Relating magnitudes: the brain's code for proportions

The calculating brain

The human brain possesses neural networks and mechanisms enabling the representation of numbers, basic arithmetic operations, and mathematical reasoning. Without the ability to represent numerical quantity and perform calculations, our scientifically and technically advanced culture would not exist. However, the origins of numerical abilities are grounded in an intuitive understanding of quantity deeply rooted in biology. Nevertheless, more advanced symbolic arithmetic skills require a cultural background with formal mathematical education. In the past two decades, cognitive neuroscience has seen significant progress in understanding the workings of the calculating brain through various methods and model systems. This review begins by exploring the mental and neuronal representations of nonsymbolic numerical quantity and then progresses to symbolic representations acquired in childhood. During arithmetic operations (addition, subtraction, multiplication, and division), these representations are processed and transformed according to arithmetic rules and principles, leveraging different mental strategies and types of arithmetic knowledge that can be dissociated in the brain. Although it was once believed that number processing and calculation originated from the language faculty, it is now evident that mathematical and linguistic abilities are primarily processed independently in the brain. Understanding how the healthy brain processes numerical information is crucial for gaining insights into debilitating numerical disorders, including acquired conditions like acalculia and learning-related calculation disorders such as developmental dyscalculia.

Parallel processes of temporal control in the supplementary motor area and the frontoparietal circuit

Durations in the several seconds' range are cognitively accessible during active timing. Functional neuroimaging studies suggest the engagement of the basal ganglia (BG) and supplementary motor area (SMA). However, their functional relevance and arrangement remain unclear because non-timing cognitive processes temporally coincide with the active timing. To examine the potential contamination by parallel processes, we introduced a sensory control and a motor control to the duration-reproduction task. By comparing their hemodynamic functions, we decomposed the neural activities in multiple brain loci linked to different cognitive processes. Our results show a dissociation of two cortical neural circuits: the SMA for both active timing and motor preparation, followed by a prefrontal-parietal circuit related to duration working memory. We argue that these cortical processes represent duration as the content but at different levels of abstraction, while the subcortical structures, including the BG and thalamus, provide the logistic basis of timing by coordinating the temporal framework across brain structures.

Questions About Quantifiers: Symbolic and Nonsymbolic Quantity Processing by the Brain

One approach to understanding how the human cognitive system stores and operates with quantifiers such as "some," "many," and "all" is to investigate their interaction with the cognitive mechanisms for estimating and comparing quantities from perceptual input (i.e., nonsymbolic quantities). While a potential link between quantifier processing and nonsymbolic quantity processing has been considered in the past, it has never been discussed extensively. Simultaneously, there is a long line of research within the field of numerical cognition on the relationship between processing exact number symbols (such as "3" or "three") and nonsymbolic quantity. This accumulated knowledge can potentially be harvested for research on quantifiers since quantifiers and number symbols are two different ways of referring to quantity information symbolically. The goal of the present review is to survey the research on the relationship between quantifiers and nonsymbolic quantity processing mechanisms and provide a set of research directions and specific questions for the investigation of quantifier processing.

Decoding of spatial proportions using somatosensory feedback in sighted and visually impaired children

BACKGROUND AND PURPOSE

Humans can naturally operate with ratios of continuous magnitudes (proportions). We asked if sighted children (S) and visually impaired children (VI) can discriminate proportions via somatosensory feedback.

PROCEDURES

Children formed a proportion by tracing a pair of straight lines with their finger, and compared this proportion with a second proportion resulting from the tracing of another pair of lines.

MAIN FINDINGS

Performance was 68% in S, thus significantly lower (p < 0.001) compared to VI (75%). Tracing velocity (p < 0.01) and trial-to-trial variability of tracing velocity (p < 0.05) was higher in S compared to VI.

CONCLUSIONS

Operating with proportions solely from somatosensory feedback is possible, thus tracing lines might support learning in mathematics education. Kinematic variables point to the reason for the difference between S and VI, in that higher trial-to-trial variability in velocity in S leads to biased estimation of absolute line lengths.

Comparison of visual quantities in untrained neural networks

The ability to compare quantities of visual objects with two distinct measures, proportion and difference, is observed even in newborn animals. However, how this function originates in the brain, even before visual experience, remains unknown. Here, we propose a model in which neuronal tuning for quantity comparisons can arise spontaneously in completely untrained neural circuits. Using a biologically inspired model neural network, we find that single units selective to proportions and differences between visual quantities emerge in randomly initialized feedforward wirings and that they enable the network to perform quantity comparison tasks. Notably, we find that two distinct tunings to proportion and difference originate from a random summation of monotonic, nonlinear neural activities and that a slight difference in the nonlinear response function determines the type of measure. Our results suggest that visual quantity comparisons are primitive types of functions that can emerge spontaneously before learning in young brains.

