Exploring the origin of the number-size congruency effect: Sensitivity or response bias?

Number, size, and space associated in a common system by distinct mechanisms

The spatial numerical association of response codes (SNARC) effect demonstrates that people respond faster to small numbers with their left hand and faster to large numbers with their right hand. The size congruity effect (SCE) refers to the fact that congruent trials between numerical values and physical sizes are faster than incongruent trials. Previous studies have found that the SNARC effect and SCE are independent when magnitudes or sizes are processed explicitly. This study aimed to explore whether number, size, and space are common and distinct mechanisms using an implicit parity judgment task. The results showed that the SNARC effect, SCE, and SNARC-like effect all co-existed. Furthermore, there was a significant interaction between the SNARC effect and SCE, in which the SNARC effect in the SCE-congruent condition was larger than in the SCE-incongruent condition, whereas SCE merely emerged in the SNARC-compatible trials. However, participants responded to small numbers in large size faster than to large numbers in small size with the left hand in SCE-incongruent trials, which reflected that number-space mapping (SNARC effect) was stronger than size-space mapping (SNARC-like effect). These findings provide new evidence for A Theory of Magnitude (ATOM), which suggests that number, size, and space are associated with a common generalized magnitude system through distinct mechanisms.

Space-time interference: The asymmetry we get out is the asymmetry we put in

Temporal judgments are more affected by space than vice versa. This asymmetry has often been interpreted as primacy of spatial representations over temporal ones. This interpretation is in line with conceptual metaphor theory that humans conceptualize time by spatial metaphors, but is inconsistent with the assumption of a common neuronal magnitude system. Here we review the accumulating evidence for a genuinely symmetric interference between time and space and discuss potential explanations as to why asymmetric interference can arise, both with respect to the interaction between spatial size and temporal duration, and the interaction between traveled distance and travel time. Contrary to the view of hierarchical representations of time and space, our review suggests that asymmetric interference can be explained on the basis of working memory processes and the aspect of speed inherent in dynamic stimuli. We conclude that the asymmetry we often get out (space affects time more than vice versa) is a consequence of the asymmetry we put in (by using biased paradigms and stimuli facilitating spatial processing).

Effect of egocentric and allocentric reference frames on spatial-numerical associations

From an embodied view of cognition, sensorimotor mechanisms are strongly involved in abstract processing, such as Arabic number meanings. For example, spatial cognition can influence number processing. These spatial-numerical associations (SNAs) have been deeply explored since the seminal SNAs of response code (SNARC) effect (i.e., faster left/right sided responses to small/large magnitude numbers, respectively). Although these SNAs along the transverse plane (left-to-right axis) have been extensively studied in cognitive sciences, no systematic assessment of other planes of the tridimensional space has been afforded. Moreover, there is no evidence of how SNAs organise themselves throughout the changes in spatial body-reference frames (egocentric and allocentric). Hence, this study aimed to explore how SNAs organise themselves along the transverse and sagittal planes when egocentric and allocentric changes are processed during body displacements in the environment. In the first experiment, the results revealed that, when the participants used an egocentric reference, SNAs were observed only along the sagittal plane. In a second experiment that used an allocentric reference, the reversed pattern of results was observed: SNAs were present only along the transverse plane of the body. Overall, these findings suggest that, depending on the spatial reference frames of the body, SNAs are strongly flexible.

Numerical values modulate size perception

The link between various codes of magnitude and their interactions has been studied extensively for many years. In the current study, we examined how the physical and numerical magnitudes of digits are mapped into a combined mental representation. In two psychophysical experiments, participants reported the physically larger digit among two digits. In the identical condition, participants compared digits of an identical value (e.g., "2" and "2"); in the different condition, participants compared digits of distinct numerical values (i.e., "2" and "5"). As anticipated, participants overestimated the physical size of a numerically larger digit and underestimated the physical size of a numerically smaller digit. Our results extend the shared-representation account of physical and numerical magnitudes.

