# Justin Halberda's research while affiliated with Johns Hopkins University and other places

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## Publications (120)

A sentence like every circle is blue might be understood in terms of individuals and their properties (e.g., for each thing that is a circle, it is blue) or in terms of a relation between groups (e.g., the blue things include the circles). Relatedly, theorists can specify the contents of universally quantified sentences in first-order or second-ord...

Exact arithmetic abilities require symbolic numerals, which constitute a precise representation of quantities, such as the Arabic digits. Numerical thinking, however, also engages an intuitive non-linguistic number sense, the Approximate Number System (ANS). The ANS allows us to discriminate quantities, approximate arithmetic transformations, and e...

What are the developmental foundations of logical thought? Here we find that 2.5-year-old toddlers (N=36) can reason using a Disjunctive Inference (i.e., A OR B, NOT A, THEREFORE B) across three contexts, which argues that domain-general logical reasoning may be in place from as early as the third year of life.

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 o...

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 b...

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 o...

Quantificational determiners have meanings that are "conservative" in the following sense: in sentences, repeating a determiner's internal argument within its external argument is logically insignificant. Using a verification task to probe which sets (or properties) of entities are represented when participants evaluate sentences, we test the predi...

We investigated whether training the Approximate Number System (ANS) would transfer to improved arithmetic performance in 7-8 year olds compared to a control group. All children participated in Pre- and Post-Training assessments of exact symbolic arithmetic (additions and subtractions) and approximate symbolic arithmetic abilities (a novel test). D...

Supplementary materials to: Ferres-Forga, N., & Halberda, J. (2020). Approximate number system discrimination training for 7-8 year olds improves approximate, but not exact, arithmetics, and only in children with low pre-training arithmetic scores. [Tests, instructions, and results]. Journal of Numerical Cognition, 6(3), 275-303. https://doi.org/10...

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 estimat...

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, leavin...

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 contin...

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...

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 te...

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...

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.

Research suggests that individual differences in math abilities correlate with approximate representations of quantity that are supported by a primitive Approximate Number System (ANS). However, relatively little research has addressed the direction of this association in early childhood. Here we examined the development of the ANS and math ability...

Language is a sub-component of human cognition. One important, though often unattained goal for both cognitive scientists and linguists is to explicate how the meanings of words and sentences relate to the more general, non-linguistic, cognitive systems that are used to evaluate whether sentences are true or false. In the present paper, we explore...

We investigated whether the surface area and convex hull of a dot stimulus could explain responses to numerosity in the same stimuli. We found that people responded differently when they attended to the number of dots than if they were attending to the size or spacing of the same stimuli.

Playing action video games has previously been linked to improvements in attentional control, the ability to choose what to attend and what to ignore that relies on a frontoparietal network of the brain. Here we asked whether action video game training would impact a range of mathematical abilities that rely on similar brain regions. Twenty-four ad...

Crosscutting concepts such as scale, proportion, and quantity are recognised by U.S. science standards as a potential vehicle for students to integrate their scientific and mathematical knowledge; yet, U.S. students and adults trail their international peers in scale and measurement estimation. Culturally based knowledge of scale such as measuremen...

Abstract 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 b...

In unequal societies the effectiveness of formal education depends on the socioeconomic status (SES) of students. Studies have shown that poverty affects the development of the brain in ways that might compromise future learning, thus increasing the differences between groups with different SES. Interest is growing in the development of tools that...

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 correlationa...

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...

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 i...

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...

Full Data Set With Scores on ANS and Verbal Number Knowledge Measures.
(XLSX)

What is the relationship between our intuitive sense of number (e.g., when estimating how many marbles are in a jar), and our intuitive sense of other quantities, including time (e.g., when estimating how long it has been since we last ate breakfast)? Recent work in cognitive, developmental, comparative psychology, and computational neuroscience ha...

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 prop...

To what extent are storage and manipulation abilities in visual working memory (VWM) in/dependent? We answer this question by investigating whether factors known to influence VWM storage (memory load, object complexity, precision of representations) produce similar vs. different costs in VWM manipulation. To this end, we developed a novel task that...

Humans can represent number either exactly - using their knowledge of exact numbers as supported by language, or approximately - using their approximate number system (ANS). Adults can map between these two systems - they can both translate from an approximate sense of the number of items in a brief visual display to a discrete number word estimate...

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, spa...

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 ca...

