Rochel Gelman

Preverbal and verbal counting and computation

We describe the preverbal system of counting and arithmetic reasoning revealed by experiments on numerical representations in animals. In this system, numerosities are represented by magnitudes, which are rapidly but inaccurately generated by the Meck and Church (1983) preverbal counting mechanism. We suggest the following. (1) The preverbal counting mechanism is the source of the implicit principles that guide the acquisition of verbal counting. (2) The preverbal system of arithmetic computation provides the framework for the assimilation of the verbal system. (3) Learning to count involves, in part, learning a mapping from the preverbal numerical magnitudes to the verbal and written number symbols and the inverse mappings from these symbols to the preverbal magnitudes. (4) Subitizing is the use of the preverbal counting process and the mapping from the resulting magnitudes to number words in order to generate rapidly the number words for small numerosities. (5) The retrieval of the number facts, which plays a central role in verbal computation, is mediated via the inverse mappings from verbal and written numbers to the preverbal magnitudes and the use of these magnitudes to find the appropriate cells in tabular arrangements of the answers. (6) This model of the fact retrieval process accounts for the salient features of the reaction time differences and error patterns revealed by experiments on mental arithmetic. (7) The application of verbal and written computational algorithms goes on in parallel with, and is to some extent guided by, preverbal computations, both in the child and in the adult.

First principles organize attention to and learning about relevant data: Number and the animate-inanimate distinction as examples

Early cognitive development benefits from nonilnguistic representations of skeietai sets of domain-specific principles and complementary domain-relevant doto obstroction processes. The principles outline the domain, identify relevant inputs, and structure coherently what is learned. Knowledge acquisition within the domoin is a faint function of such domain-specific principles and domain-general learning mechanisms. Two examples of early learning illustrate this. Skeietol preverboi counting principles help children sort different linguistic strings into those that function OS the conventional count-word OS opposed to labels for obfects in the child’s linguistic community. Skeletal causal principles, working with complementary perceptual processes that abstract information obout biological and nonbiological conditions and patterns of movement, leod to the rapid ocquisition of knowledge about the animate-inanimate dlstinction. By 3 years of age children con say whether photographs of unfamiliar nonmammoiion animals, mommois, statues, and wheeled obfectr portray objects capable or incopabie of self-generated motion. They also generate answers to questions about the insides of animate items more reodiiy than ones about the insides of inanimate items. Although these children already are ortlcuiote about motters relevant to a theory of action, their ilmited knowledge of growth iiiustrotes that early skeletal principies do not rule out the need to acquire new principles, in thls case ones that underlie a biological account of animacy (Carey, 1985).

Nonverbal Counting in Humans: The Psychophysics of Number Representation

In a nonverbal counting task derived from the animal literature, adult human subjects repeatedly attempted to produce target numbers of key presses at rates that made vocal or subvocal counting difficult or impossible. In a second task, they estimated the number of flashes in a rapid, randomly timed sequence. Congruent with the animal data, mean estimates in both tasks were proportional to target values, as was the variability in the estimates. Converging evidence makes it unlikely that subjects used verbal counting or time durations to perform these tasks. The results support the hypothesis that adult humans share with nonverbal animals a system for representing number by magnitudes that have scalar variability (a constant coefficient of variation). The mapping of numerical symbols to mental magnitudes provides a formal model of the underlying nonverbal meaning of the symbols (a model of numerical semantics).

Numerical abstraction by human infants

Across several experiments, 6- to 8-month-old human infants were found to detect numerical correspondences between sets of entities presented in different sensory modalities and bearing no natural relation to one another. At the basis of this ability, we argue, is a sensitivity to numerosity, an abstract property of collections of objects and events. Our findings provide evidence that the emergence of the earliest numerical abilities does not depend upon the development of language or complex actions, or upon cultural experience with number.

