Hilary Barth

The construction of large number representations in adults

What is the nature of our mental representation of quantity? We find that human adults show no performance cost of comparing numerosities across vs. within visual and auditory stimulus sets, or across vs. within simultaneous and sequential sets. In addition, reaction time and performance in such tasks are determined by the ratio of the numerosities to be compared; absolute set size has no effect. These findings suggest that modality-specific stimulus properties undergo a non-iterative transformation into representations of quantity that are independent of the modality or format of the stimulus.

Non-symbolic arithmetic in adults and young children

Five experiments investigated whether adults and preschool children can perform simple arithmetic calculations on non-symbolic numerosities. Previous research has demonstrated that human adults, human infants, and non-human animals can process numerical quantities through approximate representations of their magnitudes. Here we consider whether these non-symbolic numerical representations might serve as a building block of uniquely human, learned mathematics. Both adults and children with no training in arithmetic successfully performed approximate arithmetic on large sets of elements. Success at these tasks did not depend on non-numerical continuous quantities, modality-specific quantity information, the adoption of alternative non-arithmetic strategies, or learned symbolic arithmetic knowledge. Abstract numerical quantity representations therefore are computationally functional and may provide a foundation for formal mathematics.

The Development of Numerical Estimation: Evidence against a Representational Shift.

How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations. Here we present a new theoretically motivated and empirically supported account of the development of numerical estimation, based on the idea that number-line estimation tasks entail judgments of proportion. We extend existing models of perceptual proportion judgment to the case of abstract numerical magnitude. Two experiments provide support for these models; three likely sources of developmental change in childrens estimation performance are identified and discussed. This work demonstrates that proportion-judgment models provide a unified account of estimation patterns that have previously been explained in terms of a developmental shift from logarithmic to linear representations of number.

rTMS over the intraparietal sulcus disrupts numerosity processing.

It has been widely argued that the intraparietal sulcus (IPS) is involved in tasks that evoke representations of numerical magnitude, among other cognitive functions. However, the causal role of this parietal region in processing symbolic and non-symbolic numerosity has not been established. The current study used repetitive Transcranial Magnetic Stimulation (rTMS) to the left and right IPS to investigate the effects of temporary deactivations of these regions on the capacity to represent symbolic (Arabic numbers) and non-symbolic (arrays of dots) numerosities. We found that comparisons of both symbolic and non-symbolic numerosities were impaired after rTMS to the left IPS but enhanced by rTMS to the right IPS. A signature effect of numerical distance was also found: greater impairment (or lesser facilitation) when comparing numerosities of similar magnitude. The reverse pattern of impairment and enhancement was found in a control task that required judging an analogue stimulus property (ellipse orientation) but no numerosity judgements. No rTMS effects for the numerosity tasks were found when stimulating an area adjacent but distinct from the IPS, the left and right angular gyrus. These data suggest that left IPS is critical for processing symbolic and non-symbolic numerosity; this processing may thus depend on common neural mechanisms, which are distinct from mechanisms supporting the processing of analogue stimulus properties.

Adults' number-line estimation strategies: evidence from eye movements.

Although the development of number-line estimation ability is well documented, little is known of the processes underlying successful estimators’ mappings of numerical information onto spatial representations during these tasks. We tracked adults’ eye movements during a number-line estimation task to investigate the processes underlying number-to-space translation, with three main results. First, eye movements were strongly related to the target number’s location, and early processing measures directly predicted later estimation performance. Second, fixations and estimates were influenced by the size of the first number presented, indicating that adults calibrate their estimates online. Third, adults’ number-line estimates demonstrated patterns of error consistent with the predictions of psychophysical models of proportion estimation, and eye movement data predicted the specific error patterns we observed. These results support proportion-based accounts of number-line estimation and suggest that adults’ translation of numerical information into spatial representations is a rapid, online process.

