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.

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.

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.

Approximate Number Sense in Students With Severe Hearing Loss: A Modality-Neutral Cognitive Ability.

The Approximate Number System (ANS) allows humans and non-human animals to estimate large quantities without counting. It is most commonly studied in visual contexts (i.e., with displays containing different numbers of dots), although the ANS may operate on all approximate quantities regardless of modality (e.g., estimating the number of a series of auditory tones). Previous research has shown that there is a link between ANS and mathematics abilities, and that this link is resilient to differences in visual experience (Kanjlia et al., 2018). However, little is known about the function of the ANS and its relationship to mathematics abilities in the absence of other types of sensory input. Here, we investigated the acuity of the ANS and its relationship with mathematics abilities in a group of students from the Sichuan Province in China, half of whom were deaf. We found, consistent with previous research, that ANS acuity improves with age. We found that mathematics ability was predicted by Non-verbal IQ and Inhibitory Control, but not visual working memory capacity or Attention Network efficiencies. Even above and beyond these predictors, ANS ability still accounted for unique variance in mathematics ability. Notably, there was no interaction with hearing, which indicates that the role played by the ANS in explaining mathematics competence is not modulated by hearing capacity. Finally, we found that age, Non-verbal IQ and Visual Working Memory capacity were predictive of ANS performance when controlling for other factors. In fact, although students with hearing loss performed slightly worse than students with normal hearing on the ANS task, hearing was no longer significantly predictive of ANS performance once other factors were taken into account. These results indicate that the ANS is able to develop at a consistent pace with other cognitive abilities in the absence of auditory experience, and that its relationship with mathematics ability is not contingent on sensory input from hearing.

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.

Linguistic meanings as cognitive instructions.

Natural languages like English connect pronunciations with meanings. Linguistic pronunciations can be described in ways that relate them to our motor system (e.g., to the movement of our lips and tongue). But how do linguistic meanings relate to our nonlinguistic cognitive systems? As a case study, we defend an explicit proposal about the meaning of most by comparing it to the closely related more: whereas more expresses a comparison between two independent subsets, most expresses a subset-superset comparison. Six experiments with adults and children demonstrate that these subtle differences between their meanings influence how participants organize and interrogate their visual world. In otherwise identical situations, changing the word from most to more affects preferences for picture-sentence matching (experiments 1-2), scene creation (experiments 3-4), memory for visual features (experiment 5), and accuracy on speeded truth judgments (experiment 6). These effects support the idea that the meanings of more and most are mental representations that provide detailed instructions to conceptual systems.

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.

Location- and object-based attention enhance number estimation.

Humans and non-humans can extract an estimate of the number of items in a collection very rapidly, raising the question of whether attention is necessary for this process. Visual attention operates in various modes, showing selectivity both to spatial location and to objects. Here, we tested whether each form of attention can enhance number estimation, by measuring whether presenting a visual cue to increase attentional engagement will lead to a more accurate and precise representation of number, both when attention is directed to location and when it is directed to objects. Results revealed that enumeration of a collection of dots in the location previously cued led to faster, more precise, and more accurate judgments than enumeration in un-cued locations, and a similar benefit was seen when the cue and collection appeared on the same object. This work shows that like many other perceptual tasks, numerical estimation may be enhanced by the spread of active attention inside a pre-cued object.

Effects of Visual Training of Approximate Number Sense on Auditory Number Sense and School Math Ability.

Research with children and adults suggests that people's math performance is predicted by individual differences in an evolutionarily ancient ability to estimate and compare numerical quantities without counting (the approximate number system or ANS). However, previous work has almost exclusively used visual stimuli to measure ANS precision, leaving open the possibility that the observed link might be driven by aspects of visuospatial competence, rather than the amodal ANS. We addressed this possibility in an ANS training study. Sixty-eight 6-year-old children participated in a 5-week study that either trained their visual ANS ability or their phonological awareness (an active control group). Immediately before and after training, we assessed children's visual and auditory ANS precision, as well as their symbolic math ability and phonological awareness. We found that, prior to training, children's precision in a visual ANS task related to their math performance - replicating recent studies. Importantly, precision in an auditory ANS task also related to math performance. Furthermore, we found that children who completed visual ANS training showed greater improvements in auditory ANS precision than children who completed phonological awareness training. Finally, children in the ANS training group showed significant improvements in math ability but not phonological awareness. These results suggest that the link between ANS precision and school math ability goes beyond visuospatial abilities and that the modality-independent ANS is causally linked to math ability in early childhood.

Age and Species Comparisons of Visual Mental Manipulation Ability as Evidence for its Development and Evolution.