The developmental relationship between nonsymbolic and symbolic fraction abilities

A fundamental research question in quantitative cognition concerns the developmental relationship between nonsymbolic and symbolic quantitative abilities. This study examined this developmental relationship in abilities to process nonsymbolic and symbolic fractions. There were 99 6th graders (M_age = 11.86 years), 101 10th graders (M_age = 15.71 years), and 102 undergraduate and graduate students (M_age = 21.97 years) participating in this study, and their nonsymbolic and symbolic fraction abilities were measured with nonsymbolic and symbolic fraction comparison tasks, respectively. Nonsymbolic and symbolic fraction abilities were significantly correlated in all age groups even after controlling for the ability to process nonsymbolic absolute quantity and general cognitive abilities, including working memory and inhibitory control. Moreover, the strength of nonsymbolic-symbolic correlations was higher in 6th graders than in 10th graders and adults. These findings suggest a weakened association between nonsymbolic and symbolic fraction abilities during development, and this developmental pattern may be related with participants' increasing proficiency in symbolic fractions.

Leveraging Math Cognition to Combat Health Innumeracy

Rational numbers (i.e., fractions, percentages, decimals, and whole-number frequencies) are notoriously difficult mathematical constructs. Yet correctly interpreting rational numbers is imperative for understanding health statistics, such as gauging the likelihood of side effects from a medication. Several pernicious biases affect health decision-making involving rational numbers. In our novel developmental framework, the natural-number bias-a tendency to misapply knowledge about natural numbers to all numbers-is the mechanism underlying other biases that shape health decision-making. Natural-number bias occurs when people automatically process natural-number magnitudes and disregard ratio magnitudes. Math-cognition researchers have identified individual differences and environmental factors underlying natural-number bias and devised ways to teach people how to avoid these biases. Although effective interventions from other areas of research can help adults evaluate numerical health information, they circumvent the core issue: people's penchant to automatically process natural-number magnitudes and disregard ratio magnitudes. We describe the origins of natural-number bias and how researchers may harness the bias to improve rational-number understanding and ameliorate innumeracy in real-world contexts, including health. We recommend modifications to formal math education to help children learn the connections among natural and rational numbers. We also call on researchers to consider individual differences people bring to health decision-making contexts and how measures from math cognition might identify those who would benefit most from support when interpreting health statistics. Investigating innumeracy with an interdisciplinary lens could advance understanding of innumeracy in theoretically meaningful and practical ways.

Reduced choice-confidence in negative numerals

Negative numbers are central in math. However, they are abstract, hard to learn, and manipulated slower than positive numbers regardless of math ability. It suggests that confidence, namely the post-decision estimate of being correct, should be lower than positives. We asked participants to pick the larger single-digit numeral in a pair and collected their implicit confidence with button pressure (button pressure was validated with three empirical signatures of confidence). We also modeled their choices with a drift-diffusion decision model to compute the post-decision estimate of being correct. We found that participants had relatively low confidence with negative numerals. Given that participants compared with high accuracy the basic base-10 symbols (0–9), reduced confidence may be a general feature of manipulating abstract negative numerals as they produce more uncertainty than positive numerals per unit of time.

Circling around number: People can accurately extract numeric values from circle area ratios

It has long been known that people have the ability to estimate numerical quantities without counting. A standard account is that people develop a sense of the size of symbolic numbers by learning to map symbolic numbers (e.g., 6) to their corresponding numerosities (e.g. :::) and concomitant approximate magnitude system (ANS) representations. However, we here demonstrate that adults are capable of extracting fractional numerical quantities from non-symbolic visual ratios (i.e., labeling a ratio of two circle areas with the appropriate symbolic fraction). Not only were adult participants able to perform this task, but they were remarkably accurate: linear regressions on median estimates yielded slopes near 1, and accounted for 97% of the variability. Participants also performed at least as well on line-estimation and ratio-estimation tasks using non-numeric circular stimuli as they did in earlier experiments using non-symbolic numerosities, which are frequently considered to be numeric stimuli. We discuss results as consistent with accounts suggesting that non-symbolic ratios have the potential to act as a reliable and stable ground for symbolic number, even when composed of non-numeric stimuli.

Real models: The limits of behavioural evidence for understanding the ANS

Clarke and Beck use behavioural evidence to argue that (1) approximate ratio computations are sufficient for claiming that the approximate number system (ANS) represents the rationals, and (2) the ANS does not represent the reals. We argue that pure behaviour is a poor litmus test for this problem, and that we should trust the psychophysical models that place ANS representations within the reals.

Ratio-based perceptual foundations for rational numbers, and perhaps whole numbers, too?

Clarke and Beck suggest that the ratio processing system (RPS) may be a component of the approximate number system (ANS), which they suggest represents rational numbers. We argue that available evidence is inconsistent with their account and advocate for a two-systems view. This implies that there may be many access points for numerical cognition - and that privileging the ANS may be a mistake.

