Stella F. Lourenco

General Magnitude Representation in Human Infants

Behavioral demonstrations of reciprocal interactions among the dimensions of space, number, and time, along with evidence of shared neural mechanisms in posterior parietal cortex, are consistent with a common representational code for general magnitude information. Although much recent speculation has concerned the developmental origins of a system of general magnitude representation, direct evidence in preverbal infants is lacking. Here we show that 9-month-olds transfer associative learning across magnitude dimensions. For example, if shown that larger objects were black and had stripes and that smaller objects were white and had dots, infants expected the same color-pattern mapping to hold for numerosity (i.e., greater numerosity: black with stripes; smaller numerosity: white with dots) and duration (i.e., longer-lasting objects: black with stripes; shorter-lasting objects: white with dots). Cross-dimensional transfer occurred bidirectionally for all combinations of size, numerosity, and duration. These results provide support for the existence of an early-developing and prelinguistic general magnitude system, whereby representations of magnitude information are (at least partially) abstracted from the specific dimensions.

Socioeconomic Status Modifies the Sex Difference in Spatial Skill

We examined whether the male spatial advantage varies across children from different socioeconomic (SES) groups. In a longitudinal study, children were administered two spatial tasks requiring mental transformations and a syntax comprehension task in the fall and spring of second and third grades. Boys from middle-and high-SES backgrounds outperformed their female counterparts on both spatial tasks, whereas boys and girls from a low-SES group did not differ in their performance level on these tasks. As expected, no sex differences were found on the verbal comprehension task. Prior studies have generally been based on the assumption that the male spatial advantage reflects ability differences in the population as a whole. Our finding that the advantage is sensitive to variations in SES provides a challenge to this assumption, and has implications for a successful explanation of the sex-related difference in spatial skill.

Nonsymbolic number and cumulative area representations contribute shared and unique variance to symbolic math competence

Humans and nonhuman animals share the capacity to estimate, without counting, the number of objects in a set by relying on an approximate number system (ANS). Only humans, however, learn the concepts and operations of symbolic mathematics. Despite vast differences between these two systems of quantification, neural and behavioral findings suggest functional connections. Another line of research suggests that the ANS is part of a larger, more general system of magnitude representation. Reports of cognitive interactions and common neural coding for number and other magnitudes such as spatial extent led us to ask whether, and how, nonnumerical magnitude interfaces with mathematical competence. On two magnitude comparison tasks, college students estimated (without counting or explicit calculation) which of two arrays was greater in number or cumulative area. They also completed a battery of standardized math tests. Individual differences in both number and cumulative area precision (measured by accuracy on the magnitude comparison tasks) correlated with interindividual variability in math competence, particularly advanced arithmetic and geometry, even after accounting for general aspects of intelligence. Moreover, analyses revealed that whereas number precision contributed unique variance to advanced arithmetic, cumulative area precision contributed unique variance to geometry. Taken together, these results provide evidence for shared and unique contributions of nonsymbolic number and cumulative area representations to formally taught mathematics. More broadly, they suggest that uniquely human branches of mathematics interface with an evolutionarily primitive general magnitude system, which includes partially overlapping representations of numerical and nonnumerical magnitude.

The Approximate Number System and its Relation to Early Math Achievement: Evidence from the Preschool Years

Humans rely on two main systems of quantification; one is nonsymbolic and involves approximate number representations (known as the approximate number system or ANS), and the other is symbolic and allows for exact calculations of number. Despite the pervasiveness of the ANS across development, recent studies with adolescents and school-aged children point to individual differences in the precision of these representations that, importantly, have been shown to relate to symbolic math competence even after controlling for general aspects of intelligence. Such findings suggest that the ANS, which humans share with nonhuman animals, interfaces specifically with a uniquely human system of formal mathematics. Other findings, however, point to a less straightforward picture, leaving open questions about the nature and ontogenetic origins of the relation between these two systems. Testing children across the preschool period, we found that ANS precision correlated with early math achievement but, critically, that this relation was nonlinear. More specifically, the correlation between ANS precision and math competence was stronger for children with lower math scores than for children with higher math scores. Taken together, our findings suggest that early-developing connections between the ANS and mathematics may be fundamentally discontinuous. Possible mechanisms underlying such nonlinearity are discussed.

