Size congruity effects with two-digit numbers: Expanding the number line?

Numerical comparisons of exponential expressions: The saliency of the base component

Exponential expressions represent series that grow at a fast pace such as carbon pollution and the spread of disease. Despite their importance, people tend to struggle with these expressions. In two experiments, participants chose the larger of two exponential expressions as quickly and accurately as possible. We manipulated the distance between the base/power components and their compatibility. In base-power compatible pairs, both the base and power of one expression were larger than the other (e.g., 23 vs. 34), while in base-power incompatible pairs, the base of one expression was larger than the base in the other expression but the relation between the power components of the two expressions was reversed (e.g., 32 vs. 24). Moreover, while in the first experiment the larger power always led to the larger result, in the second experiment we introduced base-result congruent pairs as well. Namely, the larger base led to the larger result. Our results showed a base-power compatibility effect, which was also larger for larger power distances (Experiments 1-2). Furthermore, participants processed the base-result congruent pairs faster and more accurately than the power-result congruent pairs (Experiment 2). These findings suggest that while both the base and power components are processed when comparing exponential expressions, the base is more salient. This exemplifies an incorrect processing of the syntax of exponential expressions, where the power typically has a larger mathematical contribution to the result of the expression.

Place-value and physical size converge in automatic processing of multi-digit numbers

Previous research has shown that multi-digit number processing is modulated by both place-value and physical size of the digits. By pitting place-value against physical size, the present study examined whether one of the attributes had a greater impact on the automatic processing of multi-digit numbers. In three experiments, participants were presented with two-digit number pairs that appeared in frames. They were instructed to select the larger frame while ignoring the numbers within the frames. Importantly, we manipulated the physical size of the digits (i.e., both decade/unit digits were physically larger) within the frames, the unit-decade compatibility (i.e., the relationship between the numerical values of both decade and unit digits was consistent or inconsistent), and the congruity between the numerical values of the decade digits and the frames' physical size (i.e., decade-value-frame-size congruity). In Experiment 1, where all pairs were unit-decade compatible, a decade-value-frame-size congruity effect emerged for pairs with physically larger decade, but not unit, digits. However, when adding unit-decade incompatible pairs (Experiments 2-3), in unit-decade compatible pairs, there was a decade-value-frame-size congruity effect regardless of the digits' physical size. In contrast, in unit-decade incompatible pairs, there was no decade-value-frame-size congruity effect, even when the physically larger digit (i.e., unit) contradicted the place-value information, presumably due to the cancellation of the opposing influences of the digits' physical sizes their place-values. Overall, these findings suggest that place-value and physical size are intertwined in the Hindu-Arabic numerical system and are processed as one.

Why fractions are difficult? Modeling optimal and sub-optimal integration strategies of numerators and denominators by educated adults

Many children and educated adults experience difficulties in understanding and manipulating fractions. In this study, we argue that a major cause of this challenge is rooted in the need to integrate information from two separate informational sources (i.e., denominator and numerator) according to a normative arithmetic rule (i.e., division). We contend that in some tasks, the correct arithmetic rule is replaced by an inadequate (sub-optimal) operation (e.g., multiplication), which leads to inaccurate representation of fractions. We tested this conjecture by applying two rigorous models of information integration : (a) functional measurement (Experiments 1-3) and (b) conjoint measurement (Experiment 4-5) to data from number-to-line and comparative judgment tasks. These allowed us to compare participants' integration strategies with that of an ideal-observer model. Functional measurement analyses on data from the number-to-line task, revealed that participants could represent the global magnitude of proper and improper fractions quite accurately and combine the fractions' components according to an ideal-observer model. However, conjoint measurement analyses on data from the comparative judgment task, showed that most participants combined these fractions' components according to a sub-optimal (saturated) observer model, that is inconsistent with an ideal-observer (additive) model. These results support the view that educated adults are capable of extracting multiple types of representations of fractions depending on the task at-hand. These representations can be either accurate and conform with normative arithmetic or approximated and inconsistent with normative arithmetic. The latter may lead to the observed difficulties people experience with fractions.

