presented at different spatial locations (e.g., Knops, Viarouge et al, 2009;Masson & Pesenti, 2014). Such methods leave open the possibility that the observed biases are artifacts of polarity correspondence (e.g., left-less/right-more, a type of stimulus-response compatibility; Proctor & Cho, 2006), rather than evidence of an MNL (see also Santens & Gevers, 2008).…”

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“…Mentally summing the values of successive cards, and thereby computing the total value of one's hand, is a prerequisite for making this decision wisely. Research suggests that such mental arithmetic may rely on an internal spatial representation of numerical magnitude (e.g., Knops, Viarouge, & Dehaene, 2009;Marghetis, Nuñez, & Bergen, 2014;McCrink, Dehaene, & DehaeneLambertz, 2007; for a review, see Fischer & Shaki, 2014). For example, when pointing to an arithmetic solution on a visually presented number line, participants are biased leftward on subtraction problems and rightward on addition problems (Pinhas & Fischer, 2008).…”

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Gamble on gaze: Eye movements reflect the numerical value of blackjack hands

2016

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We used a novel task-a blackjack game that naturally involves mental summation of numerical values-to investigate the role of attention in the mental number line (MNL) and to provide insight into the ecological validity of this representational format. By analyzing the spatial position of participants' spontaneous, task-irrelevant eye movements, we avoided some of the limitations of previous research on the MNL, in which the findings could be attributed to taskspecific factors such as the use of overt spatial cues. In two experiments, we found that eye movements along the horizontal axis reflected the overall numerical value of participants' hands, with smaller-value hands eliciting fixations toward the left of the screen and larger-value hands eliciting fixations toward the right. This pattern held even when controlling for the number of cards in the hand and the value of the card most recently dealt-suggesting that the effects were driven by mental summation of values, not merely by the processing of serial order or individual numbers. Vertical eye movements, in contrast, reflected hand value less reliably. In showing that spontaneous eye movements along the horizontal axis track the magnitude of internally computed sums in an ecologically relevant task, our findings provide evidence for a dynamic MNL that supports magnitude-driven shifts of attention and that may be recruited during everyday forms of numerical reasoning.

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Dynamic representations underlying symbolic and nonsymbolic calculation: Evidence from the operational momentum effect

Cited by 149 publications

References 47 publications

A role for attentional reorienting during approximate multiplication and division

A role for attentional reorienting during approximate multiplication and division

Understanding economical outcomes with the mental number line

Gamble on gaze: Eye movements reflect the numerical value of blackjack hands

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