Recent work has shown that people are sensitive to coarse differences in network topology, including network features like “holes,” “crosses,” and “T-junctions.” Even children as young as 4 years old will readily distinguish between items that differ slightly in their network topology. But how robust is this sensitivity? Here, we evaluate whether people are not only sensitive to differences in the presence or absence of certain topological features, but also to their exact spatial arrangement. In a first experiment, we show that people distinguish figures which possess all the same topological features as other figures in a set if the features differ in spatial arrangement. In a second experiment, we show that people also match figures based on exact spatial arrangement. Finally, we show that memory encodes the correct relational structure of the figures: People are more likely to falsely indicate having seen an item if it shared the precise arrangement of topological features of other items they had seen (compared to a closely matched item which had the same features arranged in a different way). Combined, these results bolster the theory that people intuitively appreciate the precise spatial arrangement of topological features.

Understanding how non-numerical visual features systematically distort numerosity perception holds promise for unveiling the processes that give rise to our visual number sense. Recent studies show that increasing visual coherence systematically increases perceived numerosity, with this effect strengthening over development (DeWind et al., 2020; Qu, Bonner, et al., 2024; Qu et al., 2022). Here, we investigate the cognitive mechanisms underlying the coherence illusion from a view of perceptual decision processes. Specifically, we applied a drift diffusion model (DDM) to a previously described dataset from participants aged 5-30 tested in an ordinal numerical comparison task with color entropy systematically manipulated (Qu et al., 2022). By jointly modeling choice data and response times, we decomposed numerical discrimination performance into distinct decision components: the speed of numerical evidence accumulation (drift rate), the amount of evidence required for a decision (boundary separation), and the response bias reflecting a prior tendency of selecting one side over the other. We found that color coherence affected only the drift rate but not response bias or boundary separation, demonstrating that color coherence distorts numerical calculation through biased accumulation of evidence of quantity. Moreover, the impact of coherence on the drift rate coefficient increased with age as quantitative information is accumulated more efficiently over development. Our results offer a framework for understanding how numerical illusions arise from perceptual decision-making dynamics.

There are many ways to describe and represent the visuospatial world. A space can be described by its euclidean properties-the size of objects, the angles of boundaries, the distances between them. A space can also be described in nonspatial terms: One could explain the layout of a city by the order of its streets. Somewhere in between, topological representations-such as those commonly depicted in public-transit maps-capture coarse relational structure without precise euclidean detail, offering a relatively efficient, low-dimensional way of capturing spatial content. Here, we ask whether human adults quickly and automatically perceive such relations. In six experiments, we show that differences in simple topological features influence a range of visual tasks from object matching to number estimation to visual search. We discuss the possibility that topological relations are a kind of visual primitive that supports visuospatial representation.

A core aim of developmental cognitive science is to uncover the basic building blocks of human thought. For instance, work revealing that even young children, adults without formal education, and distant animal species are sensitive to basic Euclidean properties indicates that humans may be endowed with some primitive understanding of Euclidean geometry. But what about other forms of geometry? Here, we explore children's sensitivity to topological spatial forms. We show that children, like adults, spontaneously distinguish and match items in accordance with their topological relations. As well, we show that children's judgments about object similarity are remarkably consistent with adults', indicating stability in object concepts throughout the lifespan. Finally, we compare children's sensitivity to various topological forms with their sensitivity to geometric properties like curvature, perpendicularity, and symmetry, and find that while there is some variability in performance across all the features tested, overall performance for geometric vs. topological is comparable. Collectively, these findings suggest that even young children have an intuitive understanding of topological relations and suggest that topological relations may be among the building blocks of human visuospatial representation.