Distinct neural-behavioral correspondence within face processing and attention networks for the composite face effect

The composite face effect (CFE) is recognized as a hallmark for holistic face processing, but our knowledge remains sparse about its cognitive and neural loci. Using functional magnetic resonance imaging with independent localizer and complete composite face task, we here investigated its neural-behavioral correspondence within face processing and attention networks. Complementing classical comparisons, we adopted a dimensional reduction approach to explore the core cognitive constructs of the behavioral CFE measurement. Our univariate analyses found an alignment effect in regions associated with both the extended face processing network and attention networks. Further representational similarity analyses based on the Euclidian distances among all experimental conditions were used to identify cortical regions with reliable neural-behavioral correspondences. Multidimensional scaling and hierarchical clustering analyses for neural-behavioral correspondence data revealed two principal components underlying the behavioral CFE effect, which fit best to the neural responses in the bilateral insula and medial frontal gyrus. These findings highlight the distinct neurocognitive contributions of both face processing and attentional networks to the behavioral CFE outcome, which bridge the gaps between face recognition and attentional control models.

Can the Stroop effect serve as the gold standard of conflict monitoring and control? A conceptual critique

The Stroop effect has been a key to the assay of selective attention since the time of the epoch-making study by J.R. Stroop almost a century ago. However, recent work based on computational modeling and recording of brain activations ignored the primary meaning of the Stroop effect as a measure of selectivity-with the Stroop test losing its raison d'être. Espousing the new framework, numerous studies in the past 20 years conceived performance in the Stroop task in terms of conflict-induced adjustments governed by central control on a trial-to-trial basis. In the face of this tsunami, we try to convince the reader that the Stroop effect cannot serve as a testing ground for conflict-monitoring and control, because these constructs are fundamentally unsuited to serve as a candidate theory of Stroop processes. A range of problems are discussed that singly and collectively pose grave doubts regarding the validity of a control and conflict monitoring account in the Stroop domain. We show how the key notion of conflict is misconstrued in conflict-monitoring models. Due to space limitations and for sake of wider accessibility, our treatment here cannot be technical.

How facial aging affects perceived gender: Insights from maximum likelihood conjoint measurement

Conjoint measurement was used to investigate the joint influence of facial gender and facial age on perceived gender (Experiment 1) and perceived age (Experiment 2). A set of 25 faces was created, covarying independently five levels of gender (from feminine to masculine) and five levels of age (from young to old). Two independent groups of observers were presented with all possible pairs of faces from this set and compared which member of the pair appeared as more masculine (Experiment 1) or older (Experiment 2). Three nested models of the contribution of gender and age to judgment (i.e., independent, additive, and saturated) were fit to the data using maximum likelihood. The results showed that both gender and age contributed to the perceived gender and age of the faces according to a saturated observer model. In judgments of gender (Experiment 1), female faces were perceived as more masculine as they became older. In judgments of age (Experiment 2), young faces (age 20 and 30) were perceived as older as they became more masculine. Taken together, the results entail that: (a) observers integrate facial gender and age information when judging either of the dimensions, and that (b) cues for femininity and cues for aging are negatively correlated. This correlation exerts stronger influence on female faces, and can explain the success of cosmetics in concealing signs of aging and exaggerating sexually dimorphic features.

Conjoint measurement of physical size and numerical magnitude: Numerals do not automatically activate their semantic meaning

I applied the methodology known as maximum likelihood conjoint measurement (MLCM) to diagnose how numerical magnitude affects the perception of physical size of numerals in the size congruity effect (SCE). Traditional theories of the SCE argue for automatic activation of numerical magnitude, and therefore the obligatory interaction in perception between number and size. Participants in the current study were presented with pairs of numerals varying on physical size and numerical magnitude. They judged which member of the pair was physically larger, while ignoring numerical magnitude. Three nested observer models (i.e. independent, additive, and saturated) of the contribution of physical size and numerical magnitude to perceived size were fit to the data using maximum likelihood. The independent observer model exhibited the best fit for the majority of observers. These results cast doubts on the view that numerical magnitude is activated automatically in judgments of physical size of numerals.

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.