Three Perceptual Tools for Seeing and Understanding Visualized Data

The visual system evolved and develops to process the scenes, faces, and objects of the natural world, but people adapt this powerful system to process data within an artificial world of visualizations. To extract patterns in data from these artificial displays, viewers appear to use at least three perceptual tools, including a tool that extracts global statistics, one that extracts shapes within the data, and one that produces sentence-like comparisons. A better understanding of the power, limits, and deployment of these tools would lead to better guidelines for designing effective data displays.

Bias and noise in proportion estimation: A mixture psychophysical model

The importance of proportional reasoning has long been recognized by psychologists and educators, yet we still do not have a good understanding of how humans mentally represent proportions. In this paper we present a psychophysical model of proportion estimation, extending previous approaches. We assumed that proportion representations are formed by representing each magnitude of a proportion stimuli (the part and its complement) as Gaussian activations in the mind, which are then mentally combined in the form of a proportion. We next derived the internal representation of proportions, including bias and internal noise parameters -capturing respectively how our estimations depart from true values and how variable estimations are. Methodologically, we introduced a mixture of components to account for contaminating behaviors (guessing and reversal of responses) and framed the model in a hierarchical way. We found empirical support for the model by testing a group of 4th grade children in a spatial proportion estimation task. In particular, the internal density reproduced the asymmetries (skewedness) seen in this and in previous reports of estimation tasks, and the model accurately described wide variations between subjects in behavior. Bias estimates were in general smaller than by using previous approaches, due to the model's capacity to absorb contaminating behaviors. This property of the model can be of especial relevance for studies aimed at linking psychophysical measures with broader cognitive abilities. We also recovered higher levels of noise than those reported in discrimination of spatial magnitudes and discuss possible explanations for it. We conclude by illustrating a concrete application of our model to study the effects of scaling in proportional reasoning, highlighting the value of quantitative models in this field of research.

More than the sum of its parts: Exploring the development of ratio magnitude versus simple magnitude perception

Humans perceptually extract quantity information from our environments, be it from simple stimuli in isolation, or from relational magnitudes formed by taking ratios of pairs of simple stimuli. Some have proposed that these two types of magnitude are processed by a common system, whereas others have proposed separate systems. To test these competing possibilities, the present study examined the developmental trajectories of simple and relational magnitude discrimination and relations among these abilities for preschoolers (n = 42), 2nd-graders (n = 31), 5th-graders (n = 29), and adults (n = 32). Participants completed simple magnitude and ratio discrimination tasks in four different nonsymbolic formats, using dots, lines, circles, and irregular blobs. All age cohorts accurately discriminated both simple and ratio magnitudes. Discriminability differed by format such that performance was highest with line and lowest with dot stimuli. Moreover, developmental trajectories calculated for each format were similar across simple and ratio discriminations. Although some characteristics were similar for both types of discrimination, ratio acuity in a given format was more closely related with ratio acuities in alternate formats than to within-format simple magnitude acuity. Results demonstrate that ratio magnitude processing shares several similarities to simple magnitude processing, but is also substantially different.

Grasp Actually: An Evolutionist Argument for Enactivist Mathematics Education

What evolutionary account explains our capacity to reason mathematically? Identifying the biological provenance of mathematical thinking would bear on education, because we could then design learning environments that simulate ecologically authentic conditions for leveraging this universal phylogenetic inclination. The ancient mechanism coopted for mathematical activity, I propose, is our fundamental organismic capacity to improve our sensorimotor engagement with the environment by detecting, generating, and maintaining goal-oriented perceptual structures regulating action, whether actual or imaginary. As such, the phenomenology of grasping a mathematical notion is literally that – gripping the environment in a new way that promotes interaction. To argue for the plausibility of my thesis, I first survey embodiment literature to implicate cognition as constituted in perceptuomotor engagement. Then, I summarize findings from a design-based research project investigating relations between learning to move in new ways and learning to reason mathematically about these conceptual choreographies. As such, the project proposes educational implications of enactivist evolutionary biology.

The numerator bias exists in millions of real-world comparisons

Fractions are crucial, from math and science education to daily activities, but they are hard. A puzzling aspect of fractions is that people over-rely on the numerator when comparing a pair of fractions. Previous work has considered this numerator bias mostly as a reasoning mishap. Still, in a vast amount of pairwise comparisons, across many real-world domains, not just education textbooks, we report a high prior probability that the larger fraction has the larger numerator, and, for a relevant case, we provide formal arguments why. The existence of such a regularity suggests that the numerator bias may reflect a rational adaptation that detects and exploits likely events. In a pair of visual-proportion tasks (discrete and continuous fractions), we confirm that the numerator bias in participants adapts to experimented regularities. Even though weak education and math abilities play a role, adaptation to informative priors outside the classroom poses a challenge to educators, learners, and decision-makers.