A sensorimotor perspective on numerical cognition

Numbers are present in every part of modern society and the human capacity to use numbers is unparalleled in other species. Understanding the mental and neural representations supporting this capacity is of central interest to cognitive psychology, neuroscience, and education. Embodied numerical cognition theory suggests that beyond the seemingly abstract symbols used to refer to numbers, their underlying meaning is deeply grounded in sensorimotor experiences, and that our specific understanding of numerical information is shaped by actions related to our fingers, egocentric space, and experiences with magnitudes in everyday life. We propose a sensorimotor perspective on numerical cognition in which number comprehension and numerical proficiency emerge from grounding three distinct numerical core concepts: magnitude, ordinality, and cardinality.

The value of banknotes: relevance of size, colour and design

In the current study, we evaluate the relevance of three physical features when people retrieve the monetary value of banknotes. To this end, three monetary comparison tasks were designed in which in each trial a pair of banknotes were presented and participants selected the one with higher monetary value. In each task, a different banknote feature (size, colour and design) was examined and a congruent and an incongruent condition (the value of the physical feature corresponded or not to its actual value, respectively) were compared to a neutral condition (no information about the physical feature was provided). We found a pattern of facilitation and interference effects which suggests that size is the most relevant physical feature for accessing the monetary value of banknotes followed by colour. However, the availability of a variety of designs across banknotes seemed not to facilitate the performance of the task, but rather the opposite, hindering the monetary comparison task.

Sub-cortical areas process physical size but not numerical value

A robust finding in the numerical cognition literature is that the physical size and the numerical value of two to-be-compared digits interact, resulting in a size congruity effect (SiCE). The current study focuses on the possible role of prestriate areas in a digit comparison Stroop-like task. In the visual pathway, prestriate areas refer to regions from the retina up to the primary visual cortex (V1). I hypothesized that processing of physical size, but not numerical value, begins already in prestriate areas. This is because physical size is a basic visual feature while the numerical value of a symbol is a learned convention that should be retrieved from long-term memory. Adult participants compared the size or the numerical value of two digits. Without participants' awareness, I projected the digits either to the same eye, or each digit to a different eye. The latter type of presentation prevents prestriate areas from taking part in comparing the digits. Therefore, slower a response time under this condition hints at the involvement of prestriate areas. Evidence confirmed the initial hypothesis, demonstrating slower performance when the stimuli are segregated between the eyes but only for physical size comparisons. This finding suggests that at least the initial processing of physical size, when relevant, is done before, and by different neural substrate than numerical value. The implications of the study and future directions are discussed.

Task and information conflicts in the numerical Stroop task

Studies of the Stroop color‐word task have provided evidence for the existence of two conflicts: (1) an early task conflict between noting the relevant color and reading afforded by the irrelevant word (or word‐like stimuli), and (2) a late information conflict between the information provided by the word and the information provided by the color. Measurements of pupil changes, in addition to reaction time (RT), have extended understanding regarding these two conflicts. The current work examines the generalizability of such understanding. We ask whether similar processes work in the comparative judgment of numbers (e.g., in the numerical Stroop task). We present two experiments that support and extend the knowledge gained in the word‐color context to numerical processing. Similar to results with the Stroop color‐word task, we found a dissociation between RT and pupillometry and an early task conflict followed by an information conflict.

Recent Stroop color‐word studies have indicated the existence of an early task conflict followed by an information conflict. The current experiments used pupillometry to show the existence of these two conflicts in a numerical Stroop‐like task. Accordingly, our research extends and generalizes the two‐conflict notion beyond color‐word processing.

Spatial attention shifts contribute to the size congruity effect

The size congruity effect in a numerical Stroop task shows that magnitude judgments of two numbers are faster and more accurate when the numerically larger number also appears in a physically larger size, indicating the interaction between numerical and physical magnitudes. It has recently been suggested that spatial shifts of attention between the two numbers may contribute to the size congruity effect. However, a complete line of evidence for the attentional attribution to the size congruity effect remains to be established. Therefore, the present study aimed to provide further demonstrations for the idea that spatial shifts of attention contribute to the size congruity effect during magnitude judgments regarding either the numerical or physical dimension of two numbers. Participants were sequentially or simultaneously presented with a pair of single-digit Arabic numbers whose numerical and physical magnitudes varied independently. They were instructed to perform a magnitude judgment regarding the numerical value or physical size of the paired numbers. Across three experiments, we consistently found that the size congruity effect was reduced or eliminated when number pairs were presented sequentially compared to when they were presented simultaneously. Because in the sequential presentation mode the paired numbers were successively presented at central fixation and therefore spatial attention shifts should be completely precluded by the central presentation of number stimuli, the present findings support the notion that spatial shifts of attention between numbers in the simultaneous presentation mode play an important role in generating the size congruity effect for both numerical and physical tasks.

