Observers Efficiently Extract the Minimal and Maximal Element in Perceptual Magnitude Sets: Evidence for a Bipartite Format.

The mind represents abstract magnitude information, including time, space, and number, but in what format is this information stored? We show support for the bipartite format of perceptual magnitudes, in which the measured value on a dimension is scaled to the dynamic range of the input, leading to a privileged status for values at the lowest and highest end of the range. In six experiments with college undergraduates, we show that observers are faster and more accurate to find the endpoints (i.e., the minimum and maximum) than any of the inner values, even as the number of items increases beyond visual short-term memory limits. Our results show that length, size, and number are represented in a dynamic format that allows for comparison-free sorting, with endpoints represented with an immediately accessible status, consistent with the bipartite model of perceptual magnitudes. We discuss the implications for theories of visual search and ensemble perception.

Emergence of the Link Between the Approximate Number System and Symbolic Math Ability.

Experimentally manipulating Approximate Number System (ANS) precision has been found to influence children's subsequent symbolic math performance. Here in three experiments (N = 160; 81 girls; 3-5 year old) we replicated this effect and examined its duration and developmental trajectory. We found that modulation of 5-year-olds' ANS precision continued to affect their symbolic math performance after a 30-min delay. Furthermore, our cross-sectional investigation revealed that children 4.5 years and older experienced a significant transfer effect of ANS manipulation on math performance, whereas younger children showed no such transfer, despite experiencing significant changes in ANS precision. These findings support the existence of a causal link between nonverbal numerical approximation and symbolic math performance that first emerges during the preschool years.

Better together: Multiple lines of evidence for a link between approximate and exact number representations: A reply to Merkley, Matejko, and Ansari.

The results of our recent experiments suggest that temporarily modulating children's approximate number system (ANS) precision leads to a domain-specific change in their symbolic math performance (Journal of Experimental Child Psychology, 2016, Vol. 147, pp. 82-99). We interpreted these results as evidence for a causal relationship between ANS precision and symbolic math. In a commentary on our work, Merkley, Matejko, and Ansari argue that our methodology limits the interpretation of our results, primarily because our experiments did not meet the criteria for an intervention study as set out by What Works Clearinghouse and others. Here, we clarify the goals and limitations of our study and emphasize the variety of approaches to demonstrating causality. We argue that our goal was not to design and test an intervention or to compare the effectiveness of different treatments. Instead, we aimed to experimentally manipulate one variable (i.e., ANS acuity) and, in a randomized sample of children, observe whether this manipulation had any statistically significant effect on a dependent variable (i.e., performance on a set of symbolic math questions). We provide further analyses to support our assertion that a temporary manipulation of ANS performance does lead to a change in math performance. These results point to a causal relationship between ANS precision and math, and they suggest that further investigation of this relationship will be fruitful.

The precision of mapping between number words and the approximate number system predicts children's formal math abilities.

Children can represent number in at least two ways: by using their non-verbal, intuitive approximate number system (ANS) and by using words and symbols to count and represent numbers exactly. Furthermore, by the time they are 5years old, children can map between the ANS and number words, as evidenced by their ability to verbally estimate numbers of items without counting. How does the quality of the mapping between approximate and exact numbers relate to children's math abilities? The role of the ANS-number word mapping in math competence remains controversial for at least two reasons. First, previous work has not examined the relation between verbal estimation and distinct subtypes of math abilities. Second, previous work has not addressed how distinct components of verbal estimation-mapping accuracy and variability-might each relate to math performance. Here, we addressed these gaps by measuring individual differences in ANS precision, verbal number estimation, and formal and informal math abilities in 5- to 7-year-old children. We found that verbal estimation variability, but not estimation accuracy, predicted formal math abilities, even when controlling for age, expressive vocabulary, and ANS precision, and that it mediated the link between ANS precision and overall math ability. These findings suggest that variability in the ANS-number word mapping may be especially important for formal math abilities.

Changing the precision of preschoolers' approximate number system representations changes their symbolic math performance.

