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1 + 2 is more than 2 + 1: Violations of commutativity and identity axioms in mental arithmetic
(2015)
Over the past decade or so, a large number of studies have revealed that conceptual meaning is sensitive to situational context. More recently, similar contextual influences have been documented in the domain of number knowledge. Here we show such context dependency in a length production task. Adult participants saw single digit addition problems of the form n1 + n2 and produced the sum by changing bi-directionally the length of a horizontally extended line, using radially arranged buttons. We found that longer lines were produced when n1 < n2 compared to n1 > n2 and that unit size increased with result size. Thus, the mathematical axioms of commutativity and identity do not seem to hold in mental addition. We discuss implications of these observations for our understanding of cognitive mechanisms involved in mental arithmetic and for situated cognition generally.
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.
Commentary
(2020)
Commentary
(2020)
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.
There is evidence both for mental number representations along a horizontal mental number line with larger numbers to the right of smaller numbers (for Western cultures) and a physically grounded, vertical representation where “more is up.” Few studies have compared effects in the horizontal and vertical dimension and none so far have combined both dimensions within a single paradigm where numerical magnitude was task-irrelevant and none of the dimensions was primed by a response dimension. We now investigated number representations over both dimensions, building on findings that mental representations of numbers and space co-activate each other. In a Go/No-go experiment, participants were auditorily primed with a relatively small or large number and then visually presented with quasi-randomly distributed distractor symbols and one Arabic target number (in Go trials only). Participants pressed a central button whenever they detected the target number and elsewise refrained from responding. Responses were not more efficient when small numbers were presented to the left and large numbers to the right. However, results indicated that large numbers were associated with upper space more strongly than small numbers. This suggests that in two-dimensional space when no response dimension is given, numbers are conceptually associated with vertical, but not horizontal space.
There is evidence both for mental number representations along a horizontal mental number line with larger numbers to the right of smaller numbers (for Western cultures) and a physically grounded, vertical representation where “more is up.” Few studies have compared effects in the horizontal and vertical dimension and none so far have combined both dimensions within a single paradigm where numerical magnitude was task-irrelevant and none of the dimensions was primed by a response dimension. We now investigated number representations over both dimensions, building on findings that mental representations of numbers and space co-activate each other. In a Go/No-go experiment, participants were auditorily primed with a relatively small or large number and then visually presented with quasi-randomly distributed distractor symbols and one Arabic target number (in Go trials only). Participants pressed a central button whenever they detected the target number and elsewise refrained from responding. Responses were not more efficient when small numbers were presented to the left and large numbers to the right. However, results indicated that large numbers were associated with upper space more strongly than small numbers. This suggests that in two-dimensional space when no response dimension is given, numbers are conceptually associated with vertical, but not horizontal space.
A large number of experimental findings from neuroscience and experimental psychology demonstrated interactions between spatial cognition and numerical cognition. In particular, many researchers posited a horizontal mental number line, where small numbers are thought of as being to the left of larger numbers. This review synthesizes work on the mental association between space and number, indicating the existence of multiple spatial mappings: recent research has found associations between number and vertical space, as well as associations between number and near/far space. We discuss number space in three dimensions with an eye on potential origins of the different number mappings, and how these number mappings fit in with our current knowledge of brain organization and brain-culture interactions. We derive novel predictions and show how this research fits into a general view of cognition as embodied, grounded and situated. (C) 2015 Elsevier Ltd. All rights reserved.
Research on problem solving offers insights into how humans process task-related information and which strategies they use (Newell and Simon, 1972; Öllinger et al., 2014). Problem solving can be defined as the search for possible changes in one's mind (Kahneman, 2003). In a recent study, Adams et al. (2021) assessed whether the predominant problem solving strategy when making changes involves adding or subtracting elements. In order to do this, they used several examples of simple problems, such as editing text or making visual patterns symmetrical, either in naturalistic settings or on-line. The essence of the authors' findings is a strong preference to add rather than subtract elements across a diverse range of problems, including the stabilizing of artifacts, creating symmetrical patterns, or editing texts. More specifically, they succeeded in demonstrating that “participants were less likely to identify advantageous subtractive changes when the task did not (vs. did) cue them to consider subtraction, when they had only one opportunity (vs. several) to recognize the shortcomings of an additive search strategy or when they were under a higher (vs. lower) cognitive load” (Adams et al., 2021, p. 258).
Addition and subtraction are generally defined as de-contextualized mathematical operations using abstract symbols (Russell, 1903/1938). Nevertheless, understanding of both symbols and operations is informed by everyday activities, such as making or breaking objects (Lakoff and Núñez, 2000; Fischer and Shaki, 2018). The universal attribution of “addition bias” or “subtraction neglect” to problem solving activities is perhaps a convenient shorthand but it overlooks influential framing effects beyond those already acknowledged in the report and the accompanying commentary (Meyvis and Yoon, 2021).
Most importantly, while Adams et al.'s study addresses an important issue, their very method of verbally instructing participants, together with lack of control over several known biases, might render their findings less than conclusive. Below, we discuss our concerns that emerged from the identified biases, namely those regarding the instructions and the experimental materials. Moreover, we refer to research from mathematical cognition that provides new insights into Adams et al.'s findings.
Research on problem solving offers insights into how humans process task-related information and which strategies they use (Newell and Simon, 1972; Öllinger et al., 2014). Problem solving can be defined as the search for possible changes in one's mind (Kahneman, 2003). In a recent study, Adams et al. (2021) assessed whether the predominant problem solving strategy when making changes involves adding or subtracting elements. In order to do this, they used several examples of simple problems, such as editing text or making visual patterns symmetrical, either in naturalistic settings or on-line. The essence of the authors' findings is a strong preference to add rather than subtract elements across a diverse range of problems, including the stabilizing of artifacts, creating symmetrical patterns, or editing texts. More specifically, they succeeded in demonstrating that “participants were less likely to identify advantageous subtractive changes when the task did not (vs. did) cue them to consider subtraction, when they had only one opportunity (vs. several) to recognize the shortcomings of an additive search strategy or when they were under a higher (vs. lower) cognitive load” (Adams et al., 2021, p. 258).
Addition and subtraction are generally defined as de-contextualized mathematical operations using abstract symbols (Russell, 1903/1938). Nevertheless, understanding of both symbols and operations is informed by everyday activities, such as making or breaking objects (Lakoff and Núñez, 2000; Fischer and Shaki, 2018). The universal attribution of “addition bias” or “subtraction neglect” to problem solving activities is perhaps a convenient shorthand but it overlooks influential framing effects beyond those already acknowledged in the report and the accompanying commentary (Meyvis and Yoon, 2021).
Most importantly, while Adams et al.'s study addresses an important issue, their very method of verbally instructing participants, together with lack of control over several known biases, might render their findings less than conclusive. Below, we discuss our concerns that emerged from the identified biases, namely those regarding the instructions and the experimental materials. Moreover, we refer to research from mathematical cognition that provides new insights into Adams et al.'s findings.