Humans typically represent numbers and quantities along a left-to-right continuum. Early perspectives attributed number-space association to culture; however, recent evidence in newborns and animals challenges this hypothesis. We investigate whether the length of an array of dots influences spatial bias in rhesus macaques. We designed a touch-screen task that required monkeys to remember the location of a target. At test, monkeys maintained high performance with arrays of 2, 4, 6, or 10 dots, regardless of changes in the array's location, spacing, and length. Monkeys remembered better left targets with 2-dot arrays and right targets with 6- or 10-dot arrays. Replacing the 10-dot array with a long bar, yielded more accurate performance with rightward locations, consistent with an underlying left-to-right oriented magnitude code. Our study supports the hypothesis of a spatially oriented mental magnitude line common to humans and animals, countering the idea that this code arises from uniquely human cultural learning.

Does the visual system adapt to number? For more than fifteen years, most have assumed that the answer is an unambiguous “yes”. Against this prevailing orthodoxy, we recently took a critical look at the phenomenon, questioning its existence on both empirical and theoretical grounds, and providing an alternative explanation for extant results (the old news hypothesis). We subsequently received two critical responses. Burr, Anobile, and Arrighi rejected our critiques wholesale, arguing that the evidence for number adaptation remains overwhelming. Durgin questioned our old news hypothesis — preferring instead a theory about density adaptation he has championed for decades — but also highlighted several ways in which our arguments do pose serious challenges for proponents of number adaptation. Here, we reply to both. We first clarify our position regarding number adaptation. Then, we respond to our critics' concerns, highlighting seven reasons why we remain skeptical about number adaptation. We conclude with some thoughts about where the debate may head from here.

Number perception emerges from multiple stages of visual processing. Understanding how systematic biases in number perception occur within a hierarchy of increasingly complex feature representations helps uncover the multi-stage processing underlying our visual number sense. Recent work demonstrated that reducing coherence of low-level visual attributes such as color and orientation systematically reduces perceived number. Here we ask when in the visual processing hierarchy coherence affects numerosity perception and specifically whether the coherence effect is exclusive to low-level visual features or instead whether it can be driven by contextual or semantic relationships. We tested adults in an ordinal numerical comparison task with contextual coherence mathematically manipulated using a statistical model of visual object co-occurrence. Across several experiments, we found that arrays with high contextual coherence were perceived as numerically larger than arrays with low contextual coherence. This contextual coherence effect was not attenuated even when we reduced objects to texforms (unrecognizable images that preserve mid-level visual features) or removed semantic content from the images through box scrambling and diffeomorphic warping. Together, these results suggest that visual coherence derived from natural statistics of object co-occurrence systematically alters perceived numerosity at low-level visual processing, even before later stages at which items can be explicitly categorized and identified.

It is often assumed that adaptation — a temporary change in sensitivity to a perceptual dimension following exposure to that dimension — is a litmus test for what is and is not a “primary visual attribute”. Thus, papers purporting to find evidence of number adaptation motivate a claim of great significance: That number is something that can be seen in much the way that canonical visual features, like color, contrast, size, and speed, can. Fifteen years after its reported discovery, number adaptation's existence seems to be nearly undisputed, with dozens of papers documenting support for the phenomenon. The aim of this paper is to offer a counterweight — to critically assess the evidence for and against number adaptation. After surveying the many reasons for thinking that number adaptation exists, we introduce several lesser-known reasons to be skeptical. We then advance an alternative account — the old news hypothesis — which can accommodate previously published findings while explaining various (otherwise unexplained) anomalies in the existing literature. Next, we describe the results of eight pre-registered experiments which pit our novel old news hypothesis against the received number adaptation hypothesis. Collectively, the results of these experiments undermine the number adaptation hypothesis on several fronts, whilst consistently supporting the old news hypothesis. More broadly our work raises questions about the status of adaptation itself as a means of discerning what is and is not a visual attribute.

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known "connectedness illusion" to provide evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers.

