Accumulation of non-numerical evidence during nonsymbolic number processing in the brain: An fMRI study.

Behavioral evidence has shown that when performing a nonsymbolic number comparison task (e.g., deciding which of two dot arrays contains more dots), participants' responses are sensitive to affected by both numerical (e.g., number of items) and non-numerical magnitudes (i.e., area, density, etc.). Thus far it is unclear what brain circuits support this process of accumulating non-numerical variables during nonsymbolic number processing. To investigate this, 21 adult participants were asked to engage in a dot comparison task. To measure the neural correlates of accumulating numerical and non-numerical variables, we manipulated the number of the non-numerical magnitudes that were congruent (correlated with number) or incongruent (anticorrelated with number). In a control task, participants were asked to choose the darker of two gray rectangles (brightness task). The tasks were matched in terms of their difficulty. The results of a whole brain analysis for regions sensitive to the congruity of numerical and non-numerical magnitudes revealed a region in the right inferior frontal gyrus (rIFG). Activation in this region was found to be correlated with the relative congruency of numerical and non-numerical magnitudes. In contrast, this region was not modulated by difficulty of the brightness control task. Accordingly in view of these findings, we suggest that the rIFG supports the accumulation of non-numerical magnitudes that are positively correlated with number. Therefore taken together, this study reveals a brain region whose pattern of activity is influenced by the congruency between numerical and non-numerical variables during nonsymbolic number judgments. Hum Brain Mapp 38:4908-4921, 2017. © 2017 Wiley Periodicals, Inc.

Symbol-value association and discrimination in the archerfish.

One of the most important aspects of mathematical cognition in humans is the ability to symbolically represent magnitudes and quantities. In the last 20 years it has been shown that not only humans but also other primates, birds and dolphins can use symbolic representation of quantities. However, it remains unclear to what extent this ability is spread across the animal kingdom. Here, by training archerfish to associate variable amounts of rewards with different geometric shapes, we show for the first time that lower vertebrates can also associate a value with a symbol and make a decision that maximizes their food intake based on this information. In addition, the archerfish is able to understand up to four different quantities and organize them mentally in an ordinal manner, similar to observations in higher vertebrates. These findings point in the direction of the existence of an approximate magnitude system in fish.

Beyond comparison: The influence of physical size on number estimation is modulated by notation, range and spatial arrangement.

Can physical size affect number estimation? Previous studies have shown that physical size influences non-symbolic numerosity in comparison tasks (e.g. which of two dots is larger). The current study investigated the conditions under which physical size can affect numerosity estimation. We employed a line mapping task in order to avoid the context of comparison and the need to provide a verbal label to estimate a quantity. Adult participants were briefly presented with the digits 2-8 or groups of 2-8 dots in 3 different physical sizes and were asked to estimate the position of a presented numerosity on a vertical line from 0 to 10. Physical size affected number estimation only above the subitizing range (i.e., >4) and only for non-symbolic numbers (e.g. dot arrays). Presenting non-symbolic numbers as canonical arrangements (like on a game die) reduced the effect of the physical size in the counting range (5-9). Accordingly, we suggest that the effect of task-irrelevant physical size on performance is modulated by the ability of participants to provide an accurate estimate of number: when the estimated number is easier to perceive (i.e., subitizing range or canonical arrangements), the influence of the physical size is smaller compared to when it is more difficult to give an accurate estimate of number (i.e., counting range, random arrangement). By doing so, we describe the factors that modulate the effect of physical size on number processing and provide another example of the important role continuous properties, such as physical size, play in non-symbolic number processing.

Toward an integrative approach to numerical cognition.

In response to the commentaries, we have refined our suggested model and discussed ways in which the model could be further expanded. In this context, we have elaborated on the role of specific continuous magnitudes. We have also found it important to devote a section to evidence considered the "smoking gun" of the approximate number system theory, including cross-modal studies, animal studies, and so forth. Lastly, we suggested some ways in which the scientific community can promote more transparent and collaborative research by using an open science approach, sharing both raw data and stimuli. We thank the contributors for their enlightening comments and look forward to future developments in the field.

From "sense of number" to "sense of magnitude": The role of continuous magnitudes in numerical cognition.