From early in life, humans have access to an approximate number system (ANS) that supports an intuitive sense of numerical quantity. Previous work in both children and adults suggests that individual differences in the precision of ANS representations correlate with symbolic math performance. However, this work has been almost entirely correlational in nature. Here we tested for a causal link between ANS precision and symbolic math performance by asking whether a temporary modulation of ANS precision changes symbolic math performance. First, we replicated a recent finding that 5-year-old children make more precise ANS discriminations when starting with easier trials and gradually progressing to harder ones, compared with the reverse. Next, we show that this brief modulation of ANS precision influenced children's performance on a subsequent symbolic math task but not a vocabulary task. In a supplemental experiment, we present evidence that children who performed ANS discriminations in a random trial order showed intermediate performance on both the ANS task and the symbolic math task, compared with children who made ordered discriminations. Thus, our results point to a specific causal link from the ANS to symbolic math performance.

PsiMLE: A maximum-likelihood estimation approach to estimating psychophysical scaling and variability more reliably, efficiently, and flexibly.

A simple and popular psychophysical model-usually described as overlapping Gaussian tuning curves arranged along an ordered internal scale-is capable of accurately describing both human and nonhuman behavioral performance and neural coding in magnitude estimation, production, and reproduction tasks for most psychological dimensions (e.g., time, space, number, or brightness). This model traditionally includes two parameters that determine how a physical stimulus is transformed into a psychological magnitude: (1) an exponent that describes the compression or expansion of the physical signal into the relevant psychological scale (ß), and (2) an estimate of the amount of inherent variability (often called internal noise) in the Gaussian activations along the psychological scale (σ). To date, linear slopes on log-log plots have traditionally been used to estimate ß, and a completely separate method of averaging coefficients of variance has been used to estimate σ. We provide a respectful, yet critical, review of these traditional methods, and offer a tutorial on a maximum-likelihood estimation (MLE) and a Bayesian estimation method for estimating both ß and σ [PsiMLE(ß,σ)], coupled with free software that researchers can use to implement it without a background in MLE or Bayesian statistics (R-PsiMLE). We demonstrate the validity, reliability, efficiency, and flexibility of this method through a series of simulations and behavioral experiments, and find the new method to be superior to the traditional methods in all respects.

The reliability and internal consistency of one-shot and flicker change detection for measuring individual differences in visual working memory capacity.

We investigated the psychometric properties of the one-shot change detection task for estimating visual working memory (VWM) storage capacity-and also introduced and tested an alternative flicker change detection task for estimating these limits. In three experiments, we found that the one-shot whole-display task returns estimates of VWM storage capacity (K) that are unreliable across set sizes-suggesting that the whole-display task is measuring different things at different set sizes. In two additional experiments, we found that the one-shot single-probe variant shows improvements in the reliability and consistency of K estimates. In another additional experiment, we found that a one-shot whole-display-with-click task (requiring target localization) also showed improvements in reliability and consistency. The latter results suggest that the one-shot task can return reliable and consistent estimates of VWM storage capacity (K), and they highlight the possibility that the requirement to localize the changed target is what engenders this enhancement. Through a final series of four experiments, we introduced and tested an alternative flicker change detection method that also requires the observer to localize the changing target and that generates, from response times, an estimate of VWM storage capacity (K). We found that estimates of K from the flicker task correlated with estimates from the traditional one-shot task and also had high reliability and consistency. We highlight the flicker method's ability to estimate executive functions as well as VWM storage capacity, and discuss the potential for measuring multiple abilities with the one-shot and flicker tasks.

Number sense across the lifespan as revealed by a massive Internet-based sample.

It has been difficult to determine how cognitive systems change over the grand time scale of an entire life, as few cognitive systems are well enough understood; observable in infants, adolescents, and adults; and simple enough to measure to empower comparisons across vastly different ages. Here we address this challenge with data from more than 10,000 participants ranging from 11 to 85 years of age and investigate the precision of basic numerical intuitions and their relation to students' performance in school mathematics across the lifespan. We all share a foundational number sense that has been observed in adults, infants, and nonhuman animals, and that, in humans, is generated by neurons in the intraparietal sulcus. Individual differences in the precision of this evolutionarily ancient number sense may impact school mathematics performance in children; however, we know little of its role beyond childhood. Here we find that population trends suggest that the precision of one's number sense improves throughout the school-age years, peaking quite late at ∼30 y. Despite this gradual developmental improvement, we find very large individual differences in number sense precision among people of the same age, and these differences relate to school mathematical performance throughout adolescence and the adult years. The large individual differences and prolonged development of number sense, paired with its consistent and specific link to mathematics ability across the age span, hold promise for the impact of educational interventions that target the number sense.