Response modality-dependent categorical choice representations for vibrotactile comparisons

Previous electrophysiological studies in monkeys and humans suggest that premotor regions are the primary loci for the encoding of perceptual choices during vibrotactile comparisons. However, these studies employed paradigms wherein choices were inextricably linked with the stimulus order and selection of manual movements. It remains largely unknown how vibrotactile choices are represented when they are decoupled from these sensorimotor components of the task. To address this question, we used fMRI-MVPA and a variant of the vibrotactile frequency discrimination task which enabled the isolation of choice-related signals from those related to stimulus order and selection of the manual decision reports. We identified the left contralateral dorsal premotor cortex (PMd) and intraparietal sulcus (IPS) as carrying information about vibrotactile choices. Our finding provides empirical evidence for an involvement of the PMd and IPS in vibrotactile decisions that goes above and beyond the coding of stimulus order and specific action selection. Considering findings from recent studies in animals, we speculate that the premotor region likely serves as a temporary storage site for information necessary for the specification of concrete manual movements, while the IPS might be more directly involved in the computation of choice. Moreover, this finding replicates results from our previous work using an oculomotor variant of the task, with the important difference that the informative premotor cluster identified in the previous work was centered in the bilateral frontal eye fields rather than in the PMd. Evidence from these two studies indicates that categorical choices in human vibrotactile comparisons are represented in a response modality-dependent manner.

Weber's Law-based perception and the stability of animal groups

Group living animals form aggregations and flocks that remain cohesive in spite of internal movements of individuals. This is possible because individual group members repeatedly adjust their position and motion in response to the position and motion of other group members. Here, we develop a theoretical approach to address the question, what general features-if any-underlie the interaction rules that mediate group stability in animals of all species? We do so by considering how the spatial organization of a group would change in the complete absence of interactions. Without interactions, a group would disperse in a way that can be easily characterized in terms of Fick's diffusion equations. We can hence address the inverse theoretical problem of finding the individual-level interaction responses that are required to counterbalance diffusion and to preserve group stability. We show that an individual-level response to neighbour densities in the form of Weber's Law (a 'universal' law describing the functioning of the sensory systems of animals of all species) results in an 'anti-diffusion' term at the group level. On short timescales, this anti-diffusion restores the initial group configuration in a way that is reminiscent of methods for image deblurring in image processing. We also show that any non-homogeneous, spatial density distribution can be preserved over time if individual movement patterns have the form of a Weber's Law response. Weber's Law describes the fundamental functioning of perceptual systems. Our study indicates that it is also a necessary-but not sufficient-feature of collective interactions in stable animal groups.

Symbolic fractions elicit an analog magnitude representation in school-age children

A fundamental question about fractions is whether they are grounded in an abstract nonsymbolic magnitude code similar to that postulated for whole numbers. Mounting evidence suggests that symbolic fractions could be grounded in mechanisms for perceiving nonsymbolic ratio magnitudes. However, systematic examination of such mechanisms in children has been lacking. We asked second- and fifth-grade children (prior to and after formal instructions with fractions, respectively) to compare pairs of symbolic fractions, nonsymbolic ratios, and mixed symbolic-nonsymbolic pairs. This paradigm allowed us to test three key questions: (a) whether children show an analog magnitude code for rational numbers, (b) whether that code is compatible with mental representations of symbolic fractions, and (c) how formal education with fractions affects the symbolic-nonsymbolic relation. We examined distance effects as a marker of analog ratio magnitude processing and notation effects as a marker of converting across numerical codes. Second and fifth graders' reaction times and error rates showed classic distance and notation effects. Nonsymbolic ratios were processed most efficiently, with mixed and symbolic notations being relatively slower. Children with more formal instruction in symbolic fractions had a significant advantage in comparing symbolic fractions but had a smaller advantage for nonsymbolic ratio stimuli. Supplemental analyses showed that second graders relied on numerator distance more than holistic distance and that fifth graders relied on holistic fraction magnitude distance more than numerator distance. These results suggest that children have a nonsymbolic ratio magnitude code and that symbolic fractions can be translated into that magnitude code.

No calculation necessary: Accessing magnitude through decimals and fractions

Research on how humans understand the relative magnitude of symbolic fractions presents a unique case of the symbol-grounding problem with numbers. Specifically, how do people access a holistic sense of rational number magnitude from decimal fractions (e.g. 0.125) and common fractions (e.g. 1/8)? Researchers have previously suggested that people cannot directly access magnitude information from common fraction notation, but instead must use a form of calculation to access this meaning. Questions remain regarding the nature of calculation and whether a division-like conversion to decimals is a necessary process that permits access to fraction magnitudes. To test whether calculation is necessary to access fractions magnitudes, we carried out a series of six parallel experiments in which we examined how adults access the magnitude of rational numbers (decimals and common fractions) under varying task demands. We asked adult participants to indicate which of two fractions was larger in three different conditions: decimal-decimal, fraction-fraction, and mixed decimal-fraction pairs. Across experiments, we manipulated two aspects of the task demands. 1) Response windows were limited to 1, 2 or 5 s, and 2) participants either did or did not have to identify when the two stimuli were the same magnitude (catch trials). Participants were able to successfully complete the task even at a response window of 1 s and showed evidence of holistic magnitude processing. These results indicate that calculation strategies with fractions are not necessary for accessing a sense of a fractions meaning but are strategic routes to magnitude that participants may use when granted sufficient time. We suggest that rapid magnitude processing with fractions and decimals may occur by mapping symbolic components onto common amodal mental representations of rational numbers.