In this review, we are pitting two theories against each other: the more accepted theory, the number sense theory, suggesting that a sense of number is innate and non-symbolic numerosity is being processed independently of continuous magnitudes (e.g., size, area, and density); and the newly emerging theory suggesting that (1) both numerosities and continuous magnitudes are processed holistically when comparing numerosities and (2) a sense of number might not be innate. In the first part of this review, we discuss the number sense theory. Against this background, we demonstrate how the natural correlation between numerosities and continuous magnitudes makes it nearly impossible to study non-symbolic numerosity processing in isolation from continuous magnitudes, and therefore, the results of behavioral and imaging studies with infants, adults, and animals can be explained, at least in part, by relying on continuous magnitudes. In the second part, we explain the sense of magnitude theory and review studies that directly demonstrate that continuous magnitudes are more automatic and basic than numerosities. Finally, we present outstanding questions. Our conclusion is that there is not enough convincing evidence to support the number sense theory anymore. Therefore, we encourage researchers not to assume that number sense is simply innate, but to put this hypothesis to the test and consider whether such an assumption is even testable in the light of the correlation of numerosity and continuous magnitudes.

One tamed at a time: A new approach for controlling continuous magnitudes in numerical comparison tasks.

Non-symbolic stimuli (i.e., dot arrays) are commonly used to study numerical cognition. However, in addition to numerosity, non-symbolic stimuli entail continuous magnitudes (e.g., total surface area, convex-hull, etc.) that correlate with numerosity. Several methods for controlling for continuous magnitudes have been suggested, all with the same underlying rationale: disassociating numerosity from continuous magnitudes. However, the different continuous magnitudes do not fully correlate; therefore, it is impossible to disassociate them completely from numerosity. Moreover, relying on a specific continuous magnitude in order to create this disassociation may end up in increasing or decreasing numerosity saliency, pushing subjects to rely on it more or less, respectively. Here, we put forward a taxonomy depicting the relations between the different continuous magnitudes. We use this taxonomy to introduce a new method with a complimentary Matlab toolbox that allows disassociating numerosity from continuous magnitudes and equating the ratio of the continuous magnitudes to the ratio of the numerosity in order to balance the saliency of numerosity and continuous magnitudes. A dot array comparison experiment in the subitizing range showed the utility of this method. Equating different continuous magnitudes yielded different results. Importantly, equating the convex hull ratio to the numerical ratio resulted in similar interference of numerical and continuous magnitudes.

The symbol-grounding problem in numerical cognition: A review of theory, evidence, and outstanding questions.

How do numerical symbols, such as number words, acquire semantic meaning? This question, also referred to as the "symbol-grounding problem," is a central problem in the field of numerical cognition. Present theories suggest that symbols acquire their meaning by being mapped onto an approximate system for the nonsymbolic representation of number (Approximate Number System or ANS). In the present literature review, we first asked to which extent current behavioural and neuroimaging data support this theory, and second, to which extent the ANS, upon which symbolic numbers are assumed to be grounded, is numerical in nature. We conclude that (a) current evidence that has examined the association between the ANS and number symbols does not support the notion that number symbols are grounded in the ANS and (b) given the strong correlation between numerosity and continuous variables in nonsymbolic number processing tasks, it is next to impossible to measure the pure association between symbolic and nonsymbolic numerosity. Instead, it is clear that significant cognitive control resources are required to disambiguate numerical from continuous variables during nonsymbolic number processing. Thus, if there exists any mapping between the ANS and symbolic number, then this process of association must be mediated by cognitive control. Taken together, we suggest that studying the role of both cognitive control and continuous variables in numerosity comparison tasks will provide a more complete picture of the symbol-grounding problem.

Automaticity of Conceptual Magnitude.

What is bigger, an elephant or a mouse? This question can be answered without seeing the two animals, since these objects elicit conceptual magnitude. How is an object's conceptual magnitude processed? It was suggested that conceptual magnitude is automatically processed; namely, irrelevant conceptual magnitude can affect performance when comparing physical magnitudes. The current study further examined this question and aimed to expand the understanding of automaticity of conceptual magnitude. Two different objects were presented and participants were asked to decide which object was larger on the screen (physical magnitude) or in the real world (conceptual magnitude), in separate blocks. By creating congruent (the conceptually larger object was physically larger) and incongruent (the conceptually larger object was physically smaller) pairs of stimuli it was possible to examine the automatic processing of each magnitude. A significant congruity effect was found for both magnitudes. Furthermore, quartile analysis revealed that the congruity was affected similarly by processing time for both magnitudes. These results suggest that the processing of conceptual and physical magnitudes is automatic to the same extent. The results support recent theories suggested that different types of magnitude processing and representation share the same core system.

