Operational Momentum in Multiplication and Division?

Abstract

Biases are commonly seen in numerical cognition. The operational momentum (OM) effect shows that responses to addition and subtraction problems are biased in the whole-number direction of the operation. It is not known if this bias exists for other arithmetic operations. To determine whether OM exists in scalar operations, we measured response bias in adults performing symbolic (Arabic digits) and non-symbolic (dots) multiplication and division problems. After seeing two operands, with either a multiplication (×) or division (÷) sign, participants chose among five response choices. Both non-random performance profiles and the significant contribution of both operands in a multiple regression analysis predicting the chosen values, suggest that adults were able to use numerical information to approximate the outcomes in both notations, though they were more accurate on symbolic problems. Performance on non-symbolic problems was influenced by the size of the correct choice relative to alternatives. Reminiscent of the bias in addition and subtraction, we found a significant response bias for non-symbolic problems. Non-symbolic multiplication problems were overestimated and division problems were underestimated. These results indicate that operational momentum is present in non-symbolic multiplication and division. Given the influence of the size of the correct choice relative to alternatives, an interaction between heuristic bias and approximate calculation is possible.

A role for attentional reorienting during approximate multiplication and division

“…Only division, which was underestimated, showed a response bias significantly different from zero. In sum, we replicated the results from Katz and Knops (2014), showing a significant OM-effect for non-symbolic multiplication and division but no OM-effect for symbolic operations, despite having encouraged approximate calculation by omitting the correct response from the symbolic response alternatives. …”

“…If participants responded non-randomly, then range and rank should have a significant effect on response choice (Katz & Knops, 2014). Indeed, we found that rank and range interacted for both multiplication and division in both symbolic and non-symbolic notation.…”

“…replicating Katz and Knops (2014). Second, we explored the underlying mechanisms of the OM effect by testing the association of the OM effect with visuo-spatial attention as measured by a Posner paradigm.…”

“…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.…”

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

“…Support for this account has come from the influence of numbers on shifts of spatial attention (e.g., Fischer, Castel, Dodd, & Pratt, 2003); from identifying common brain parietal structures that are involved in addition and subtraction, as well as in directed saccadic eye movements (Knops, Thirion, Hubbard, Michel, & Dehaene, 2009); from observing OM in eye-fixation behavior (Klein et al, 2014); and from linking OM with attentional control measures (Knops et al, 2013). In comparison to this impressive support, the "compression account" of OM (McCrink et al, 2007; see also Chen & Verguts, 2012), which postulates that participants apply a linear transformation on the compressed mental number line representation, which in turn leads to over-or underestimating the outcome, has received less empirical support (Knops, Dehaene, Berteletti, & Zorzi, 2014;Knops et al, 2013; see also Katz & Knops, 2014).…”