Variability signatures distinguish verbal from nonverbal counting for both large and small numbers

Humans appear to share with animals a nonverbal counting process. In a nonverbal counting condition, subjects pressed a key a numeral-specified number of times, while saying “the” at every press. The mean number of presses increased as a power function of the target number, with a constant coefficient of variation (c.v.), both within and beyond the proposed subitizing range (1–4 or 5), suggesting small numbers are represented on the same continuum as larger numbers and subject to the same noise process (scalar variability). By contrast, when subjects counted their presses out loud as fast as they could, the c.v. decreased as the inverse square root of the target value (binomial variability instead of scalar variability). The unexpected power-law relation between target value and mean number of presses in nonverbal counting suggests a new hypothesis about the development of the function relating number symbols to mental magnitudes.

Conservation Acquisition: A Problem of Learning to Attend to Relevant Attributes.

Abstract Five-year-old children who failed on conservation tests of length, number, mass, and liquid amount were given discrimination learning set (LS) training on length and number tasks. Posttests of conservation showed near perfect specific (length and number), and approximately 60% nonspecific (mass and liquid amount) transfer of training. This effect was durable as measured 2–3 weeks later. Analyses of LS learning results and the effects of other training conditions support the hypothesis that young children fail to conserve because of inattention to relevant quantitative relationships and attention to irrelevant features in classical conservation tests.

Preschoolers' counting: Principles before skill

Three- to 5-year-old children participated in one of 4 counting experiments. On the assumption that performance demands can mask the young childs implicit knowledge of the counting principles, 3 separate experiments assessed a childs ability to detect errors in a puppets application of the one-one, stable-order and cardinal count principles. In a fourth experiment children counted in different conditions designed to vary performance demands. Since children in the errror-detection experiments did not have to do the counting, we predicted excellent performance even on set sizes beyond the range a young child counts accurately. That they did well on these experiments supports the view that errors in counting—at least for set sizes up to 20—reflect performance demands and not the absence of implicit knowledge of the counting principles. In the final experiment, where children did the counting themselves, set size did affect their success. So did some variations in conditions, the most difficult of which was the one where children had to count 3-dimensional objects which were under a plexiglass cover. We expected that this condition would interfere with the childs tendency to point and touch objects in order to keep separate items which have been counted from those which have not been counted.

Conceptual competence and children's counting☆☆☆

Abstract A framework is presented for characterizing competence for cognitive tasks, with a detailed hypothesis about competence for counting by typical 5-year-old children. It is proposed that competence has three main components that are called conceptual, procedural, and utilizational competence. Conceptual competence, which is discussed in greatest detail in this article, is the implicit understanding of general principles of the domain. Procedural competence is understanding of general principles of action and takes the form of planning heuristics. Utilizational competence is understanding of relations between features of a task setting and requirements of performance. A characterization of conceptual competence for counting is presented, in the form of action schemata that constitute understanding of counting principles such as cardinality, one-to-one correspondence, and order. This hypothesis about competence is connected explicitly to a detailed analysis of performance in counting tasks. The connection is provided by derivations of planning nets for procedures that are included in process models that simulate childrens performance.

Language in the two-year old ☆

Abstract Two stages in the vocabulary development of two-year-olds are reported. In the earlier Receptive stage, the child says many fewer nouns than he understands and says no verbs at all although he understands many. The child then begins to close the comprehension/production gap, entering a Productive stage in which he says virtually all the nouns he understands plus his first verbs. Frequency and length of word combinations correlate with these vocabulary stages.

Logical Capacity of Very Young Children: Number Invariance Rules.

GELMAN, ROCHEL. Logical Capacity of Very Young Children: Number Invariance Rules. CHILD DEVELOPMENT, 1972, 43, 75-90. Children 3-6 years of age, when given an identification task where number was redundant to length or density, solved the task on the basis of number. Surreptitious subtraction or addition elicited strong surprise as well as search behavior whereas displacements did not. Children who noticed the change in number or length and density gave unambiguous explanations of the nature of the intervening operations and were able to indicate how to reverse the effect. These findings are taken to show young children can treat small numbers as invariant. The results are discussed in terms of why children of the same age fail to conserve number in the standard conservation task and how complex number concepts might develop.