Itsy bitsy spider?: Valence and self-relevance predict size estimation.

The current study explored the role of valence and self-relevance in size estimation of neutral and aversive animals. In Experiment 1, participants who were highly fearful of spiders and participants with low fear of spiders rated the size and unpleasantness of spiders and other neutral animals (birds and butterflies). We found that although individuals with both high and low fear of spiders rated spiders as highly unpleasant, only the highly fearful participants rated spiders as larger than butterflies. Experiment 2 included additional pictures of wasps (not self-relevant, but unpleasant) and beetles. The results of this experiment replicated those of Experiment 1 and showed a similar bias in size estimation for beetles, but not for wasps. Mediation analysis revealed that in the high-fear group both relevance and valence influenced perceived size, whereas in the low-fear group only valence affected perceived size. These findings suggest that the effect of highly relevant stimuli on size perception is both direct and mediated by valence.

Asymmetric Processing of Numerical and Nonnumerical Magnitudes in the Brain: An fMRI Study.

It is well established that, when comparing nonsymbolic magnitudes (e.g., dot arrays), adults can use both numerical (i.e., the number of items) and nonnumerical (density, total surface areas, etc.) magnitudes. It is less clear which of these magnitudes is more salient or processed more automatically. In this fMRI study, we used a nonsymbolic comparison task to ask if different brain areas are responsible for the automatic processing of numerical and nonnumerical magnitudes, when participants were instructed to attend to either the numerical or the nonnumerical magnitudes of the same stimuli. An interaction of task (numerical vs. nonnumerical) and congruity (congruent vs. incongruent) was found in the right TPJ. Specifically, this brain region was more strongly activated during numerical processing when the nonnumerical magnitudes were negatively correlated with numerosity (incongruent trials). In contrast, such an interference effect was not evident during nonnumerical processing when the task-irrelevant numerical magnitude was incongruent. In view of the role of the right TPJ in the control of stimulus-driven attention, we argue that these data demonstrate that the processing of nonnumerical magnitudes is more automatic than that of numerical magnitudes and that, therefore, the influence of numerical and nonnumerical variables on each other is asymmetrical.

Numerosity processing is context driven even in the subitizing range: An fMRI study.

Numerical judgments are involved in almost every aspect of our daily life. They are carried out so efficiently that they are often considered to be automatic and innate. However, numerosity of non-symbolic stimuli is highly correlated with its continuous properties (e.g., density, area), and so it is hard to determine whether numerosity and continuous properties rely on the same mechanism. Here we examined the behavioral and neuronal mechanisms underlying such judgments. We scanned subjects' hemodynamic responses to a numerosity comparison task and to a surface area comparison task. In these tasks, numerical and continuous magnitudes could be either congruent or incongruent. Behaviorally, an interaction between the order of the tasks and the relevant dimension modulated the congruency effects. Continuous magnitudes always interfered with numerosity comparison. Numerosity, on the other hand, interfered with the surface area comparison only when participants began with the numerosity task. Hemodynamic activity showed that context (induced by task order) determined the neuronal pathways in which the dimensions were processed. Starting with the numerosity task led to enhanced activity in the right hemisphere, while starting with the continuous task led to enhanced left hemisphere activity. Continuous magnitudes processing relied on activation of the frontal eye field and the post-central gyrus. Processing of numerosities, on the other hand, relied on deactivation of these areas, suggesting active suppression of the continuous dimension. Accordingly, we suggest that numerosities, even in the subitizing range, are not always processed automatically; their processing depends on context and task demands.

Comparing performance in discrete and continuous comparison tasks.

The approximate number system (ANS) theory suggests that all magnitudes, discrete (i.e., number of items) or continuous (i.e., size, density, etc.), are processed by a shared system and comply with Weber's law. The current study reexamined this notion by comparing performance in discrete (comparing numerosities of dot arrays) and continuous (comparisons of area of squares) tasks. We found that: (a) threshold of discrimination was higher for continuous than for discrete comparisons; (b) while performance in the discrete task complied with Weber's law, performance in the continuous task violated it; and (c) performance in the discrete task was influenced by continuous properties (e.g., dot density, dot cumulative area) of the dot array that were not predictive of numerosities or task relevant. Therefore, we propose that the magnitude processing system (MPS) is actually divided into separate (yet interactive) systems for discrete and continuous magnitude processing. Further subdivisions are discussed. We argue that cooperation between these systems results in a holistic comparison of magnitudes, one that takes into account continuous properties in addition to numerosities. Considering the MPS as two systems opens the door to new and important questions that shed light on both normal and impaired development of the numerical system.