Space perception and body morphology: extent of near space scales with arm length

Numerous studies have found that the near space immediately surrounding the body is represented differently than more distant space. In a previous study, we found a gradual shift in attentional bias (on a line bisection task) between near and far space (Longo and Lourenco in Neuropsychologia 44:977–981, 2006). The present study concerns the possibility that arm length relates systematically to the rate at which this gradual shift between near and far space occurs. Participants bisected lines using a laser pointer at eight distances (within and beyond arm’s reach), and the rate of shift was estimated by the slope of the least-squares regression line. A negative correlation was found between the slopes and arm length; participants with longer arms showed a more gradual shift in bias with increasing distance than those with shorter arms. These results suggest that, while near space cannot be considered categorically as that within arm’s reach, there is a systematic relation between the extent (“size”) of near space and arm length. Arm length may constitute an intrinsic metric for the representation of near space.

Near space and its relation to claustrophobic fear

It is well established that the near space immediately surrounding the body (also known as peripersonal space) is represented differently than the space farther away. When bisecting horizontal lines, for example, neurologically-healthy adults show a slight leftward bias (known as pseudoneglect) in near space; this attentional bias, however, transitions rightward in far space. Recent research has used the rate at which this shift occurs to quantify the extent (i.e., size) of near space, showing consistent individual differences that relate to arm length. Here we examined whether the size of near space relates to individual differences in claustrophobic fear, as measured by reported anxiety of enclosed spaces and physically restrictive situations. Trait feelings of claustrophobic fear predicted the size of near space, even after accounting for the relation to arm length. Specifically, people with larger near spaces reported higher rates of claustrophobic fear than people with smaller near spaces. These results are consistent with a defensive function of near space representation and suggest that an over-projection of near space may play an important role in the etiology of claustrophobia.

Orienting numbers in mental space: Horizontal organization trumps vertical

While research on the spatial representation of number has provided substantial evidence for a horizontally oriented mental number line, recent studies suggest vertical organization as well. Directly comparing the relative strength of horizontal and vertical organization, however, we found no evidence of spontaneous vertical orientation (upward or downward), and horizontal trumped vertical when pitted against each other (Experiment 1). Only when numbers were conceptualized as magnitudes (as opposed to nonmagnitude ordinal sequences) did reliable vertical organization emerge, with upward orientation preferred (Experiment 2). Altogether, these findings suggest that horizontal representations predominate, and that vertical representations, when elicited, may be relatively inflexible. Implications for spatial organization beyond number, and its ontogenetic basis, are discussed.

Origins and Development of Generalized Magnitude Representation

Among the most fundamental of mental capacities is the ability to represent magnitude information such as physical size, numerosity, and duration. Accumulating evidence suggests that such cues are processed as part of a general magnitude system with shared more versus less representational structure. Here we review recent research with young children and preverbal infants suggesting that this system is operational from early in human life and may be far more general than currently believed. We present data suggesting that from early in development the representation of magnitude extends across modality (e.g., vision and audition) and beyond the “big three” dimensions of spatial extent, number, and time. We also speculate about particular properties of the general magnitude system, including the potentially special role of space in grounding magnitude information.

Common spatial organization of number and emotional expression: A mental magnitude line

Converging behavioral and neural evidence suggests that numerical representations are mentally organized in left-to-right orientation. Here we show that this format of spatial organization extends to emotional expression. In Experiment 1, right-side responses became increasingly faster as number (represented by Arabic numerals) or happiness (depicted in facial stimuli) increased, for judgments completely unrelated to magnitude. Additional experiments suggest that magnitude (i.e., more/less relations), not valence (i.e., positive/negative), underlies left-to-right orientation of emotional expression (Experiment 2), and that this orientation accommodates to the context-relevant emotion (e.g., happier faces are more rightward when judged on happiness, but more leftward when judged on angriness; Experiment 3). These findings show that people automatically extract magnitude from a variety of stimuli, representing such information in common left-to-right format, perhaps reflecting a mental magnitude line. We suggest that number is but one dimension in a hyper-general representational system uniting disparate dimensions of magnitude and likely subserved by common neural mechanisms in posterior parietal cortex.

The Representation of Geometric Cues in Infancy

There is evidence that, from an early age, humans are sensitive to spatial information such as simple landmarks and the size of objects. This study concerns the ability to represent a particular kind of spatial information, namely, the geometry of an enclosed layout-an ability present in older children, adults, and nonhuman animals (e.g., Cheng, 1986; Hermer & Spelke, 1996). Using a looking-time procedure, 4.5- to 6.5-month-olds were tested on whether they could distinguish among the corners of an isosceles triangle. On each trial, the target corner was marked by a red dot. The stimulus (triangle with dot) appeared from different orientations across trials, ensuring that only cues related to the triangle itself could be used to differentiate the corners. When orientations were highly variable, infants discriminated the unique corner (i.e., the corner with the smaller angle and two equal-length sides) from a nonunique corner; they could not discriminate between the two nonunique corners. With less variable orientations, however, infants did discriminate between the nonunique corners of the isosceles triangle. Implications for how infants represent geometric cues are discussed.