All mobile organisms forage for resources, choosing how and when to search for new opportunities by comparing current returns with the average for the environment. In humans, nomadic lifestyles favouring exploration have been associated with genetic mutations implicated in attention deficit hyperactivity disorder (ADHD), inviting the hypothesis that this condition may impact foraging decisions in the general population. Here we tested this pre-registered hypothesis by examining how human participants collected resources in an online foraging task. On every trial, participants chose either to continue to collect rewards from a depleting patch of resources or to replenish the patch. Participants also completed a well-validated ADHD self-report screening assessment at the end of sessions. Participants departed resource patches sooner when travel times between patches were shorter than when they were longer, as predicted by optimal foraging theory. Participants whose scores on the ADHD scale crossed the threshold for a positive screen departed patches significantly sooner than participants who did not meet this criterion. Participants meeting this threshold for ADHD also achieved higher reward rates than individuals who did not. Our findings suggest that ADHD attributes may confer foraging advantages in some environments and invite the possibility that this condition may reflect an adaptation favouring exploration over exploitation.

Topology is the branch of mathematics that seeks to understand and describe spatial relations. A number of studies have examined the human perception of topology — in particular, whether adults and young children perceive and differentiate objects based on features like closure, boundedness, and emptiness. Topology is about more than ‘wholes and holes’, however; it also offers an efficient language for representing network structure. Topological maps, common for subway systems across the world, are an example of how effective this language can be. Inspired by this idea, here we examine ‘intuitive network topology’. We first show that people readily differentiate objects based on several different features of topological networks. We then show that people both remember and match objects in accordance with their topology, over and above substantial variation in their surface features. These results demonstrate that humans possess an intuitive understanding for the basic topological features of networks, and hint at the possibility that topology may serve as a format for representing relations in the mind.

BACKGROUND:A hallmark of the approximate number system (ANS) is ratio dependence. Previous work identified specific event-related potentials (ERPs) that are modulated by numerical ratio throughout the lifespan. In adults, ERP ratio dependence was correlated with the precision of the numerical judgments with individuals who make more precise judgments showing larger ratio-dependent ERP effects. The current study evaluated if this relationship generalizes to preschoolers. METHOD:ERPs were recorded from 56 4.5 to 5.5-year-olds while they compared the numerosity of two sequentially presented dot arrays. Nonverbal numerical precision, often called ANS acuity, was assessed using a similar behavioral task. RESULTS:Only children with high ANS acuity exhibited a P2p ratio-dependent effect onsetting ∼250 ms after the presentation of the comparison dot array. Furthermore, P2p amplitude positively correlated with ANS acuity across tasks. CONCLUSION:Results demonstrate developmental continuity between preschool years and adulthood in the neural basis of the ANS.

Francis Galton first reported that humans mentally organize numbers from left to right on a mental number line (1880). This spatial-numerical association was long considered to result from writing and reading habits. More recently though, newborns and animals showed a left-to-right oriented spatial numerical association challenging the primary role assigned to culture in determining the link between number and space. Despite growing evidence supporting the intrinsic association between number and space in different species, its adaptive value is still largely unknown. Here we tested for an advantage in identification of left versus right target positions in 3- to 6-year-old children. Children watched as a toy was hidden under one of 10 linearly arranged identical cups and were then asked to help a stuffed animal retrieve the toy. On each trial, the toy was hidden in the 2nd, 3rd, or 4th cup, from the left or right. To prevent children from staring at the target cup, they were asked to pick up the stuffed animal from under their chair after witnessing the hiding of the toy and then to help the stuffed animal find the toy. Older children were more accurate than younger children. Children exhibited a serial position effect, with performance higher for more exterior targets. Remarkably, children also showed a left bias: they remembered the left targets better than the right targets. Only the youngest children were dramatically influenced by the location of the experimenter during search. Additional analyses support the hypothesis that children used a left-to-right oriented searching strategy in this spatial/ordinal task.

