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

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.

Heuristics and biases in the mental manipulation of magnitudes: Evidence from length and time production

There is a debate about whether and why we overestimate addition and underestimate subtraction results (Operational Momentum or OM effect). Spatial-attentional accounts of OM compete with a model which postulates that OM reflects a weighted combination of multiple arithmetic heuristics and biases (AHAB). This study addressed this debate with the theoretically diagnostic distinction between zero problems (e.g., 3 + 0, 3 − 0) and non-zero problems (e.g., 2 + 1, 4 − 1) because AHAB, in contrast to all other accounts, uniquely predicts reverse OM for the latter problem type. In two tests (line-length production and time production), participants indeed produced shorter lines and under-estimated time intervals in non-zero additions compared with subtractions. This predicted interaction between operation and problem type extends OM to non-spatial magnitudes and highlights the strength of AHAB regarding different problem types and modalities during the mental manipulation of magnitudes. They also suggest that OM reflects methodological details, whereas reverse OM is the more representative behavioural signature of mental arithmetic.

A role for attentional reorienting during approximate multiplication and division

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.

Dividing attention increases operational momentum

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.