Processing symbolic magnitude information conveyed by number words and by scalar adjectives

Humans not only process and compare magnitude information such as size, duration, and number perceptually, but they also communicate about these properties using language. In this respect, a relevant class of lexical items are so-called scalar adjectives like “big,” “long,” “loud,” and so on which refer to magnitude information. It has been proposed that humans use an amodal and abstract representation format shared by different dimensions, called the generalised magnitude system (GMS). In this paper, we test the hypothesis that scalar adjectives are symbolic references to GMS representations, and, therefore, GMS gets involved in processing their meaning. Previously, a parallel hypothesis on the relation between number symbols and GMS representations has been tested with the size congruity paradigm. The results of these experiments showed interference between the processing of number symbols and the processing of physical (font-) size. In the first three experiments of the present study (total N = 150), we used the size congruity paradigm and the same/different task to look at the potential interaction between physical size magnitude and numerical magnitude expressed by number words. In the subsequent three experiments (total N = 149), we looked at a parallel potential interaction between physical size magnitude and scalar adjective meaning. In the size congruity paradigm, we observed interference between the processing of the numerical value of number words and the meaning of scalar adjectives, on the one hand, and physical (font-) size, on the other hand, when participants had to judge the number words or the adjectives (while ignoring physical size). No interference was obtained for the reverse situation, i.e., when participants judged the physical font size (while ignoring numerical value or meaning). The results of the same/different task for both number words and scalar adjectives strongly suggested that the interference that was observed in the size congruity paradigm was likely due to a response conflict at the decision stage of processing rather than due to the recruitment of GMS representations. Taken together, it can be concluded that the size congruity paradigm does not provide evidence in support the hypothesis that GMS representations are used in the processing of number words or scalar adjectives. Nonetheless, the hypothesis we put forward about scalar adjectives is still is a promising potential line of research. We make a number of suggestions for how this hypothesis can be explored in future studies.

The distance effect on discrimination ability and response bias during magnitude comparison in a go/no-go task

The distance effect is the change in the performance during numerical magnitude comparison, depending on the numerical distance between the compared numbers (Moyer & Landauer, Nature, 215[5109], 1519-1520, 1967). This effect is generally accepted as evidence for the mental number line (MNL) hypothesis, which proposes that the mental representation of the numbers align in an increasing linear (or monotone) order. The majority of studies investigating the distance effect are focused on the reaction time (RT) findings, which show slower responses for closer numbers. In the present study, we examined the distance effect by applying signal detection theory (SDT) to a magnitude comparison task. We aimed to reveal whether discrimination ability and the response bias measures were affected by the location of numbers on the MNL. To accomplish this, we developed a magnitude comparison task using a go/no-go procedure in which participants performed a magnitude comparison based on a reference number (i.e., 5). Results revealed a substantial distance effect in both sensitivity and response bias measures-a better discrimination performance for far numbers, and a larger response bias for close numbers. In addition, an RT distribution analysis revealed that the distance effect seems to originate mainly from slower responses. Based on the current data, we suggest that sensitivity and response bias measures could offer comprehensive information in the understanding of number-based decisions.

Response-irrelevant number, duration, and extent information triggers the SQARC effect: Evidence from an implicit paradigm

Spatial-Numerical Association of Response Codes (SNARC) and Spatial-Quantity Association of Response Codes (SQARC) effects are evident when people produce faster left-sided responses to smaller numbers, sizes, and durations and faster right-sided responses to larger numbers, sizes, and durations. SQARC effects have typically been demonstrated in paradigms where the explicit processing of quantity information is required for successful task completion. The current study tested whether the implicit presentation of task-irrelevant magnitude information could trigger a SQARC effect as has been demonstrated previously when task-irrelevant information triggers a SNARC effect. In Experiment 1, participants (n = 20) made orientation judgements for triangles varying in numerosity and physical extent. In Experiment 2, participants (n = 20) made orientation judgements for triangles varying in numerosity and for a triangle preceded by a delay of varying duration. SNARC effects were observed for the numerosity conditions of Experiments 1 and 2 replicating Mitchell et al. SQARC effects were also demonstrated for physical extent and for duration. These findings demonstrate that SQARC effects can be implicitly triggered by the presentation of the task-irrelevant magnitude.