Training nonsymbolic proportional reasoning in children and its effects on their symbolic math abilities

Our understanding of proportions can be both symbolic, as when doing calculations in school mathematics, or intuitive, as when folding a bed sheet in half. While an understanding of symbolic proportions is crucial for school mathematics, the cognitive foundations of this ability remain unclear. Here we implemented a computerized training game to test a causal link from intuitive (nonsymbolic) to symbolic proportional reasoning and other math abilities in 4th grade children. An experimental group was trained in nonsymbolic proportional reasoning (PR) with continuous extents, and an active control group was trained on a remarkably similar nonsymbolic magnitude comparison. We found that the experimental group improved at nonsymbolic PR across training sessions, showed near transfer to a paper-and-pencil nonsymbolic PR test, transfer to symbolic proportions, and far transfer to geometry. The active control group showed only a predicted far transfer to geometry. In a second experiment, these results were replicated with an independent cohort of children. Overall this study extends previous correlational evidence, suggesting a functional link between nonsymbolic PR on one hand and symbolic PR and geometry on the other.

The Relational SNARC: Spatial Representation of Nonsymbolic Ratios

Recent research in numerical cognition has begun to systematically detail the ability of humans and nonhuman animals to perceive the magnitudes of nonsymbolic ratios. These relationally defined analogs to rational numbers offer new potential insights into the nature of human numerical processing. However, research into their similarities with and connections to symbolic numbers remains in its infancy. The current research aims to further explore these similarities by investigating whether the magnitudes of nonsymbolic ratios are associated with space just as symbolic numbers are. In two experiments, we found that responses were faster on the left for smaller nonsymbolic ratio magnitudes and faster on the right for larger nonsymbolic ratio magnitudes. These results further elucidate the nature of nonsymbolic ratio processing, extending the literature of spatial-numerical associations to nonsymbolic relative magnitudes. We discuss potential implications of these findings for theories of human magnitude processing in general and how this general processing relates to numerical processing.

Processing symbolic and non-symbolic proportions: Domain-specific numerical and domain-general processes in intraparietal cortex

Previous studies on the processing of fractions and proportions focused mainly on the processing of their overall magnitude information in the intraparietal sulcus (IPS). However, the IPS is also associated with domain-general cognitive functions beyond processing overall magnitude, which may nevertheless be involved in operating on magnitude information of proportions. To pursue this issue, the present study aimed at investigating whether there is a shared neural correlate for proportion processing in the intraparietal cortex beyond overall magnitude processing and how part-whole relations are processed on the neural level. Across four presentation formats (i.e., fractions, decimals, dot patterns, and pie charts) we observed a shared neural substrate in bilateral inferior parietal cortex, slightly anterior and inferior to IPS areas recently found for overall magnitude proportion processing. Nevertheless, when evaluating the neural correlates of part-whole processing (i.e., contrasting fractions, dot patterns, and pie charts vs. decimals), we found wide-spread activation in fronto-parietal brain areas. These results indicate involvement of domain-general cognitive processes in part-whole processing beyond processing the overall magnitude of proportions. The dissociation between proportions involving part-whole relations and decimals was further substantiated by a representational similarity analysis, which revealed common neural processing for fractions, pie charts, and dot patterns, possibly representing their bipartite part-whole structure. In contrast, decimals seemed to be processed differently on the neural level, possibly reflecting missing processes of actual proportion calculation in decimals.

The Development of Nonsymbolic Probability Judgments in Children

Two experiments were designed to investigate the developmental trajectory of children's probability approximation abilities. In Experiment 1, results revealed 6- and 7-year-old children's (N = 48) probability judgments improve with age and become more accurate as the distance between two ratios increases. Experiment 2 replicated these findings with 7- to 12-year-old children (N = 130) while also accounting for the effect of the size and the perceived numerosity of target objects. Older children's performance suggested the correct use of proportions for estimating probability; but in some cases, children relied on heuristic shortcuts. These results suggest that children's nonsymbolic probability judgments show a clear distance effect and that the acuity of probability estimations increases with age.

