The Representation of Geometric Cues in Infancy

135

111

Kant argued that Euclidean geometry is synthesized on the basis of an a priori intuition of space. This proposal inspired much behavioral research probing whether spatial navigation in humans and animals conforms to the predictions of Euclidean geometry. However, Euclidean geometry also includes concepts that transcend the perceptible, such as objects that are infinitely small or infinitely large, or statements of necessity and impossibility. We tested the hypothesis that certain aspects of nonperceptible Euclidian geometry map onto intuitions of space that are present in all humans, even in the absence of formal mathematical education. Our tests probed intuitions of points, lines, and surfaces in participants from an indigene group in the Amazon, the Mundurucu, as well as adults and age-matched children controls from the United States and France and younger US children without education in geometry. The responses of Mundurucu adults and children converged with that of mathematically educated adults and children and revealed an intuitive understanding of essential properties of Euclidean geometry. For instance, on a surface described to them as perfectly planar, the Mundurucu's estimations of the internal angles of triangles added up to ∼180 degrees, and when asked explicitly, they stated that there exists one single parallel line to any given line through a given point. These intuitions were also partially in place in the group of younger US participants. We conclude that, during childhood, humans develop geometrical intuitions that spontaneously accord with the principles of Euclidean geometry, even in the absence of training in mathematics. mathematical cognition | spatial cognition | culture Geometry . . . does not deal with natural solids, but with ideal, absolutely invariable ones. These ideal bodies are complete fabrications of our mind, and experience only provides an occasion that incites us to draw it out.Henri Poincaré, L'espace et la géométrie (1)

Section: Discussionmentioning

confidence: 51%

Flexible intuitions of Euclidean geometry in an Amazonian indigene group

Izard

Pica

Spelke

et al. 2011

135

111

“…Cross-cultural research with indigenous populations and studies using visual attention paradigms with infants suggest that they are also among the most psychologically intuitive. Knowledge of core geometric concepts (e.g., points and lines) and the ability to perform basic (nonsymbolic) arithmetic computations (e.g., addition and subtraction) have been documented in cultures without access to formal math instruction (2, 7) and even in preverbal infants (57,58). Such intuitions, it is believed, rest on perceptual properties of the visual system and universal spatial experiences.…”

Section: Discussionmentioning

confidence: 99%

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

Lourenco

Bonny

Fernandez

et al. 2012

Self Cite

184

163

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.analog magnitude | Weber's law | estimation | nonsymbolic magnitude precision | mathematical cognition H ow do humans come to understand mathematics? A common view is that the mental capacity for formal mathwhich includes access to symbolic notations of number, knowledge of quantitative concepts, and the implementation of computational operations-builds on a set of core abilities such as an intuitive, nonverbal sense of numerosity (1-5). Also known as the approximate number system (ANS), this nonsymbolic sense of numerical magnitude is shared with nonhuman animals (6) and is widespread across cultures (7). Unlike the acquisition of symbolic number (e.g., Arabic digits) and formal math concepts, which are learned via explicit instruction and allow for exact quantification, the ANS may be innate (8) and is characteristically "noisy," with variance increasing linearly as a function of the absolute numerical value (9). This imprecision can be modeled as overlapping Gaussian distributions along an internal continuum (10) and is captured by Weber's law, which holds that subjective differences in intensity are proportional to the objective ratios between values. When people compare numerical values under conditions that prevent counting or that do not...

“…Children's sensitivity to distance and directional relations in a navigation task predicted their use of a map to find targets in a 3D array with surfaces at distinct distances; their sensitivity to properties of object shapes in a form analysis task predicted their use of the same map to find targets in a 3D array with corners of distinct angles. This pattern of findings provides evidence that in their untutored interpretations of symbolic maps, children flexibly recruit the core geometric representations that emerge in infancy (10,11,30,31), are shared by other animals (2,7,32,33), and are used by children and adults throughout their lives.…”

mentioning

confidence: 84%

Core foundations of abstract geometry

Dillon

Huang

Spelke

2013

103

Human adults from diverse cultures share intuitions about the points, lines, and figures of Euclidean geometry. Do children develop these intuitions by drawing on phylogenetically ancient and developmentally precocious geometric representations that guide their navigation and their analysis of object shape? In what way might these early-arising representations support laterdeveloping Euclidean intuitions? To approach these questions, we investigated the relations among young children's use of geometry in tasks assessing: navigation; visual form analysis; and the interpretation of symbolic, purely geometric maps. Children's navigation depended on the distance and directional relations of the surface layout and predicted their use of a symbolic map with targets designated by surface distances. In contrast, children's analysis of visual forms depended on the size-invariant shape relations of objects and predicted their use of the same map but with targets designated by corner angles. Even though the two map tasks used identical instructions and map displays, children's performance on these tasks showed no evidence of integrated representations of distance and angle. Instead, young children flexibly recruited geometric representations of either navigable layouts or objects to interpret the same spatial symbols. These findings reveal a link between the early-arising geometric representations that humans share with diverse animals and the flexible geometric intuitions that give rise to human knowledge at its highest reaches. Although young children do not appear to integrate core geometric representations, children's use of the abstract geometry in spatial symbols such as maps may provide the earliest clues to the later construction of Euclidean geometry.spatial cognition | mathematical cognition | map reading A bstract concepts of formal geometry underlie a wide range of human achievements, but their source has been debated for millennia (1). Human abilities to navigate the environment and to recognize objects develop early and are shared across diverse animal species. In recent years, intensive study at levels from neurons to cognition (2-5) has illuminated the geometric information guiding these abilities in animals from insects to vertebrates (6-8) and in humans from infants to adults (9-14). When navigating, humans and animals represent their position by encoding the distances and directions of extended surfaces in the terrain rather than the angles at which surfaces meet (15, 16). In contrast, humans and animals represent objects by encoding the angles and relative lengths defining 3D part structures or 2D shapes rather than their absolute sizes or the directional relations that distinguish a form from its mirror image (17, 18). Despite the pervasiveness and power of these core geometric representations, neither in isolation is adequate to support abstract geometric intuitions, which require an integrated representation of distance and angle (13,19,20). Still, these two sets of core representations together may ...