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

Abstract

Number processing evokes spatial biases, both when dealing with single digits and in more complex mental calculations. Here we investigated whether these two biases have a common origin, by examining their flexibility. Participants pointed to the locations of arithmetic results on a visually presented line with an inverted, right-to-left number arrangement. We found directionally opposite spatial biases for mental arithmetic and for a parity task administered both before and after the arithmetic task. We discuss implications of this dissociation in our results for the task-dependent cognitive representation of numbers.

Competing Biases in Mental Arithmetic: When Division Is More and Multiplication Is Less

“…Reverse OM was previously reported by Knops et al ( 2013 ) in non-symbolic addition and subtraction, perhaps reflecting lack of arithmetic knowledge in the children tested. It was also found in a spatial pointing task with adults by (Pinhas et al ( 2015 ), see also Klein et al, 2014 ), due to reversal of the visually presented number interval. Importantly, none of the OM accounts can explain the reverse OM we found when comparing multiplication and division.…”

“…Why has it not been noticed in subtraction problems where the first operand is also bigger? Indeed, we think that anchoring sheds a fresh light on the previous puzzle of larger OM for zero compared to non-zero problems (Pinhas and Fischer, 2008 ; Pinhas et al, 2015 ): specifically, the computation of OM for zero problems is based on comparing problems with identical first operands (e.g., 3 + 0 vs. 3–0), thus equating the anchoring effect. In contrast, computing OM for non-zero problems with controlled outcomes (e.g., 2 + 1 vs. 4–1) dilutes OM in the second example because its first operand is larger and its associated anchoring bias counteracts the expected bias towards smaller numbers and thus the overall OM.…”

“…This was previously shown in simple arithmetic tasks where they overestimate addition and underestimate subtraction outcomes, both with symbolic and with non-symbolic operands (McCrink et al, 2007 ; Knops et al, 2009 , 2014 ). For example, when presented with a horizontal line flanked by 0 and 10 on both sides, adults indicate the result of “4 + 2” further rightward than the result of “8–2” (Pinhas and Fischer, 2008 ; see also Pinhas et al, 2014 , 2015 ). These mental arithmetic biases are called “operational momentum” (OM) effects.…”

“…In contrast, spatial accounts of OM postulate cognitive operations on a “mental number line” (MNL), a spatially oriented representation of number concepts in ascending order (for review see Fischer and Shaki, 2014 ): The “ attentional-shift account ” suggests that OM originates from movements of attention along the MNL, causing “overshoots” in the direction associated with the arithmetic operation, namely leftward (towards smaller numbers) for subtraction and rightward (towards larger numbers) for addition (Knops et al, 2013 , 2014 ; see also Klein et al, 2014 ). Alternatively, the “ spatial-competition account ” postulates that the spatial activations induced by the operands (i.e., single numbers), the operator (i.e., plus or minus signs) and the computed outcome (Pinhas and Fischer, 2008 ; Pinhas et al, 2014 , 2015 ) together induce OM. This account acknowledges multiple sources of OM but has not yet been sufficiently elaborated to determine the relative weights of these contributions to overall OM.…”

“…Moreover, comparing across tasks can refine our understanding of any given cognitive mechanism, such as its range of applicability. For example, it is puzzling to see larger OM in zero- compared to non-zero problems (e.g., 3 + 0 vs. 2 + 1) although the second operand “zero” requires no movements along the MNL (see Pinhas and Fischer, 2008 ; Pinhas et al, 2015 ).…”