Intelligent behavior is shaped by the abilities to store and manipulate information in visual working memory. Although humans and various non-human animals demonstrate similar storage capacities, the evolution of manipulation ability remains relatively unspecified. To what extent are manipulation limits unique to humans versus shared across species? Here, we compare behavioral signatures of manipulation ability demonstrated by human adults and 6-to-8-year-old children with that of an animal separated from humans by over 300 million years of evolution: a Grey parrot (Psittacus erithacus). All groups of participants completed a variant of the "Shell Game", which required mentally updating the locations of varying set sizes of occluded objects that swapped places a number of times. The parrot not only demonstrated above-chance performance, but also outperformed children across all conditions. Indeed, the parrot's accuracy was comparable to (and slightly better than) human adults' over 12/14 set-size/number-of-swaps combinations, until four items were manipulated with 3-4 swaps, where performance decreased toward that of 6- to 8-year-olds. These results suggest that manipulation of visual working memory representations is an evolutionarily ancient ability. An important next step in this research program is establishing variability across species, and identifying the evolutionary origins (analogous or homologous) of manipulation mechanisms.

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.

Using the flicker task to estimate visual working memory storage capacity.

Studies of visual working memory (VWM) typically have used a "one-shot" change detection task to arrive at a capacity estimate of three to four objects, with additional limits imposed by the precision of the information needed for each object. Unlike the one-shot task, the flicker change detection task permits measurement of VWM capacity over time and with larger numbers of objects present in the scene, but it has rarely been used to assess the capacity of VWM. We used the flicker task to examine (a) whether capacity is close to the typical three to four items when using subtly different stimuli; (b) which dependent measure provides the most meaningful estimate of the capacity of VWM in the flicker task (response time or number of changes viewed); (c) whether capacity remains fixed at three to four items for displays containing many more objects; and (d) how VWM operates over time, with repeated opportunities to encode, retain, and compare elements in a display. Four experiments using grids of simple items varying only in luminance or color revealed a range for VWM capacity limits that was largely impervious to changes in display duration, interstimulus intervals, and array size. This estimate of VWM capacity was correlated with an estimate from the more typical one-shot task, further validating the flicker task as a tool for measuring the capacity of VWM.

Perceptual Input Is Not Conceptual Content.

Can we represent number approximately? A seductive reductionist notion is that participants in number tasks rely on continuous extent cues (e.g., area) and therefore that the representations underlying performance lack numerical content. I suggest that this notion embraces a misconception: that perceptual input determines conceptual content.

Approximate number sense correlates with math performance in gifted adolescents.

Nonhuman animals, human infants, and human adults all share an Approximate Number System (ANS) that allows them to imprecisely represent number without counting. Among humans, people differ in the precision of their ANS representations, and these individual differences have been shown to correlate with symbolic mathematics performance in both children and adults. For example, children with specific math impairment (dyscalculia) have notably poor ANS precision. However, it remains unknown whether ANS precision contributes to individual differences only in populations of people with lower or average mathematical abilities, or whether this link also is present in people who excel in math. Here we tested non-symbolic numerical approximation in 13- to 16-year old gifted children enrolled in a program for talented adolescents (the Center for Talented Youth). We found that in this high achieving population, ANS precision significantly correlated with performance on the symbolic math portion of two common standardized tests (SAT and ACT) that typically are administered to much older students. This relationship was robust even when controlling for age, verbal performance, and reaction times in the approximate number task. These results suggest that the Approximate Number System is linked to symbolic math performance even at the top levels of math performance.

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.

Visual working memory capacity increases between ages 3 and 8 years, controlling for gains in attention, perception, and executive control.

Research in adults has aimed to characterize constraints on the capacity of Visual Working Memory (VWM), in part because of the system's broader impacts throughout cognition. However, less is known about how VWM develops in childhood. Existing work has reached conflicting conclusions as to whether VWM storage capacity increases after infancy, and if so, when and by how much. One challenge is that previous studies did not control for developmental changes in attention and executive processing, which also may undergo improvement. We investigated the development of VWM storage capacity in children from 3 to 8 years of age, and in adults, while controlling for developmental change in exogenous and endogenous attention and executive control. Our results reveal that, when controlling for improvements in these abilities, VWM storage capacity increases across development and approaches adult-like levels between ages 6 and 8 years. More generally, this work highlights the value of estimating working memory, attention, perception, and decision-making components together.

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.

Acquisition of the Cardinal Principle Coincides with Improvement in Approximate Number System Acuity in Preschoolers.

Human mathematical abilities comprise both learned, symbolic representations of number and unlearned, non-symbolic evolutionarily primitive cognitive systems for representing quantities. However, the mechanisms by which our symbolic (verbal) number system becomes integrated with the non-symbolic (non-verbal) representations of approximate magnitude (supported by the Approximate Number System, or ANS) are not well understood. To explore this connection, forty-six children participated in a 6-month longitudinal study assessing verbal number knowledge and non-verbal numerical acuity. Cross-sectional analyses revealed a strong relationship between verbal number knowledge and ANS acuity. Longitudinal analyses suggested that increases in ANS acuity were most strongly related to the acquisition of the cardinal principle, but not to other milestones of verbal number acquisition. These findings suggest that experience with culture and language is intimately linked to changes in the properties of a core cognitive system.