Eye movements reveal distinct encoding patterns for number and cumulative surface area in random dot arrays.

Humans can quickly and intuitively represent the number of objects in a scene using visual evidence through the Approximate Number System (ANS). But the computations that support the encoding of visual number-the transformation from the retinal input into ANS representations-remain controversial. Two types of number encoding theories have been proposed: those arguing that number is encoded through a dedicated, enumeration computation, and those arguing that visual number is inferred from nonnumber specific visual features, such as surface area, density, convex hull, etc. Here, we attempt to adjudicate between these two theories by testing participants on both a number and a cumulative area task while also tracking their eye-movements. We hypothesize that if approximate number and surface area depend on distinct encoding computations, saccadic signatures should be distinct for the two tasks, even if the visual stimuli are identical. Consistent with this hypothesis, we find that discriminating number versus cumulative area modulates both where participants look (i.e., participants spend more time looking at the more numerous set in the number task and the larger set in the cumulative area task), and how participants look (i.e., cumulative area encoding shows fewer, longer saccades, while number encoding shows many short saccades and many switches between targets). We further identify several saccadic signatures that are associated with task difficulty and correct versus incorrect trials for both dimensions. These results suggest distinct encoding algorithms for number and cumulative area extraction, and thereby distinct representations of these dimensions.

Understanding the mapping between numerical approximation and number words: evidence from Williams syndrome and typical development.

All numerate humans have access to two systems of number representation: an exact system that is argued to be based on language and that supports formal mathematics, and an Approximate Number System (ANS) that is present at birth and appears independent of language. Here we examine the interaction between these two systems by comparing the profiles of people with Williams syndrome (WS) with those of typically developing children between ages 4 and 9 years. WS is a rare genetic deficit marked by fluent and well-structured language together with severe spatial deficits, deficits in formal math, and abnormalities of the parietal cortex, which is thought to subserve the ANS. One of our tasks, requiring approximate number comparison but no number words, revealed that the ANS precision of adolescents with WS was in the range of typically developing 2- to 4-year-olds. Their precision improved with age but never reached the level of typically developing 6- or 9-year-olds. The second task, requiring verbal number estimation using number words, revealed that the estimates produced by adolescents with WS were comparable to those of typically developing 6- and 9-year-olds, i.e. were more advanced than their ANS precision. These results suggest that ANS precision is somewhat separable from the mapping between approximate numerosities and number words, as the former can be severely damaged in a genetic disorder without commensurate impairment in the latter.

Individual differences in non-verbal number acuity correlate with maths achievement.

Human mathematical competence emerges from two representational systems. Competence in some domains of mathematics, such as calculus, relies on symbolic representations that are unique to humans who have undergone explicit teaching. More basic numerical intuitions are supported by an evolutionarily ancient approximate number system that is shared by adults, infants and non-human animals-these groups can all represent the approximate number of items in visual or auditory arrays without verbally counting, and use this capacity to guide everyday behaviour such as foraging. Despite the widespread nature of the approximate number system both across species and across development, it is not known whether some individuals have a more precise non-verbal 'number sense' than others. Furthermore, the extent to which this system interfaces with the formal, symbolic maths abilities that humans acquire by explicit instruction remains unknown. Here we show that there are large individual differences in the non-verbal approximation abilities of 14-year-old children, and that these individual differences in the present correlate with children's past scores on standardized maths achievement tests, extending all the way back to kindergarten. Moreover, this correlation remains significant when controlling for individual differences in other cognitive and performance factors. Our results show that individual differences in achievement in school mathematics are related to individual differences in the acuity of an evolutionarily ancient, unlearned approximate number sense. Further research will determine whether early differences in number sense acuity affect later maths learning, whether maths education enhances number sense acuity, and the extent to which tertiary factors can affect both.

A Continuity in Logical Development: Domain-General Disjunctive Inference by Toddlers.