Children bring intuitive arithmetic knowledge to the classroom before formal instruction in mathematics begins. For example, children can use their number sense to add, subtract, compare ratios, and even perform scaling operations that increase or decrease a set of dots by a factor of 2 or 4. However, it is currently unknown whether children can engage in a true division operation before formal mathematical instruction. Here we examined the ability of 6- to 9-year-old children and college students to perform symbolic and non-symbolic approximate division. Subjects were presented with non-symbolic (dot array) or symbolic (Arabic numeral) dividends ranging from 32 to 185, and non-symbolic divisors ranging from 2 to 8. Subjects compared their imagined quotient to a visible target quantity. Both children (Experiment 1 N = 89, Experiment 2 N = 42) and adults (Experiment 3 N = 87) were successful at the approximate division tasks in both dots and numeral formats. This was true even among the subset of children that could not recognize the division symbol or solve simple division equations, suggesting intuitive division ability precedes formal division instruction. For both children and adults, the ability to divide non-symbolically mediated the relation between Approximate Number System (ANS) acuity and symbolic math performance, suggesting that the ability to calculate non-symbolically may be a mechanism of the relation between ANS acuity and symbolic math. Our findings highlight the intuitive arithmetic abilities children possess before formal math instruction.

Animals show vast numerical competence in tasks that require both ordinal and cardinal numerical representations, but few studies have addressed whether animals can identify the numerical middle in a sequence. Two rhesus monkeys (Macaca mulatta) learned to select the middle dot in a horizontal sequence of three dots on a touchscreen. When subsequently presented with longer sequences composed of 5, 7 or 9 items, monkeys transferred the middle rule. Accuracy decreased as the length of the sequence increased. In a second test, we presented monkeys with asymmetrical sequences composed of nine items, where the numerical and spatial middle were distinct and both monkeys selected the numerical middle over the spatial middle. Our results demonstrate that rhesus macaques can extract an abstract numerical rule to bisect a discrete set of items.

Numerical illusions may provide a powerful window into the mechanisms that give rise to our visual number sense. Recent research has shown that similarly oriented elements appear more numerous than randomly oriented elements in an array. Here we examine whether the orientation coherence illusion is a more general byproduct of the effect of entropy on numerical information-processing. Participants engaged in an ordinal numerical comparison task where the color entropy of arrays was manipulated. We found that arrays with low color entropy were perceived as more numerous than arrays with high color entropy (Experiments 1 and 2), suggesting that the coherence illusion on numerosity perception is not specific to a particular visual property (e.g., orientation) but instead that the entropy of visual arrays more generally affects numerical processing. In Experiment 3, we explored the developmental trajectory of the color entropy effect in children aged 5 to 17 and found that the strength of the coherence illusion increases into adulthood, raising intriguing questions as to how perceptual experiences influence the progression of this numerosity illusion. We consider a recently proposed resource-rational model as a framework for understanding the entropy effect on numerosity perception under an information-theoretic perspective.

The ability to map number words to their corresponding quantity representations is a gatekeeper for children's future math success (Spaepen et al., 2018). Without number word knowledge at school entry, children are at greater risk for developing math learning difficulties (Chu et al., 2019). In the present study, we used functional magnetic resonance imaging (fMRI) to examine the neural basis for processing the meaning of spoken number words and its developmental trajectory in 4- to 10-year-old children, and in adults. In a number word-quantity mapping paradigm, participants listened to number words while simultaneously viewing quantities that were congruent or incongruent to the number word they heard. Whole brain analyses revealed that adults showed a neural congruity effect with greater neural activation for incongruent relative to congruent trials in anterior cingulate cortex (ACC) and left intraparietal sulcus (LIPS). In contrast, children did not show a significant neural congruity effect. However, a region of interest analysis in the child sample demonstrated age-related increases in the neural congruity effect, specifically in the LIPS. The positive correlation between neural congruity in LIPS and age was stronger in children who were already attending school, suggesting that developmental changes in LIPS function are experience-dependent.