Decoding vibrotactile choice independent of stimulus order and saccade selection during sequential comparisons

Decision-making in the somatosensory domain has been intensively studied using vibrotactile frequency discrimination tasks. Results from human and monkey electrophysiological studies from this line of research suggest that perceptual choices are encoded within a sensorimotor network. These findings, however, rely on experimental settings in which perceptual choices are inextricably linked to sensory and motor components of the task. Here, we devised a novel version of the vibrotactile frequency discrimination task with saccade responses which has the crucial advantage of decoupling perceptual choices from sensory and motor processes. We recorded human fMRI data from 32 participants while they performed the task. Using a whole-brain searchlight multivariate classification technique, we identify the left lateral prefrontal cortex and the oculomotor system, including the bilateral frontal eye fields (FEF) and intraparietal sulci, as representing vibrotactile choices. Moreover, we show that the decoding accuracy of choice information in the right FEF correlates with behavioral performance. Not only are these findings in remarkable agreement with previous work, they also provide novel fMRI evidence for choice coding in human oculomotor regions, which is not limited to saccadic decisions, but pertains to contexts where choices are made in a more abstract form.

Task Constraints Affect Mapping From Approximate Number System Estimates to Symbolic Numbers

The Approximate Number System (ANS) allows individuals to assess nonsymbolic numerical magnitudes (e.g., the number of apples on a tree) without counting. Several prominent theories posit that human understanding of symbolic numbers is based – at least in part – on mapping number symbols (e.g., 14) to their ANS-processed nonsymbolic analogs. Number-line estimation – where participants place numerical values on a bounded number-line – has become a key task used in research on this mapping. However, some research suggests that such number-line estimation tasks are actually proportion judgment tasks, as number-line estimation requires people to estimate the magnitude of the to-be-placed value, relative to set upper and lower endpoints, and thus do not so directly reflect magnitude representations. Here, we extend this work, assessing performance on nonsymbolic tasks that should more directly interface with the ANS. We compared adults’ (n = 31) performance when placing nonsymbolic numerosities (dot arrays) on number-lines to their performance with the same stimuli on two other tasks: Free estimation tasks where participants simply estimate the cardinality of dot arrays, and ratio estimation tasks where participants estimate the ratio instantiated by a pair of arrays. We found that performance on these tasks was quite different, with number-line and ratio estimation tasks failing to the show classic psychophysical error patterns of scalar variability seen in the free estimation task. We conclude the constraints of tasks using stimuli that access the ANS lead to considerably different mapping performance and that these differences must be accounted for when evaluating theories of numerical cognition. Additionally, participants showed typical underestimation patterns in the free estimation task, but were quite accurate on the ratio task. We discuss potential implications of these findings for theories regarding the mapping between ANS magnitudes and symbolic numbers.

The Posterior Parietal Cortex in Adaptive Visual Processing

Although the primate posterior parietal cortex (PPC) has been largely associated with space, attention, and action-related processing, a growing number of studies have reported the direct representation of a diverse array of action-independent nonspatial visual information in the PPC during both perception and visual working memory. By describing the distinctions and the close interactions of visual representation with space, attention, and action-related processing in the PPC, here I propose that we may understand these diverse PPC functions together through the unique contribution of the PPC to adaptive visual processing and form a more integrated and structured view of the role of the PPC in vision, cognition, and action.

Finding numbers in the brain

After listing functional constraints on what numbers in the brain must do, I sketch the two's complement fixed-point representation of numbers because it has stood the test of time and because it illustrates the non-obvious ways in which an effective coding scheme may operate. I briefly consider its neurobiological implementation. It is easier to imagine its implementation at the cell-intrinsic molecular level, with thermodynamically stable, volumetrically minimal polynucleotides encoding the remembered numbers, than at the circuit level, with plastic synapses encoding them.This article is part of a discussion meeting issue 'The origins of numerical abilities'.

A Tale of Two Visual Systems: Invariant and Adaptive Visual Information Representations in the Primate Brain

Visual information processing contains two opposite needs. There is both a need to comprehend the richness of the visual world and a need to extract only pertinent visual information to guide thoughts and behavior at a given moment. I argue that these two aspects of visual processing are mediated by two complementary visual systems in the primate brain-specifically, the occipitotemporal cortex (OTC) and the posterior parietal cortex (PPC). The role of OTC in visual processing has been documented extensively by decades of neuroscience research. I review here recent evidence from human imaging and monkey neurophysiology studies to highlight the role of PPC in adaptive visual processing. I first document the diverse array of visual representations found in PPC. I then describe the adaptive nature of visual representation in PPC by contrasting visual processing in OTC and PPC and by showing that visual representations in PPC largely originate from OTC.

Natural Alternatives to Natural Number: The Case of Ratio

The overwhelming majority of efforts to cultivate early mathematical thinking rely primarily on counting and associated natural number concepts. Unfortunately, natural numbers and discretized thinking do not align well with a large swath of the mathematical concepts we wish for children to learn. This misalignment presents an important impediment to teaching and learning. We suggest that one way to circumvent these pitfalls is to leverage students' non-numerical experiences that can provide intuitive access to foundational mathematical concepts. Specifically, we advocate for explicitly leveraging a) students' perceptually based intuitions about quantity and b) students' reasoning about change and variation, and we address the affordances offered by this approach. We argue that it can support ways of thinking that may at times align better with to-be-learned mathematical ideas, and thus may serve as a productive alternative for particular mathematical concepts when compared to number. We illustrate this argument using the domain of ratio, and we do so from the distinct disciplinary lenses we employ respectively as a cognitive psychologist and as a mathematics education researcher. Finally, we discuss the potential for productive synthesis given the substantial differences in our preferred methods and general epistemologies.