No power: exponential expressions are not processed automatically as such

Little is known about the mental representation of exponential expressions. The present study examined the automatic processing of exponential expressions under the framework of multi-digit numbers, specifically asking which component of the expression (i.e., the base/power) is more salient during this type of processing. In a series of three experiments, participants performed a physical size comparison task. They were presented with pairs of exponential expressions that appeared in frames that differed in their physical sizes. Participants were instructed to ignore the stimuli within the frames and choose the larger frame. In all experiments, the pairs of exponential expressions varied in the numerical values of their base and/or power component. We manipulated the compatibility between the base and the power components, as well as their physical sizes to create a standard versus nonstandard syntax of exponential expressions. Experiments 1 and 3 demonstrate that the physically larger component drives the size congruity effect, which is typically the base but was manipulated here in some cases to be the power. Moreover, Experiments 2 and 3 revealed similar patterns, even when manipulating the compatibility between base and power components. Our findings support componential processing of exponents by demonstrating that participants were drawn to the physically larger component, even though in exponential expressions, the power, which is physically smaller, has the greater mathematical contribution. Thus, revealing that the syntactic structure of an exponential expression is not processed automatically. We discuss these results with regard to multi-digit numbers research.

Decomposing the composite face effect: Evidence for non-holistic processing based on the ex-Gaussian distribution

Composite faces fuse the top and bottom halves from two different faces to create a powerful illusion of a novel face. It has been argued that composite faces are processed holistically, namely that the constituent face parts are perceived as a template, rather than independent features. This study sought to uncover the locus of the composite face effect by relating its empirical reaction time distributions to theoretical ex-Gaussian parameters. The results showed that the composite face effect for unfamiliar (Experiment 1) and familiar (Experiment 2) faces is generated by pure changes in the exponential component of the ex-Gaussian distribution. This held true for both partial and complete design measures. The exponential component has been attributed to working memory and attentional processes. The results suggest the involvement of attentional and working memory processes in the composite face effect and in the perception of faces in general. They cast doubts on the holistic nature of face processing. The results also provide important constraints on future computational theories of the effect.

Reclaiming the Stroop Effect Back From Control to Input-Driven Attention and Perception

According to a growing consensus, the Stroop effect is understood as a phenomenon of conflict and cognitive control. A tidal wave of recent research alleges that incongruent Stroop stimuli generate conflict, which is then managed and resolved by top-down cognitive control. We argue otherwise: control studies fail to account for major Stroop results obtained over a century-long history of research. We list some of the most compelling developments and show that no control account can serve as a viable explanation for major Stroop phenomena and that there exist more parsimonious explanations for other Stroop related phenomena. Against a wealth of studies and emerging consensus, we posit that data-driven selective attention best accounts for the gamut of existing Stroop results. The case for data-driven attention is not new: a mere twenty-five years ago, the Stroop effect was considered “the gold standard” of attention (MacLeod, 1992). We identify four pitfalls plaguing conflict monitoring and control studies of the Stroop effect and show that the notion of top-down control is gratuitous. Looking at the Stroop effect from a historical perspective, we argue that the recent paradigm change from stimulus-driven selective attention to control is unwarranted. Applying Occam’s razor, the effects marshaled in support of the control view are better explained by a selectivity of attention account. Moreover, many Stroop results, ignored in the control literature, are inconsistent with any control account of the effect.

Can we perceive two colors at the same time? A direct test of Huang and Pashler's (2007) Boolean map theory of visual attention

Can observers access two spatially separated color targets (e.g., red and green) at the same time (i.e., in parallel)? According to the Boolean map theory of visual attention (Huang & Pashler, Psychological Review, 114(3), 599-631, 2007), access to two different features that belong to the same dimension (e.g., red and green targets) is limited and therefore can be held only in a serial fashion. The current study proposes a strong test of the Boolean map theory of attention through the application of two of the most rigorous stochastic approaches to response times modeling-the system factorial technology (Townsend & Nozawa, 1995) and the logical rule models (Fifić, Little, & Nosofsky, Psychological Review, 117, 309-348, 2010). These approaches allowed identification of serial, parallel, and coactive architectures in the processing of multicolor targets. The results showed that multiple-color targets are processed serially when observers are required to process all the targets in the display (i.e., an exhaustive stopping rule), and in parallel or coactively when observers can terminate the search when one of the targets is found (i.e., self-terminating stopping rule). These results are generally inconsistent with predictions of the Boolean map theory. They highlight the role of stopping rules in multicolor visual search, as well as the flexibility of the attentional system in shifting between processing architectures.