Categorizing digits and the mental number line

Following the classical work of Moyer and Landauer (1967), experimental studies investigating the way in which humans process and compare symbolic numerical information regularly used one of two experimental designs. In selection tasks, two numbers are presented, and the task of the participant is to select (for example) the larger one. In classification tasks, a single number is presented, and the participant decides if it is smaller or larger than a predefined standard. Many findings obtained with these paradigms fit in well with the notion of a mental analog representation, or an Approximate Number System (ANS; e.g., Piazza 2010). The ANS is often conceptualized metaphorically as a mental number line, and data from both paradigms are well accounted for by diffusion models based on the stochastic accumulation of noisy partial numerical information over time. The present study investigated a categorization paradigm in which participants decided if a number presented falls into a numerically defined central category. We show that number categorization yields a highly regular, yet considerably more complex pattern of decision times and error rates as compared to the simple monotone relations obtained in traditional selection and classification tasks. We also show that (and how) standard diffusion models of number comparison can be adapted so as to account for mean and standard deviations of all RTs and for error rates in considerable quantitative detail. We conclude that just as traditional number comparison, the more complex process of categorizing numbers conforms well with basic notions of the ANS.

Compatibility between object size and response side in grasping: the left hand prefers smaller objects, the right hand prefers larger objects

It has been proposed that the brain processes quantities such as space, size, number, and other magnitudes using a common neural metric, and that this common representation system reflects a direct link to motor control, because the integration of spatial, temporal, and other quantity-related information is fundamental for sensorimotor transformation processes. In the present study, we examined compatibility effects between physical stimulus size and spatial (response) location during a sensorimotor task. Participants reached and grasped for a small or large object with either their non-dominant left or their dominant right hand. Our results revealed that participants initiated left hand movements faster when grasping the small cube compared to the large cube, whereas they initiated right hand movements faster when grasping the large cube compared to the small cube. Moreover, the compatibility effect influenced the timing of grip aperture kinematics. These findings indicate that the interaction between object size and response hand affects the planning of grasping movements and supports the notion of a strong link between the cognitive representation of (object) size, spatial (response) parameters, and sensorimotor control.

The number-weight illusion

When objects are manually lifted to compare their weight, then smaller objects are judged to be heavier than larger objects of the same physical weights: the classical size-weight illusion (Gregory, 2004). It is also well established that increasing numerical magnitude is strongly associated with increasing physical size: the number-size congruency effect e.g., (Besner & Coltheart Neuropsychologia, 17, 467-472 1979); Henik & Tzelgov Memory & Cognition, 10, 389-395 1982). The present study investigates the question suggested by combining these two classical effects: if smaller numbers are associated with smaller size, and objects of smaller size appear heavier, then are numbered objects (balls) of equal weight and size also judged as heavier when they carry smaller numbers? We present two experiments testing this hypothesis for weight comparisons of numbered (1 to 9) balls of equal size and weight, and report results which largely conform to an interpretation in terms of a new "number-weight illusion".

Compatibility between Physical Stimulus Size and Left-right Responses: Small is Left and Large is Right

According to a theory of magnitude (ATOM, Walsh, 2003, 2015), the cognitive representations of quantity, time, and space share a general magnitude code. Interestingly though, research has largely ignored the relationship between physical (stimulus) size and spatial (response) location. We conducted two experiments investigating compatibility effects between physical stimulus size and left-right responses. In both experiments, right-handed participants responded to a small or a large square stimulus by pressing a left or a right key. In Experiment 1, size was the relevant stimulus feature and we varied the S-R mapping within participants. Results revealed a strong compatibility effect: Performance was better with the compatible mapping (small-left and large-right) than with the incompatible mapping (large-left and small-right). In Experiment 2, participants responded to stimulus color, which varied independently of stimulus size, by pressing a left or right key. Results showed a congruency effect that mirrored the compatibility effect of Experiment 1. The results of our experiments suggest a strong relationship between the cognitive representation of physical (stimulus) size and response location in right-handers. The findings support the notion of a general magnitude code, as proposed in ATOM.