Children’s multiplicative transformations of discrete and continuous quantities

Recent studies have documented an evolutionarily primitive, early emerging cognitive system for the mental representation of numerical quantity (the analog magnitude system). Studies with nonhuman primates, human infants, and preschoolers have shown this system to support computations of numerical ordering, addition, and subtraction involving whole number concepts prior to arithmetic training. Here we report evidence that this system supports childrens predictions about the outcomes of halving and perhaps also doubling transformations. A total of 138 kindergartners and first graders were asked to reason about the quantity resulting from the doubling or halving of an initial numerosity (of a set of dots) or an initial length (of a bar). Controls for dot size, total dot area, and dot density ensured that children were responding to the number of dots in the arrays. Prior to formal instruction in symbolic multiplication, division, or rational number, halving (and perhaps doubling) computations appear to be deployed over discrete and possibly continuous quantities. The ability to apply simple multiplicative transformations to analog magnitude representations of quantity may form a part of the toolkit that children use to construct later concepts of rational number.

Motion-Based Mechanisms of Illusory Contour Synthesis

Neurophysiological studies and computational models of illusory contour formation have focused on contour orientation as the underlying determinant of illusory contour shape in both static and moving displays. Here, we report a class of motion-induced illusory contours that demonstrate the existence of novel mechanisms of illusory contour synthesis. In a series of experiments, we show that the velocity of contour terminations and the direction of motion of a partially occluded figure regulate the perceived shape and apparent movement of illusory contours formed from moving image sequences. These results demonstrate the existence of neural mechanisms that reconstruct occlusion relationships from both real and inferred image velocities, in contrast to the static geometric mechanisms that have been the focus of studies to date.

Minimal-group membership influences children's responses to novel experience with group members

Children, like adults, tend to prefer ingroup over outgroup individuals, but how this group bias affects childrens processing of information about social groups is not well understood. In this study, 5- and 6-year-old children were assigned to artificial groups. They observed instances of ingroup and outgroup members behaving in either a positive (egalitarian) or a negative (stingy) manner. Observations of positive ingroup and negative outgroup behaviors reliably reduced childrens liking of novel outgroup members, while observations of negative ingroup and positive outgroup behaviors had little effect on liking ratings. In addition, children successfully identified the more generous group only when the ingroup was egalitarian and the outgroup stingy. These data provide compelling evidence that children treat knowledge of and experiences with ingroups and outgroups differently, and thereby differently interpret identical observations of ingroup versus outgroup members.

Active (not passive) spatial imagery primes temporal judgements

Previous research has shown that primes that induce particular spatial perspectives can influence temporal judgements. However, most studies have used priming stimuli that involve both spatial and motor language and imagery. Here we ask whether the motor content of these stimuli plays an important role in their ability to serve as effective primes. A total of 198 adult participants made temporal judgements after exposure to spatial primes involving varying levels of imagined effort. Spatial primes involving imagined motor actions, but not those involving equivalent passive motions through space, successfully primed decisions about time. This suggests that motor content, rather than spatial content alone, contributes to the priming effects that arise when people make temporal judgements after exposure to particular spatial perspectives.

Non-Bayesian Contour Synthesis

Recent research has witnessed an explosive increase in models that treat percepts as optimal probabilistic inference. The ubiquity of partial camouflage and occlusion in natural scenes, and the demonstrated capacity of the visual system to synthesize coherent contours and surfaces from fragmented image data, has inspired numerous attempts to model visual interpolation processes as rational inference. Here, we report striking new forms of visual interpolation that generate highly improbable percepts. We present motion displays depicting simple occlusion sequences that elicit vivid percepts of illusory contours (ICs) in displays for which they play no necessary explanatory role. These ICs define a second, redundant occluding surface, even though all of the image data can be fully explained by an occluding surface that is clearly visible. The formation of ICs in these images therefore entails an extraordinarily improbable co-occurrence of two occluding surfaces that arise from the same local occlusion events. The perceived strength of the ICs depends on simple low-level image properties, which suggests that they emerge as the outputs of mechanisms that automatically synthesize contours from the pattern of occlusion and disocclusion of local contour segments. These percepts challenge attempts to model visual interpolation as a form of rational inference and suggest the need to consider a broader space of computational problems and/or implementation level constraints to understand their genesis.