Confidence judgments during ratio comparisons reveal a Bayesian bias

Rational numbers are essential in mathematics and decision-making but humans often and erroneously rely on the magnitude of the numerator or denominator to determine the relative size of a quotient. The source of this flawed whole number strategy is poorly understood. Here we test the Bayesian hypothesis that the human bias toward large values in the numerator or denominator of a ratio estimate is the result of higher confidence in large samples. Larger values are considered a better (more certain) instance of that ratio than the same ratio composed of smaller values. We collected confidence measures explicitly (Experiment 1) and implicitly (Experiment 2) during subjects' comparisons of non-symbolic proportions (images with arrays of orange and blue dots). We manipulated the discernibility of the fractions to control difficulty and varied the cardinality and congruency of the numerators, denominators, and ratio values (e.g. 8/20 vs. 5/10 and 16/40 vs. 10/20). The results revealed that subjects' confidence during ratio comparisons was modulated by the numerical magnitude of the fraction's components, consistent with a Bayesian perception of relative ratios. The results suggest that the large number bias could arise from greater confidence in large samples.

From continuous magnitudes to symbolic numbers: The centrality of ratio

Leibovich et al.'s theory neither accounts for the deep connections between whole numbers and other classes of number nor provides a potential mechanism for mapping continuous magnitudes to symbolic numbers. We argue that focusing on non-symbolic ratio processing abilities can furnish a more expansive account of numerical cognition that remedies these shortcomings.

Numerical Proportion Representation: A Neurocomputational Account

Proportion representation is an emerging subdomain in numerical cognition. However, its nature and its correlation with simple number representation remain elusive, especially at the theoretical level. To fill this gap, we propose a gain-field model of proportion representation to shed light on the neural and computational basis of proportion representation. The model is based on two well-supported neuroscientific findings. The first, gain modulation, is a general mechanism for information integration in the brain; the second relevant finding is how simple quantity is neurally represented. Based on these principles, the model accounts for recent relevant proportion representation data at both behavioral and neural levels. The model further addresses two key computational problems for the cognitive processing of proportions: invariance and generalization. Finally, the model provides pointers for future empirical testing.

Spontaneous, modality-general abstraction of a ratio scale

The existence of a generalized magnitude system in the human mind and brain has been studied extensively but remains elusive because it has not been clearly defined. Here we show that one possibility is the representation of relative magnitudes via ratio calculations: ratios are a naturally dimensionless or abstract quantity that could qualify as a common currency for magnitudes measured on vastly different psychophysical scales and in different sensory modalities like size, number, duration, and loudness. In a series of demonstrations based on comparisons of item sequences, we demonstrate that subjects spontaneously use knowledge of inter-item ratios within and across sensory modalities and across magnitude domains to rate sequences as more or less similar on a sliding scale. Moreover, they rate ratio-preserved sequences as more similar to each other than sequences in which only ordinal relations are preserved, indicating that subjects are aware of differences in levels of relative-magnitude information preservation. The ubiquity of this ability across many different magnitude pairs, even those sharing no sensory information, suggests a highly general code that could qualify as a candidate for a generalized magnitude representation.

Relational Priming Based on a Multiplicative Schema for Whole Numbers and Fractions

Why might it be (at least sometimes) beneficial for adults to process fractions componentially? Recent research has shown that college-educated adults can capitalize on the bipartite structure of the fraction notation, performing more successfully with fractions than with decimals in relational tasks, notably analogical reasoning. This study examined patterns of relational priming for problems with fractions in a task that required arithmetic computations. College students were asked to judge whether or not multiplication equations involving fractions were correct. Some equations served as structurally inverse primes for the equation that immediately followed it (e.g., 4 × 3/4 = 3 followed by 3 × 8/6 = 4). Students with relatively high math ability showed relational priming (speeded solution times to the second of two successive relationally related fraction equations) both with and without high perceptual similarity (Experiment 2). Students with relatively low math ability also showed priming, but only when the structurally inverse equation pairs were supported by high perceptual similarity between numbers (e.g., 4 × 3/4 = 3 followed by 3 × 4/3 = 4). Several additional experiments established boundary conditions on relational priming with fractions. These findings are interpreted in terms of componential processing of fractions in a relational multiplication context that takes advantage of their inherent connections to a multiplicative schema for whole numbers.