Children grow up surrounded by opportunities to learn (the language of their community, the movements of their body, other people's preferences and mental lives, games, social norms, etc.). Here, we find that toddlers (N = 36; age range 2.3-3.2 years) rely on a logical reasoning strategy, Disjunctive Inference (i.e., A OR B, A is ruled out, THEREFORE, B), across a variety of situations, all before they have any formal education or extensive experience with words for expressing logical meanings. In learning new words, learning new facts about a person, and finding the winner of a race, toddlers systematically consider and reject competitors before deciding who must be the winner. This suggests that toddlers may have a general-purpose logical reasoning tool that they can use in any situation.

Modeling Magnitude Discrimination: Effects of Internal Precision and Attentional Weighting of Feature Dimensions.

Given a rich environment, how do we decide on what information to use? A view of a single entity (e.g., a group of birds) affords many distinct interpretations, including their number, average size, and spatial extent. An enduring challenge for cognition, therefore, is to focus resources on the most relevant evidence for any particular decision. In the present study, subjects completed three tasks-number discrimination, surface area discrimination, and convex hull discrimination-with the same stimulus set, where these three features were orthogonalized. Therefore, only the relevant feature provided consistent evidence for decisions in each task. This allowed us to determine how well humans discriminate each feature dimension and what evidence they relied on to do so. We introduce a novel computational approach that fits both feature precision and feature use. We found that the most relevant feature for each decision is extracted and relied on, with minor contributions from competing features. These results suggest that multiple feature dimensions are separately represented for each attended ensemble of many items and that cognition is efficient at selecting the appropriate evidence for a decision.

Compromised approximate number system acuity in extremely preterm school-aged children.

AIM: The aim of this study was to compare the approximate number system acuity in children born extremely preterm aged 6 years 6 months and typically developing, age-matched peers. METHOD: This population-based follow-up study included 65 children born before 27 gestational weeks (35 males, 30 females; mean gestational age 25.4 wks [SD 1.1 wk]; mean birthweight 789 g [SD 158 g]) and 47 typically developing children (24 females, 23 males) at the age of 6 years 6 months. A battery of cognitive tests was administered, including a computerized test for measuring approximate number system acuity and tests for general cognition, working memory, processing speed, and visual attention. Approximate number system outcome measures were means of Weber fraction (w) values and response time in milliseconds. RESULTS: The 43 extremely preterm children in whom usable data were obtained performed significantly worse than the typically developing children on the approximate number system task (w=0.30 [SD 0.23] vs. 0.17 [SD 0.13]; p=0.003) and were significantly slower (response time=2934 ms [SD 1102 ms] vs 2376 ms [SD 310 ms]; p=0.002). The differences remained when adjusting for differences in other cognitive functions (p=0.03). INTERPRETATION: Preterm birth has a negative impact on an individual's ability to rapidly approximate and compare numbers of visually presented items. This deficiency is thought to be a consequence of dorsal stream dysfunction. Future studies will investigate whether this deficiency is correlated with lower mathematical proficiency in this group of children.

Numerical approximation abilities correlate with and predict informal but not formal mathematics abilities.

Previous research has found a relationship between individual differences in children's precision when nonverbally approximating quantities and their school mathematics performance. School mathematics performance emerges from both informal (e.g., counting) and formal (e.g., knowledge of mathematics facts) abilities. It remains unknown whether approximation precision relates to both of these types of mathematics abilities. In the current study, we assessed the precision of numerical approximation in 85 3- to 7-year-old children four times over a span of 2years. In addition, at the final time point, we tested children's informal and formal mathematics abilities using the Test of Early Mathematics Ability (TEMA-3). We found that children's numerical approximation precision correlated with and predicted their informal, but not formal, mathematics abilities when controlling for age and IQ. These results add to our growing understanding of the relationship between an unlearned nonsymbolic system of quantity representation and the system of mathematics reasoning that children come to master through instruction.

Is Approximate Number Precision a Stable Predictor of Math Ability?

