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Institut
Several chronometric biases in numerical cognition have informed our understanding of a mental number line (MNL). Complementing this approach, we investigated spatial performance in a magnitude comparison task. Participants located the larger or smaller number of a pair on a horizontal line representing the interval from 0 to 10. Experiments 1 and 2 used only number pairs one unit apart and found that digits were localized farther to the right with "select larger" instructions than with "select smaller" instructions. However, when numerical distance was varied (Experiment 3), digits were localized away from numerically near neighbors. This repulsion effect reveals context-specific distortions in number representation not previously noticed with chronometric measures.
There has been increasing interest in the spatial mapping of various perceptual and cognitive magnitudes, such as expanding the spatial-numerical association of response codes (SNARC) effect into domains outside of numerical cognition. Recently, De Tommaso and Prpic (Attention, Perception, & Psychophysics, 82, 2765-2773, 2020) reported in this journal that only fast tempos over 104 beats per minute have spatial associations, with more right-sided associations and faster responses for faster tempos. After discussing the role of perceived loudness and possible response strategies, we propose and recommend methodological improvements for further research.
To construct a coherent multi-modal percept, vertebrate brains extract low-level features (such as spatial and temporal frequencies) from incoming sensory signals. However, because frequency processing is lateralized with the right hemisphere favouring low frequencies while the left favours higher frequencies, this introduces asymmetries between the hemispheres. Here, we describe how this lateralization shapes the development of several cognitive domains, ranging from visuo-spatial and numerical cognition to language, social cognition, and even aesthetic appreciation, and leads to the emergence of asymmetries in behaviour. We discuss the neuropsychological and educational implications of these emergent asymmetries and suggest future research approaches.
Magnitude estimation has been studied since the beginnings of scientific psychology and constitutes a fundamental aspect of human behavior. Yet, it has apparently never been noticed that estimates depend on the spatial arrangement used. We tested 167 adults in three experiments to show that the spatial layout of stimuli and responses systematically distorts number estimation, length production, and weight reproduction performance. The direction of distortion depends on the observer's counting habits, but does not seem to reflect the use of spatially associated number concepts. Our results imply that all quantitative estimates are contaminated by a "spell of space" whenever stimuli or responses are spatially distributed.
The literature on spatial associations during number processing is dominated by the SNARC (spatial-numerical association of response codes) effect. We describe spatial biases found for single digits and pairs of numbers, first in the "original" speeded parity task and then extending the scope to encompass different tasks, a range of measures, and various populations. Then we review theoretical accounts before surveying the emerging evidence for similar spatial associations during mental arithmetic. We conclude that the mental number line hypothesis and an embodied approach are useful frameworks for further studies.
Even simple mental arithmetic is fraught with cognitive biases. For example, adding repeated numbers (so-called tie problems, e.g., 2 + 2) not only has a speed and accuracy advantage over adding different numbers (e.g., 1 + 3) but may also lead to under-representation of the result relative to a standard value (Charras et al., 2012, 2014). Does the tie advantage merely reflect easier encoding or retrieval compared to non-ties, or also a distorted result representation? To answer this question, 47 healthy adults performed two tasks, both of which indicated under-representation of tie results: In a result-to-position pointing task (Experiment 1) we measured the spatial mapping of numbers and found a left-bias for tie compared to non-tie problems. In a result-to-line-length production task (Experiment 2) we measured the underlying magnitude representation directly and obtained shorter lines for tie-compared to non-tie problems. These observations suggest that the processing benefit of tie problems comes at the cost of representational reduction of result meaning. This conclusion is discussed in the context of a recent model of arithmetic heuristics and biases.
Even before formal schooling, children map numbers onto space in a directional manner. The origin of this preliterate spatial–numerical association is still debated. We investigated the role of enculturation for shaping the directionality of the association between numbers and space, focusing on counting behavior in 3- to 5-year-old preliterate children. Two studies provide evidence that, after observing reading from storybooks (left-to-right or right-to-left reading) children change their counting direction in line with the direction of observed reading. Just observing visuospatial directional movements had no such effect on counting direction. Complementarily, we document that book illustrations, prevalent in children’s cultures, exhibit directionality that conforms to the direction of a culture’s written language. We propose that shared book reading activates spatiotemporal representations of order in young children, which in turn affect their spatial representation of numbers.