Fractions We Cannot Ignore: The Nonsymbolic Ratio Congruity Effect

Although many researchers theorize that primitive numerosity processing abilities may lay the foundation for whole number concepts, other classes of numbers, like fractions, are sometimes assumed to be inaccessible to primitive architectures. This research presents evidence that the automatic processing of nonsymbolic magnitudes affects processing of symbolic fractions. Participants completed modified Stroop tasks in which they selected the larger of two symbolic fractions while the ratios of the fonts in which the fractions were printed and the overall sizes of the compared fractions were manipulated as irrelevant dimensions. Participants were slower and less accurate when nonsymbolic dimensions of printed fractions were incongruent with the symbolic comparison decision. Results indicated a robust basic sensitivity to nonsymbolic ratios that exceeds prior conceptions about the accessibility of fraction values. Results also indicated a congruity effect for overall fraction size, contrary to findings of prior research. These findings have implications for extending theory about the nature of human number sense and mathematical cognition more generally.

Ratio abstraction over discrete magnitudes by newly hatched domestic chicks (Gallus gallus)

A large body of literature shows that non-human animals master a variety of numerical tasks, but studies involving proportional discrimination are sparse and primarily done with mature animals. Here we trained 4-day-old domestic chicks (Gallus gallus) to respond to stimuli depicting multiple examples of the proportion 4:1 when compared with the proportion 2:1. Stimuli were composed of green and red dot arrays; for the rewarded 4:1 proportion, 4 green dots for every red dot (e.g. ratios: 32:8, 12:3, and 44:11). The birds continued to discriminate when presented with new ratios at test (such as 20:5), characterized by new numbers of dots and new spatial configurations (Experiment 1). This indicates that chicks can extract the common proportional value shared by different ratios and apply it to new ones. In Experiment 2, chicks identified a specific proportion (2:1) from either a smaller (4:1) or a larger one (1:1), demonstrating an ability to represent the specific, and not relative, value of a particular proportion. Again, at test, chicks selectively responded to the previously reinforced proportion from new ratios. These findings provide strong evidence for very young animals’ ability to extract, identify, and productively use proportion information across a range of different amounts.

The neuronal code for number

Humans and non-human primates share an elemental quantification system that resides in a dedicated neural network in the parietal and frontal lobes. In this cortical network, 'number neurons' encode the number of elements in a set, its cardinality or numerosity, irrespective of stimulus appearance across sensory motor systems, and from both spatial and temporal presentation arrays. After numbers have been extracted from sensory input, they need to be processed to support goal-directed behaviour. Studying number neurons provides insights into how information is maintained in working memory and transformed in tasks that require rule-based decisions. Beyond an understanding of how cardinal numbers are encoded, number processing provides a window into the neuronal mechanisms of high-level brain functions.

Non-symbolic division in childhood

The approximate number system (ANS) underlies representations of large numbers of objects as well as the additive, subtractive, and multiplicative relationships between them. In this set of studies, 5- and 6-year-old children were shown a series of video-based events that conveyed a transformation of a large number of objects into one-half or one-quarter of the original number. Children were able to estimate correctly the outcomes to these halving and quartering problems, and they based their responses on scaling by number, not on continuous quantities or guessing strategies. Children's performance exhibited the ratio signature of the ANS. Moreover, children performed above chance on relatively early trials, suggesting that this scaling operation is easily conveyed and readily performed. The results support the existence of a flexible and substantially untrained capacity to scale numerical amounts.

Individual Differences in Nonsymbolic Ratio Processing Predict Symbolic Math Performance

What basic capacities lay the foundation for advanced numerical cognition? Are there basic nonsymbolic abilities that support the understanding of advanced numerical concepts, such as fractions? To date, most theories have posited that previously identified core numerical systems, such as the approximate number system (ANS), are ill-suited for learning fraction concepts. However, recent research in developmental psychology and neuroscience has revealed a ratio-processing system (RPS) that is sensitive to magnitudes of nonsymbolic ratios and may be ideally suited for supporting fraction concepts. We provide evidence for this hypothesis by showing that individual differences in RPS acuity predict performance on four measures of mathematical competence, including a university entrance exam in algebra. We suggest that the nonsymbolic RPS may support symbolic fraction understanding much as the ANS supports whole-number concepts. Thus, even abstract mathematical concepts, such as fractions, may be grounded not only in higher-order logic and language, but also in basic nonsymbolic processing abilities.

The number-time interaction depends on relative magnitude in the suprasecond range

Numerical representations influence temporal processing. Previous studies have consistently shown that larger numbers are perceived to last longer than smaller ones. However, whether this effect is modulated by the absolute or relative magnitudes of the numbers has yet to be fully understood. Here, participants observed single- and double-digit Arabic numerals in separate experimental blocks and reproduced stimulus duration of 600 or 1200 ms. Our results replicated previous findings that the duration of larger numbers was reproduced longer than that of smaller numbers within each digit set. Although the effect of numerical magnitude across single- and double-digit numerals was found when the numerals were presented for 600 ms, the difference was negligible when they were presented for 1200 ms, suggesting that relative magnitude is an important factor in the number-time interaction in the suprasecond range. These results suggest that contextual influence on number-time interaction may depend on the actual stimulus duration.