Span and rate of apprehension in children and adults

Addition and subtraction by human infants

Human infants can discriminate between different small numbers of items, and can determine numerical equivalence across perceptual modalities. This may indicate the possession of true numerical concepts. Alternatively, purely perceptual discriminations may underlie these abilities. This debate addresses the nature of subitization, the ability to quantify small numbers of items without conscious counting. Subitization may involve the holistic recognition of canonical perceptual patterns that do not reveal ordinal relationships between the numbers, or may instead be an iterative or 'counting' process that specifies these numerical relationships. Here I show that 5-month-old infants can calculate the results of simple arithmetical operations on small numbers of items. This indicates that infants possess true numerical concepts, and suggests that humans are innately endowed with arithmetical abilities. It also suggests that subitization is a process that encodes ordinal information, not a pattern-recognition process yielding non-numerical percepts.

Varieties of numerical abilities

This paper provides a tutorial introduction to numerical cognition, with a review of essential findings and current points of debate. A tacit hypothesis in cognitive arithmetic is that numerical abilities derive from human linguistic competence. One aim of this special issue is to confront this hypothesis with current knowledge of number representations in animals, infants, normal and gifted adults, and brain-lesioned patients. First, the historical evolution of number notations is presented, together with the mental processes for calculating and transcoding from one notation to another. While these domains are well described by formal symbol-processing models, this paper argues that such is not the case for two other domains of numerical competence: quantification and approximation. The evidence for counting, subitizing and numerosity estimation in infants, children, adults and animals is critically examined. Data are also presented which suggest a specialization for processing approximate numerical quantities in animals and humans. A synthesis of these findings is proposed in the form of a triple-code model, which assumes that numbers are mentally manipulated in an arabic, verbal or analogical magnitude code depending on the requested mental operation. Only the analogical magnitude representation seems available to animals and preverbal infants.

Subitizing: magical numbers or mere superstition?

It is widely believed that humans are endowed with a specialized numerical process, called subitizing, which enables them to apprehend rapidly and accurately the numerosity of small sets of objects. A major part of the evidence for this process is a purported discontinuity in the mean response time (RT) versus numerosity curves at about 4 elements, when subjects enumerate up to 7 or more elements in a visual display. In this article, RT data collected in a speeded enumeration experiment are subjected to a variety of statistical analyses, including several tests on the RT distributions. None of these tests reveals a significant discontinuity as numerosity increases. The data do suggest a strong stochastic dominance in RT by display numerosity, indicating that the mental effort required to enumerate does increase with each additional element in the display, both within and beyond the putative subitizing range.

Is subitizing a unique numerical ability?

Two models that predict the relation between mean enumeration time and numerosity in a speeded enumeration experiment are tested. The first is a bilinear two-process model, and the second is a log-linear single-process model. Previously, support for the bilinear model has provided evidence for the existence of a unique numerical ability called "subitizing." Both models are shown to yield close approximations to the empirical data, but at the same time to consistently violate the robust shape of these data. Two fundamental discrepancies exist: (1) Enumeration of single-element displays is unpredictably fast, both in the data reported here and elsewhere, and (2) the response-time function for multiple elements is continuously convex upward, with a significant log-quadratic component. The findings support the contention that enumeration is a capacity-limited process, but not statistically reliable change in processing character, that is, from subitizing to some other process, is evident in enumeration of displays of up to six elements.

Children's understanding of counting

This study examines the abstractness of children's mental representation of counting, and their understanding that the last number word used in a count tells how many items there are (the cardinal word principle). In the first experiment, twenty-four 2- and 3-year-olds counted objects, actions, and sounds. Children counted objects best, but most showed some ability to generalize their counting to actions and sounds, suggesting that at a very young age, children begin to develop an abstract, generalizable mental representation of the counting routine. However, when asked "how many" following counting, only older children (mean age 3.6) gave the last number word used in the count a majority of the time, suggesting that the younger children did not understand the cardinal word principle. In the second experiment (the "give-a-number" task), the same children were asked to give a puppet one, two, three, five, and six items from a pile. The older children counted the items, showing a clear understanding of the cardinal word principle. The younger children succeeded only at giving one and sometimes two items, and never used counting to solve the task. A comparison of individual children's performance across the "how-many" and "give-a-number" tasks shows strong within-child consistency, indicating that children learn the cardinal word principle at roughly 3 1/2 years of age. In the third experiment, 18 2- and 3-year-olds were asked several times for one, two, three, five, and six items, to determine the largest numerosity at which each child could succeed consistently. Results indicate that children learn the meanings of smaller number words before larger ones within their counting range, up to the number three or four. They then learn the cardinal word principle at roughly 3 1/2 years of age, and perform a general induction over this knowledge to acquire the meanings of all the number words within their counting range.

Perception of numbers by human infants

Infants are capable of discriminating, representing, and remembering particular small numbers of items. A perceptual enumeration process called subitizing, present in 2-year-olds, probably underlies this capacity. This finding indicates that some number capacity is present before the onset of verbal counting, and it suggests that verbal counting may have precursors present during infancy.