A role for attentional reorienting during approximate multiplication and division

Abstract

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...

Addition and Subtraction but Not Multiplication and Division Cause Shifts of Spatial Attention

“…At the very least, the results in the present study showed that the interaction between space and multiplication (or division) operation was – if it exists at all – weaker than those for addition and subtraction operations. These results also support the findings of Katz et al (2017) who found that the cognitive processes underlying multiplication and division are less prone to spatial biases compared to addition and subtraction. Therefore, the dissociation in our current study suggested that the cognitive mechanisms underlying multiplication and division operations might be different from those for addition and subtraction ( Katz et al, 2017 ).…”

“…These results also support the findings of Katz et al (2017) who found that the cognitive processes underlying multiplication and division are less prone to spatial biases compared to addition and subtraction. Therefore, the dissociation in our current study suggested that the cognitive mechanisms underlying multiplication and division operations might be different from those for addition and subtraction ( Katz et al, 2017 ). However, the Bayes factor analysis indicated that more data are needed to make this conclusive in future work.…”

“…Moreover, it has been suggested that single-digit multiplication performance is dominated by retrieval of multiplication tables from memory ( Verguts and Fias, 2005 ). It was plausible that the null effect of spatial-arithmetic association in multiplication or division was caused by its exact answer instead of approximate arithmetic which has a functional association with spatial attention ( Katz et al, 2017 ).…”

“…They found that in non-symbolic operations, adults preferred to select the option larger than the correct outcome for multiplication, and to select the option smaller than the correct outcome for division ( Katz and Knops, 2014 ). However, these effects were not observed in symbolic multiplication and division, presumably because participants could calculate exactly and choose the correct values ( Katz and Knops, 2014 ; Katz et al, 2017 ). Recently, contrasting results were reported by Shaki and Fischer (2017) who observed a reverse operational momentum effect with symbolic numbers.…”

Recruitment of the occipital cortex by arithmetic processing follows computational bias in the congenitally blind

“…Additions and subtractions have been thought to involve shifts of attention along a mental number line: a shift of attention toward the right (or toward larger numbers) for addition and toward the left (smaller numbers) for subtraction (Knops et al, 2009a,b;Masson and Pesenti, 2014;McCrink et al, 2007;Pinhas and Fischer, 2008). Importantly, these shifts of attention have not been reported for multiplication and division operations (Katz and Knops, 2014;Katz et al, 2017, Li et al, 2018Mathieu et al, 2016). In accordance with this operation dissociation, it has also been demonstrated that adults and even children use a variety of strategies while solving arithmetic operations (Campbell and Timm, 2000;Dehaene and Cohen, 1997;Hecht, 1999;LeFevre et al, 1996;Siegler and Shrager, 1984).…”