Comparative judgments of symbolic and non-symbolic stimuli yield different patterns of reaction times.

Are different magnitudes, such as Arabic numerals, length and area, processed by the same system? Answering this question can shed light on the building blocks of our mathematical abilities. A shared representation theory suggested that discriminability of all magnitudes complies with Weber's law. The current work examined this suggestion. We employed comparative judgment tasks to investigate different types of comparisons - conceptual comparison of numbers, physical comparison of numbers and physical comparison of different shapes. We used 8 different size ratios and plotted reaction time as a function of these ratios. Our findings suggest that the relationship between discriminability and size ratio is not always linear, as previously suggested; rather, it is modulated by the type of comparison and the type of stimuli. Hence, we suggest that the representation of magnitude is not as rigid as previously suggested; it changes as a function of task demands and familiarity with the compared stimuli.

Size before numbers: conceptual size primes numerical value.

The present work examined the influence of conceptual object size on numerical processing. In two experiments, pictures of conceptually large or small animals of equal retinal size served as prime stimuli appearing before numerically big or small integer targets. Participants were instructed to perform an unbiased parity judgment task on the target integers. When the prime's conceptual size was congruent with the target's numerical value, participants' reaction time was faster than when the two were incongruent with each other. This influence of conceptual object size on numerical value perception suggests that both types of magnitudes share similar mental representations. Our results are in accord with recent theories (e.g., Cantlon, Platt, & Brannon, 2009; Henik, Leibovich, Naparstek, Diesendruck, & Rubinsten, 2012) that emphasize the evolutionary importance of evaluation and perception of sizes to the development of the numerical system.

The importance of being relevant: modulation of magnitude representations.

The current study aims to answer two main questions. First, is there a difference between the representations of the numerical and the physical properties of visually presented numbers? Second, can the relevancy of the dimension change its representation? In a numerical Stroop task, participants were asked to indicate either the physically or the numerically larger value of two digits. The ratio between the physical sizes and the numerical values changed orthogonally from 0.1 (the largest difference) to 0.8. Reaction times (RT) were plotted as a function of both physical and numerical ratios. Trend analysis revealed that while the numerical dimension followed Weber's law regardless of task demands, the physical ratio deviated from linearity. Our results suggest that discrete and continuous magnitudes are represented by different yet interactive systems rather than by a shared representation.

Magnitude processing in non-symbolic stimuli.

Dot arrays are often used to study basic numerical skills across cultures, species and development. Researchers investigate the ability of subjects to discriminate between dot arrays, as a function of the ratio or distance between their numerosities. Such studies have contributed significantly to the number sense theory (i.e., that humans are born with the ability to process numerosities, and share this ability with various species)-possibly the most influential theory in numerical cognition literature today. However, a dot array contains, in addition to numerosity, continuous properties such as the total surface area of the dots, their density, etc. These properties are highly correlated with numerosity and therefore might influence participants' performance. Different ways in which different studies choose to deal with this confound sometimes lead to contradicting results, and in our opinion, do not completely eliminate the confound. In this work, we review these studies and suggest several possible reasons for the contradictions in the literature. We also suggest that studying continuous properties, instead of just trying to control them, may contribute to unraveling the building blocks of numerical abilities.

Inhibition of return in the archer fish.

Inhibition of return is the inhibitory tagging of recently attended locations or objects. It was previously suggested that inhibition of return is a foraging facilitator in visual search. Inhibition of return was first discovered in humans and was demonstrated also in monkeys, yet it has never been demonstrated in non-primates. Here we report the presence of inhibition of return in the archer fish, which shoots down prey on overhanging vegetation, using squirts of water spouted from its mouth. Moreover, we find similar attentional effects for fish as for human participants. Our results show that the generation of inhibition of return does not require a fully developed cortex and strengthen the view that inhibition of return functions as a foraging facilitator.