Number to me, space to you
(2022)
Recent work has shown that number concepts activate both spatial and magnitude representations. According to the social co-representation literature which has shown that participants typically represent task components assigned to others together with their own, we asked whether explicit magnitude meaning and explicit spatial coding must be present in a single mind, or can be distributed across two minds, to generate a spatial-numerical congruency effect. In a shared go/no-go task that eliminated peripheral spatial codes, we assigned explicit magnitude processing to participants and spatial processing to either human or non-human co-agents. The spatial-numerical congruency effect emerged only with human co-agents. We demonstrate an inter-personal level of conceptual congruency between space and number that arises from a shared conceptual representation not contaminated by peripheral spatial codes. Theoretical implications of this finding for numerical cognition are discussed.
A recent cross-cultural comparison (Shaki, Fischer, & Petrusic, 2009) suggested that spatially consistent processing habits for words and numbers are a necessary condition for the spatial representation of numbers (Spatial-Numerical Association of Response Codes; SNARC effect). Here we reexamine the SNARC in Israelis who read text from right to left but numbers from left to right. We show that, despite these spatially inconsistent processing habits, a SNARC effect still emerges when the response dimension is spatially orthogonal to the conflicting processing dimension. These results clarify the cognitive conditions for spatial-numerical mappings.
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.
Previous work on spatial-numerical association (SNAs) included either spatially distributed stimuli or responses. This raises the possibility that the inferred spatial nature of number concepts was a methodological artifact. We present results from a novel task that involves two categories (spatially oriented objects and number magnitudes) and dissociates spatial classification from number classification. The results reveal SNAs without inferential limitations of previous work and point to a working memory mechanism that transfers spatial coding across categories.
It has been debated whether negative number concepts are cognitively represented on the same mental number line as positive number concepts. The present study reviews this debate and identifies limitations of previous studies. A method with nonspatial stimuli and responses is applied to overcome these limitations and to document a systematic implicit association of negative numbers with left space, thus indicating a leftward extension of the mental number line. Importantly, this result only held for left-to-right counting adults. Implications for the experiential basis of abstract conceptual knowledge are discussed.
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.
Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum (OM) effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems (e.g., 3+0, 3-0) but reverse OM with non-zero problems (e.g., 2+1, 4-1). In a third experiment, we tested the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving.
Mental arithmetic shows systematic spatial biases. The association between numbers and space is well documented, but it is unknown whether arithmetic operation signs also have spatial associations and whether or not they contribute to spatial biases found in arithmetic. Adult participants classified plus and minus signs with left and right button presses under two counterbalanced response rules. Results from two experiments showed that spatially congruent responses (i.e., right-side responses for the plus sign and left-side responses for the minus sign) were responded to faster than spatially incongruent ones (i.e., left-side responses for the plus sign and right-side responses for the minus sign). We also report correlations between this novel operation sign spatial association (OSSA) effect and other spatial biases in number processing. In a control experiment with no explicit processing requirements for the operation signs there were no sign-related spatial biases. Overall, the results suggest that (a) arithmetic operation signs can evoke spatial associations (OSSA), (b) experience with arithmetic operations probably underlies the OSSA, and (c) the OSSA only partially contributes to spatial biases in arithmetic.
Reproducibility is a defining feature of science, but the extent to which it characterizes current research is unknown. We conducted replications of 100 experimental and correlational studies published in three psychology journals using high-powered designs and original materials when available. Replication effects were half the magnitude of original effects, representing a substantial decline. Ninety-seven percent of original studies had statistically significant results. Thirty-six percent of replications had statistically significant results; 47% of original effect sizes were in the 95% confidence interval of the replication effect size; 39% of effects were subjectively rated to have replicated the original result; and if no bias in original results is assumed, combining original and replication results left 68% with statistically significant effects. Correlational tests suggest that replication success was better predicted by the strength of original evidence than by characteristics of the original and replication teams.
Western adults associate small numbers with left space and large numbers with right space. Where does this pervasive spatial-numerical association come from? In this study, we first recorded directional counting preferences in adults with different reading experiences (left to right, right to left, mixed, and illiterate) and observed a clear relationship between reading and counting directions. We then recorded directional counting preferences in pre-schoolers and elementary school children from three of these reading cultures (left to right, right to left, and mixed). Culture-specific counting biases existed before reading acquisition in children as young as 3 years and were subsequently modified by early reading experience. Together, our results suggest that both directional counting and scanning activities contribute to number-space associations.
Spatial-numerical associations (SNAs) have been studied extensively in the past two decades, always requiring either explicit magnitude processing or explicit spatial-directional processing. This means that the typical finding of an association of small numbers with left or bottom space and of larger numbers with right or top space could be due to these requirements and not the conceptual representation of numbers. The present study compares explicit and implicit magnitude processing in an implicit spatial-directional task and identifies SNAs as artefacts of either explicit magnitude processing or explicit spatial-directional processing; they do not reveal spatial conceptual links. This finding requires revision of current accounts of the relationship between numbers and space.