Development of Sensitivity to Geometry in Visual Forms

Geometric form perception has been extensively studied in human children, but it has not been systematically characterized from the perspective of formal geometry. Here, we present the findings of three experiments that use a deviant detection task to test children’s and adults’ sensitivity to geometric invariants in a variety of visual displays. Children as young as 4 years of age analyzed shapes by detecting relationships of distance and angle but not by detecting the relationships that distinguish an object from its mirror image (hereafter, sense). Patterns of visual form analysis showed high invariance over development: the properties that were least detectable by children also posed the greatest difficulty for adults. In general, sensitivity to all tested properties improved with age, with an asymptote at about 12 years, before the onset of instruction in formal geometry. When presented with a carefully controlled set of forms that varied exclusively in length, angle or sense, children were found to develop sensitivity to these properties at different rates, responding first to length, then to angle, and last to sense. Between 8 and 10 years of age, moreover, children began to confer a privileged status to the relation of perpendicularity. Geometric competence therefore appears to emerge as an interplay between developmentally invariant, core intuitions and later acquired distinctions.Psycholog

Toward Exact Number: Young Children Use One-to-One Correspondence to Measure Set Identity but Not Numerical Equality

Exact integer concepts are fundamental to a wide array of human activities, but their origins are obscure. Some have proposed that children are endowed with a system of natural number concepts, whereas others have argued that children construct these concepts by mastering verbal counting or other numeric symbols. This debate remains unresolved, because it is difficult to test children’s mastery of the logic of integer concepts without using symbols to enumerate large sets, and the symbols themselves could be a source of difficulty for children. Here, we introduce a new method, focusing on large quantities and avoiding the use of words or other symbols for numbers, to study children’s understanding of an essential property underlying integer concepts: the relation of exact numerical equality. Children aged 32–36 months, who possessed no symbols for exact numbers beyond 4, were given one-to-one correspondence cues to help them track a set of puppets, and their enumeration of the set was assessed by a non-verbal manual search task. Children used one-to-one correspondence relations to reconstruct exact quantities in sets of 5 or 6 objects, as long as the elements forming the sets remained the same individuals. In contrast, they failed to track exact quantities when one element was added, removed, or substituted for another. These results suggest an alternative to both nativist and symbol-based constructivist theories of the development of natural number concepts: Before learning symbols for exact numbers, children have a partial understanding of the properties of exact numbers.Psycholog

Distinct Cerebral Pathways for Object Identity and Number in Human Infants

All humans, regardless of their culture and education, possess an intuitive understanding of number. Behavioural evidence suggests that numerical competence may be present early on in infancy. Here, we present brain-imaging evidence for distinct cerebral coding of number and object identity in 3-mo-old infants. We compared the visual event-related potentials evoked by unforeseen changes either in the identity of objects forming a set, or in the cardinal of this set. In adults and 4-y-old children, number sense relies on a dorsal system of bilateral intraparietal areas, different from the ventral occipitotemporal system sensitive to object identity. Scalp voltage topographies and cortical source modelling revealed a similar distinction in 3-mo-olds, with changes in object identity activating ventral temporal areas, whereas changes in number involved an additional right parietoprefrontal network. These results underscore the developmental continuity of number sense by pointing to early functional biases in brain organization that may channel subsequent learning to restricted brain areas

Quels sont les liens entre arithmétique et langage ? Une étude en Amazonie

http://www.editionsdelherne.com/Is calculation possible without language ? Or is the human ability for arithmetic dependent on the language faculty ? To clarify the relation between language and arihmetic, we studied numerical cognition in speakers of Munduruku, an Amazonian langauge with a very lexicon of number words. This state of affaits we claim can be understood in a model wher the distinction between performance and competence is at work

Exact and Approximate Arithmetic in an Amazonian Indigene Group

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic

Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education

Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures

The mapping of numbers onto space is fundamental to measurement and to mathematics. Is this mapping a cultural invention or a universal intuition shared by all humans regardless of culture and education? We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.Psycholog

Response to Comment on “Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures

The performance of the Mundurucu on the number-space task may exemplify a general competence for drawing analogies between space and other linear dimensions, but Mundurucu participants spontaneously chose number when other dimensions were available. Response placement may not reflect the subjective scale for numbers, but Cantlon et al.’s proposal of a linear scale with scalar variability requires questionable additional hypotheses.Psycholog

Exact Equality and Successor Function: Two Key Concepts on the Path Towards Understanding Exact Numbers

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers, as prerequisites to the concept of exact numbers: the fact that all numbers can be generated by a successor function, and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Mundurucu (an Amazonian language), and young western children (3-4 years old) understand these fundamental properties of numbers.Psycholog

Visual foundations of Euclidean Geometry

Geometry defines entities that can be physically realized in space, and our knowledge of abstract geometry may therefore stem from our representations of the physical world. Here, we focus on Euclidean geometry, the geometry historically regarded as “natural”. We examine whether humans possess representations describing visual forms in the same way as Euclidean geometry – i.e., in terms of their shape and size. One hundred and twelve participants from the U.S. (age 3–34 years), and 25 participants from the Amazon (age 5–67 years) were asked to locate geometric deviants in panels of 6 forms of variable orientation. Participants of all ages and from both cultures detected deviant forms defined in terms of shape or size, while only U.S. adults drew distinctions between mirror images (i.e. forms differing in “sense”). Moreover, irrelevant variations of sense did not disrupt the detection of a shape or size deviant, while irrelevant variations of shape or size did. At all ages and in both cultures, participants thus retained the same properties as Euclidean geometry in their analysis of visual forms, even in the absence of formal instruction in geometry. These findings show that representations of planar visual forms provide core intuitions on which humans’ knowledge in Euclidean geometry could possibly be grounded