Previous research shows that children's ability to estimate numbers of items using their Approximate Number System (ANS) predicts later math ability. To more closely examine the predictive role of early ANS acuity on later abilities, we assessed the ANS acuity, math ability, and expressive vocabulary of preschoolers twice, six months apart. We also administered attention and memory span tasks to ask whether the previously reported association between ANS acuity and math ability is ANS-specific or attributable to domain-general cognitive skills. We found that early ANS acuity predicted math ability six months later, even when controlling for individual differences in age, expressive vocabulary, and math ability at the initial testing. In addition, ANS acuity was a unique concurrent predictor of math ability above and beyond expressive vocabulary, attention, and memory span. These findings of a predictive relationship between early ANS acuity and later math ability add to the growing evidence for the importance of early numerical estimation skills.

A Shared Intuitive (Mis)understanding of Psychophysical Law Leads Both Novices and Educated Students to Believe in a Just Noticeable Difference (JND).

Humans are both the scientists who discover psychological laws and the thinkers who behave according to those laws. Oftentimes, when our natural behavior is in accord with those laws, this dual role serves us well: our intuitions about our own behavior can serve to inform our discovery of new laws. But, in cases where the laws that we discover through science do not agree with the intuitions and biases we carry into the lab, we may find it harder to believe in and adopt those laws. Here, we explore one such case. Since the founding of psychophysics, the notion of a Just Noticeable Difference (JND) in perceptual discrimination has been ubiquitous in experimental psychology-even in spite of theoretical advances since the 1950's that argue that there can be no such thing as a threshold in perceiving difference. We find that both novices and psychologically educated students alike misunderstand the JND to mean that, below a certain threshold, humans will be unable to tell which of two quantities is greater (e.g., that humans will be completely at chance when trying to judge which is heavier, a bag with 3000 grains of sand or 3001). This belief in chance performance below a threshold is inconsistent with psychophysical law. We argue that belief in a JND is part of our intuitive theory of psychology and is therefore very difficult to dispel.

Preschool acuity of the approximate number system correlates with school math ability.

Previous research shows a correlation between individual differences in people's school math abilities and the accuracy with which they rapidly and nonverbally approximate how many items are in a scene. This finding is surprising because the Approximate Number System (ANS) underlying numerical estimation is shared with infants and with non-human animals who never acquire formal mathematics. However, it remains unclear whether the link between individual differences in math ability and the ANS depends on formal mathematics instruction. Earlier studies demonstrating this link tested participants only after they had received many years of mathematics education, or assessed participants' ANS acuity using tasks that required additional symbolic or arithmetic processing similar to that required in standardized math tests. To ask whether the ANS and math ability are linked early in life, we measured the ANS acuity of 200 3- to 5-year-old children using a task that did not also require symbol use or arithmetic calculation. We also measured children's math ability and vocabulary size prior to the onset of formal math instruction. We found that children's ANS acuity correlated with their math ability, even when age and verbal skills were controlled for. These findings provide evidence for a relationship between the primitive sense of number and math ability starting early in life.

Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia).

Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD.

Rapid fast-mapping abilities in 2-year-olds.

Learning a new word consists of two primary tasks that have often been conflated into a single process: referent selection, in which a child must determine the correct referent of a novel label, and referent retention, which is the ability to store this newly formed label-object mapping in memory for later use. In addition, children must be capable of performing these tasks rapidly and repeatedly as they are frequently exposed to novel words during the course of natural conversation. Here we used a preferential pointing task to investigate 2-year-olds' (N=72) ability to infer the referent of a novel noun from a single ambiguous exposure and their ability to retain this mapping over time. Children were asked to identify the referent of a novel label on six critical trials distributed throughout the course of a 10-min study involving many familiar and novel objects. On these critical trials, images of a known object and a novel object (e.g., a ball and a nameless artifact constructed in the laboratory) appeared on two computer screens and a voice asked children to "point at the _____ [e.g., glark]." Following label onset, children were allowed only 3s during which to infer the correct referent, point at it, and potentially store this new word-object mapping. In a final posttest trial, all previously labeled novel objects appeared and children were asked to point to one of them (e.g., "Can you find the glark?"). To succeed, children needed to have initially mapped the novel labels correctly and retained these mappings over the course of the study. Despite the difficult demands of the current task, children successfully identified the target object on the retention trial. We conclude that 2-year-olds are able to fast map novel nouns during a brief single exposure under ambiguous labeling conditions.