The Size Congruity Effect Vanishes in Grasping: Implications for the Processing of Numerical Information

Judgments of the physical size in which a numeral is presented are often affected by the task-irrelevant attribute of its numerical magnitude, the Size Congruity Effect (SCE). The SCE is typically interpreted as a marker of the automatic activation of numerical magnitude. However, a growing literature shows that the SCE is not robust, a possible indication that numerical information is not always activated in an automatic fashion. In the present study, we tested the SCE via grasping by way of resolving the automaticity debate. We found results that challenge the robustness of the SCE and, consequently, the validity of the automaticity assumption. The SCE was absent when participants grasped the physically larger object of a pair of 3D wooden numerals. An SCE was still recorded when the participants perceptually indicated the general location of the larger object, but not when they grasped that object. These results highlight the importance of the sensory domain when considering the generality of a perceptual effect.

Loudness counts: Interactions between loudness, number magnitude, and space

ATOM (a theory of magnitude) suggests that magnitude information of different formats (numbers, space, and time) is processed within a generalized magnitude network. In this study we investigated whether loudness, as a possible indicator of intensity and magnitude, interacts with the processing of numbers. Small and large numbers, spoken in a quiet and a loud voice, were simultaneously presented to the left and right ear (Experiments 1a and 1b). Participants judged whether the number presented to the left or right ear was louder or larger. Responses were faster when the smaller number was spoken in a quiet voice, and the larger number in a loud voice. Thus, task-irrelevant numerical information influenced the processing of loudness and vice versa. This bi-directional link was also confirmed by classical SNARC paradigms (spatial-numerical association of response codes; Experiments 2a-2c) when participants again judged the magnitude or loudness of separately presented stimuli. In contrast, no loudness-number association was found in a parity judgment task. Regular SNARC effects were found in the magnitude and parity judgment task, but not in the loudness judgment task. Instead, in the latter task, response side was associated with loudness. Possible explanations for these results are discussed.

Bottom-up and top-down attentional contributions to the size congruity effect

The size congruity effect refers to the interaction between the numerical and physical (i.e., font) sizes of digits in a numerical (or physical) magnitude selection task. Although various accounts of the size congruity effect have attributed this interaction to either an early representational stage or a late decision stage, only Risko, Maloney, and Fugelsang (Attention, Perception, & Psychophysics, 75, 1137-1147, 2013) have asserted a central role for attention. In the present study, we used a visual search paradigm to further study the role of attention in the size congruity effect. In Experiments 1 and 2, we showed that manipulating top-down attention (via the task instructions) had a significant impact on the size congruity effect. The interaction between numerical and physical size was larger for numerical size comparison (Exp. 1) than for physical size comparison (Exp. 2). In the remaining experiments, we boosted the feature salience by using a unique target color (Exp. 3) or by increasing the display density by using three-digit numerals (Exps. 4 and 5). As expected, a color singleton target abolished the size congruity effect. Searching for three-digit targets based on numerical size (Exp. 4) resulted in a large size congruity effect, but search based on physical size (Exp. 5) abolished the effect. Our results reveal a substantial role for top-down attention in the size congruity effect, which we interpreted as support for a shared-decision account.

On the Internal Representation of Numerical Magnitude and Physical Size

A nascent idea in the numerical cognition literature – the analogical hypothesis ( Pinel, Piazza, Bihan, & Dehaene, 2004 ) – assumes a common noisy code for the representation of symbolic (e.g., numerals) and nonsymbolic (e.g., numerosity, physical size, luminance) magnitudes. The present work subjected this assumption to various tests from the perspective of General Recognition Theory (GRT; Ashby & Townsend, 1986 ) – a multidimensional extension of Signal Detection Theory ( Green & Swets, 1966 ). The GRT was applied to the dimensions of numerical magnitude and physical size with the following goals: (a) characterizing the internal representation of these dimensions in the psychological space, and (b) assessing various types of (in)dependence and separability governing the perception of these dimensions. The results revealed various violations of independence and separability with Stroop incongruent, but not with Stroop congruent stimuli. The outcome suggests that there are deep differences in architecture between Stroop congruent and incongruent stimuli that reach well beyond the semantic relationship involved.

