The Order of Magnitude: Why SNARC-like Tasks (Still) Cannot Support a Generalized Magnitude System

The influence of semantics and numerical representation on the SNARC effect

World semantics refers to how individuals comprehend and ascribe meaning to phenomena within specific sociocultural and environmental contexts. And it significantly influences how people conceptualize numerical properties, such as quantity ("how many") and order ("which"), influencing their resolution of cardinal and ordinal numerical task. In order to reveal the influence of world semantics on the joint effects of encoding spatial-numerical responses related to cardinal and ordinal numbers, this study conducted three experimental contexts with distinct numerical representations: (1) a non-symbolic context, where numbers were represented using dot arrays; (2) a symbolic context, where numbers were presented as Chinese characters; and (3) a word problem solving context, where numerical values were embedded in real-world mathematical scenarios. To investigate how numerical attributes (cardinality vs. ordinality) influence on the SNARC effect under varying level of world semantics across these three contexts. Results showed that in the non-symbolic context, the SNARC effect is primarily observed for cardinal numbers. In the symbolic context, ordinal number elicited a stronger SNARC effect. However, in the word problem context, no significant SNARC effect was observed. The findings suggest that as world semantics become more prominent, the SNARC effect diminishes, particularly in applied numerical contexts. Additionally, ordinal numbers exhibited a stronger SNARC effect than cardinal numbers, supporting the role of sequential processing in spatial-numerical associations.

A SNARC-like effect for visual speed

Numerical and nonnumerical magnitudes can be represented along a hypothetical left-to-right continuum, where smaller quantities are associated with the left side and larger quantities with the right side. However, these representations are flexible, as their intensity and direction can be modulated by various contextual cues and task demands. In four experiments, we investigated the spatial representation of visual speed. Visual speed is inherently connected to physical space and spatial directions, making it distinct from other magnitudes. With this in mind, we explored whether the spatial representation of visual speed aligns with the typical left-to-right orientation or is influenced dynamically by the movement direction of the stimuli. Participants compared the speed of random dot kinematograms to a reference speed using lateralised response keys. On each trial, all dots moved consistently in one single direction, which varied across the experiments and could also vary from trial to trial in Experiments 2 and 4. The dot movements were left-to-right (Experiment 1), random across a 360° spectrum (Experiment 2), right-to-left (Experiment 3), and random left-to-right or right-to-left (Experiment 4). The results supported a relatively stable left-to-right spatial representation of speed (Experiments 1-3), which was compromised by mutable motion directions along the horizontal axis (Experiment 4). We suggest that representing stimuli as belonging to a single set rather than different sets, may be crucial for the emergence of spatial representations of quantities.

Beyond fixed sets: boundary conditions for obtaining SNARC-like effects with continuous semantic magnitudes

Previous research has demonstrated the presence of an effect (i.e., the spatial-numerical association of response codes or SNARC) in both numerical parity and magnitude judgment tasks in which smaller numerical magnitudes are manually responded to faster on the left side and larger numerical magnitudes on the right side. Such a result has typically been attributed to a spatially based representation of numerical magnitude in long-term memory, the format of which has recently been postulated to be positional in line with learning of a canonically ordered number sequence. As a test of this view, in the current research, participants made classification judgments involving either the size (N = 88) or the living-nonliving status (N = 114) corresponding to the names of animals/objects etc. to which no learned canonical ordering of size exists. Names were taken from a very large set of 400 animals/objects etc. and each name was presented only once in an experimental session. Responses were made using left and right manual keypresses. In this work, the relation between response time and the relative sizes of the animals/objects did not differ across the left-right side of response indicating that SNARC-like effects did not occur. As such, the results suggest that space is not an inherent aspect of the long-term representation of magnitude in the brain and that some form of positional coding of magnitude is necessary for SNARC-like effects to occur.

The spatial representation of loudness in a timbre discrimination task

When participants decide whether a presented tone is loud or soft they react faster to loud tones with a top-sided response key in comparison to a bottom-sided response key and vice versa for soft tones. This effect is comparable to the well-established horizontal Spatial-Numerical Association of Response Codes (SNARC) effect and is often referred to as Spatial-Musical Association of Response Codes (SMARC) effect for loudness. The SMARC effect for loudness is typically explained by the assumption of a spatial representation or by the polarity correspondence principle. Crucially, both theories differ in the prediction of the SMARC effect when loudness is task-irrelevant. Therefore, we investigated whether the SMARC effect still occurs in a timbre discrimination task: Participants (N = 36) heard a single tone and classified its timbre with vertically arranged response keys. Additionally, the tone's loudness level varied in six levels. In case of a spatial representation, the SMARC effect should still occur while in case of polarity corresponding principle, the effect should be absent. Results showed that the SMARC effect was still present and that the differences between top-sided and bottom-sided responses were a linear function of loudness level indicating a continuous spatial representation of loudness.

Near-optimal integration of the magnitude information of time and numerosity

Magnitude information is often correlated in the external world, providing complementary information about the environment. As if to reflect this relationship, the perceptions of different magnitudes (e.g. time and numerosity) are known to influence one another. Recent studies suggest that such magnitude interaction is similar to cue integration, such as multisensory integration. Here, we tested whether human observers could integrate the magnitudes of two quantities with distinct physical units (i.e. time and numerosity) as abstract magnitude information. The participants compared the magnitudes of two visual stimuli based on time, numerosity, or both. Consistent with the predictions of the maximum-likelihood estimation model, the participants integrated time and numerosity in a near-optimal manner; the weight of each dimension was proportional to their relative reliability, and the integrated estimate was more reliable than either the time or numerosity estimate. Furthermore, the integration approached a statistical optimum as the temporal discrepancy of the acquisition of each piece of information became smaller. These results suggest that magnitude interaction arises through a similar computational mechanism to cue integration. They are also consistent with the idea that different magnitudes are processed by a generalized magnitude system.

Thinking about order: a review of common processing of magnitude and learned orders in animals

Rich behavioral and neurobiological evidence suggests cognitive and neural overlap in how quantitatively comparable dimensions such as quantity, time, and space are processed in humans and animals. While magnitude domains such as physical magnitude, time, and space represent information that can be quantitatively compared (4 "is half of" 8), they also represent information that can be organized ordinally (1→2→3→4). Recent evidence suggests that the common representations seen across physical magnitude, time, and space domains in humans may be due to their common ordinal features rather than their common quantitative features, as these common representations appear to extend beyond magnitude domains to include learned orders. In this review, we bring together separate lines of research on multiple ordinal domains including magnitude-based and learned orders in animals to explore the extent to which there is support for a common cognitive process underlying ordinal processing. Animals show similarities in performance patterns across natural quantitatively comparable ordered domains (physical magnitude, time, space, dominance) and learned orders (acquired through transitive inference or simultaneous chaining). Additionally, they show transfer and interference across tasks within and between ordinal domains that support the theory of a common ordinal representation across domains. This review provides some support for the development of a unified theory of ordinality and suggests areas for future research to better characterize the extent to which there are commonalities in cognitive processing of ordinal information generally.