The Approximate Number System (ANS) is a portion of your cognition that is active across your entire life, and is in the business of giving you a rapid and intuitive sense for numbers and their relations (e.g., how many blue versus yellow dots are on a screen, or how many voices you hear speaking at the dinner table). In this chapter, we discuss ap...

Based on the large number of objects in any visual scene, and our ever-changing goals, it is ecologically natural for humans to dynamically adjust which information and items are stored in Visual Working Memory (VWM) across views. This means that effectively using VWM in context requires observers to e.g., load and purge items from VWM, switch atte...

Basic intellectual abilities of quantity and numerosity estimation have been detected across animal species. Such abilities are referred to as ‘number sense’. For human species, individual differences in number sense are detectable early in life, persist in later development, and relate to general intelligence. The origins of these individual diffe...

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...

Measuring individual differences in children's emerging language abilities is important to researchers and clinicians alike. The 2 most widely used methods for assessing children's vocabulary both have limitations: Experimenter-administered tests are time-consuming and expensive, and parent questionnaires have only been designed for children up to...

Our internal representation of a complex visual scene relies on the dynamic processing of information in visual working memory (VWM). Though traditional methods have focused on storage limitations, here we move beyond these issues to explore cognitive abilities for dynamically manipulating information in VWM. Along the way, we discover independent...

When presented with visual scenes containing multiple items, humans can rapidly organize elements into groups and can estimate the number of items within groups. Here we systematically examine how humans define groups of items and how grouping influences numerical estimation of items. Through behavioral experiments and modeling work, we find that h...

Objects and ensembles have independently been the focus of much investigation, but it remains to be determined how they are related. We have been investigating the possibility that ensembles function as single units for visual attention and memory – similar to single objects. Here, using one-shot change detection, we find that multiple ensembles ca...

We explore individual differences in the precision of the Approximate Number System (ANS), and the trainability of this key process for apprehending the number of objects in our environment. This system, anchored in the dorsal stream, impacts many of our decisions – from trying to determine the fastest cashier line at the grocery to high-order cogn...

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.
This population-based follow-up study included 65 children born before 27 gestational weeks (35 males, 30 females; mean gestational age 25.4wks [SD 1.1wk]; mean birthweight 7...

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...

Humans share with other animals a system for thinking about numbers in an imprecise and intuitive way. The Approximate Number System (ANS) that underlies this thinking is present throughout the lifespan, is entirely nonverbal, and supports basic numerical computations like comparing, adding, and subtracting quantities. Humans, unlike other animals,...

Does making an inference lead to better learning than being instructed directly? Two experiments evaluated preschoolers' ability to learn new words, comparing their memory for words learned via inference or instruction. On Inference trials, one familiar and one novel object was presented and children were asked to Point at the [object name (i.e., p...

Increasing numbers of studies have explored human observers' ability to rapidly extract statistical descriptions from collections of similar items (e.g., the average size and orientation of a group of tilted Gabor patches). Determining whether these descriptions are generated by mechanisms that are independent from object-based sampling procedures...

Perceptual decisions are often affected not only by the evidence gathered during a trial but also by the history of preceding trials. This effect-termed perceptual hysteresis-provides evidence for how perceptual information is represented and how it is used. The present research focuses on how the difficulty of preceding trials affects subsequent o...

Many educated adults possess exact mathematical abilities in addition to an approximate, intuitive sense of number, often referred to as the Approximate Number System (ANS). Here we investigate the link between ANS precision and mathematics performance in adults by testing participants on an ANS-precision test and collecting their scores on the Sch...

From very early in life, humans can approximate the number and surface area of objects in a scene. The ability to discriminate between 2 approximate quantities, whether number or area, critically depends on the ratio between the quantities, with the most difficult ratio that a participant can reliably discriminate known as the Weber fraction. While...

Representation of visual features is subject to internal noise, which is sometimes constant (e.g., color), anisotropic (e.g., orientation) or scalar (e.g., luminance) within a dimension. A common metric of internal noise has been the coefficient of variation (CV=SD/mean), which is equivalent to a Weber fraction. For many features, SD varies linearl...

The enumeration of a small number of objects has been repeatedly shown to consist of two distinct stages dependent on the number of objects. For 1-3 objects, enumeration is fast and accurate. For more than 3 objects, enumeration is slow and less accurate. The former is coined subitizing and the later is counting. These results have led many authors...