Paying attention to attention: evidence for an attentional contribution to the size congruity effect

Understanding the mechanisms supporting our comprehension of magnitude information represents a key goal in cognitive psychology. A major phenomenon employed in the pursuit of this goal has been the physical size congruity effect-namely, the observation that comparing the relative numerical sizes of two numbers is influenced by their relative physical sizes. The standard account of the physical size congruity effect attributes it to the automatic influence of the comparison of irrelevant physical magnitudes on numerical judgments. Here we develop an alternative account of this effect on the basis of the operation of attention in the typical size congruity display and the temporal dynamics of number comparison. We also provide a test of a number of predictions derived from this alternative account by combining a physical size congruity manipulation with a manipulation designed to alter the operation of attention within the typical size congruity display (i.e., a manipulation of the relative onsets of the digits). This test provides evidence consistent with an attentional contribution to the size congruity effect. Implications for our understanding of magnitude and the interactions between attention and magnitude are discussed.

Empirical and theoretical studies on number comparison: design parameters and research questions

For well over one-hundred years, several key factors have been well established in the study of number comparison, including mental number line, numerical distance effect, and effect of sensory representation on number processing. The purpose of this article is to put some of these studies together to discuss design parameters and research questions addressed in the mental number comparison studies. Most of the studies discuss sensory representation and abstract number representation as well as degree of their interaction. In order to give the different views on a particular research question, the author classified studies under the related research questions. For example, Stroop and size congruity effect studies are addressed under this title chronologically. It was very clear that the design parameters and research question might change the interpretation of a task. It may be time to shift attention from the question of the interaction degree of sensory representation and abstract representation to a larger scope. The larger scope would be to understand the differences and similarities between different groups using a universal approach.

Digit Repetition Effect in Two-Digit Number Comparison

Previous data from two-digit number naming show that when target and prime numbers share one digit at different positions (e.g., target 28 and prime 86 or 72) an inhibition effect may be observed (Ratinckx, Brysbaert, & Fias, 2005). Such an effect has been ascribed to the mechanism of morpho-phonological transcoding from Arabic to verbal format. We evaluate the alternative hypothesis of an inhibition effect arising during the Arabic form processing when two different syntactic values have to be assigned to the same single digit. In Experiments 1 and 3, a digit repetition effect was observed in number comparison tasks, even when phonological transcoding is blocked by an articulatory suppression task. Conversely, Experiment 2 showed that no digit repetition effect can be found with number comparison of verbal written numbers. Results are compatible with an Arabic-syntactic processing stage, where units and decades are decomposed and where each single digit is activated and recognized.

Proximity and Precedence in Arithmetic

How does the physical structure of an arithmetic expression affect the computational processes engaged in by reasoners? In handwritten arithmetic expressions containing both multiplications and additions, terms that are multiplied are often placed physically closer together than terms that are added. Three experiments evaluate the role such physical factors play in how reasoners construct solutions to simple compound arithmetic expressions (such as “2 + 3 × 4”). Two kinds of influence are found: First, reasoners incorporate the physical size of the expression into numerical responses, tending to give larger responses to more widely spaced problems. Second, reasoners use spatial information as a cue to hierarchical expression structure: More narrowly spaced subproblems within an expression tend to be solved first and tend to be multiplied. Although spatial relationships besides order are entirely formally irrelevant to expression semantics, reasoners systematically use these relationships to support their success with various formal properties.

Slippery platform: The role of automatic and intentional processes in testing the effect of notation

The type of processing of numerical dimensions varies greatly and is governed by context. Considering this flexibility in tandem with a fuzzy demarcation line between automatic and intentional processes, it is suggested that testing the effect of notation should not be confined to automatic processing, in particular to passive viewing. Recent behavioral data satisfying the authors' stipulations reveal a considerable, though perhaps not exclusive, core of common abstract processing.

Automaticity for numerical magnitude of two-digit Arabic numbers in children

This paper examines the automatic processing of the numerical magnitude of two-digit Arabic numbers using a Stroop-like task in school-aged children. Second, third, and fourth graders performed physical size judgments on pairs of two-digit numbers varying on both physical and numerical dimensions. To investigate the importance of synchrony between the speed of processing of the numerical magnitude and the physical dimensions on the size congruity effect (SCE), we used masked priming: numerical magnitude was subliminally primed in half of the trials, while neutral priming was used in the other half. The results indicate a SCE in physical judgments, providing the evidence of automatic access to the magnitude of two-digit numbers in children. This effect was modulated by the priming type, as a SCE only appeared when the numerical magnitude was primed. This suggests that young children needed a relative synchronization of numerical and physical dimensions to access the magnitude of two-digit numbers automatically.