Catherine Sophian

Beyond competence: The significance of performance for conceptual development

Abstract Conceptual constraints must change with age if they are to account for childrens acquisition of kinds of knowledge that do not fall within the initial constraints. A bi-directional relation between competence and performance is therefore hypothesized, such that cognitive competences not only guide performance but also are shaped by it. This hypothesis offers a solution to the difficulties that current competence models have in accounting for developmental change. Goals are proposed as a potential source of changing constraints because they change with age, they shape what children do, and they also influence what children learn from what they do. These ideas are illustrated with examples drawn primarily from research on childrens quantitative concepts.

Perceptions of proportionality in young children: matching spatial ratios

Three experiments examined the ability of young children (4- and 5-year-olds) and adults to identify correspondences in spatial ratios. In all three experiments, children as young as 4 and 5 years of age made accurate spatial proportionality judgments. Moreover, their judgments were no less accurate when both response alternatives differed in spatial configuration from the sample than when all the configurations were alike, indicating that those judgments were based on relational information and not on the exact form of the stimuli. The findings, which suggest that the concept of ratio may have important foundations in the ways young children perceive relations within and between objects, have implications for theories of mathematical development and for methods of mathematics instruction.

How do people apprehend large numerosities

People discriminate remarkably well among large numerosities. These discriminations, however, need not entail numerical representation of the quantities being compared. This research evaluated the role of both non-numerical and numerical information in adult judgments of relative numerosity for large-numerosity spatial arrays. Results of Experiment 1 indicated that judgments of relative numerosity were affected by the amount of open space in the arrays being compared. Further, the accuracy of verbal estimates of the numerosities of the arrays made upon completion of the comparison task bore little relation to performance on that task. Experiment 2, however, showed that numerical estimates for individually presented arrays were affected in much the same way by open space within or around the edges of the array as were the comparative judgments examined in Experiment 1. The findings suggest that adults heuristically utilize non-numerical cues as well as numerical information in apprehending large numerosities.

When Three Is Less Than Two: Early Developments in Children's Understanding of Fractional Quantities.

Four experiments examined young childrens understanding of the inverse relation between the number of parts into which a quantity is to be divided and the size of each part. In Experiment 1 5-, 6-, and 7-year-old children tended to judge, incorrectly, that bigger shares would result from sharing with more, rather than fewer, recipients. In Experiment 2, 5-year-olds correctly recognized the inverse effect of additional recipients when the sharing was based on subtraction rather than on equal partitioning. In Experiment 3, a modification of the equal-sharing task from Experiment 1 designed to reduce cognitive complexity successfully elicited correct performance from 7-year-olds but not from 5-year-olds. However, 5-year-olds markedly improved when they were given a chance to compare the outcomes of sharing with different numbers of recipients. Experiment 4 corroborated and extended this evidence of learning.

Involuntary memory and the development of retrieval skills in young children

Abstract An incidental memory paradigm was used to study involuntary encoding processes and voluntary retrieval strategies in childrens memory. Preschool (mean age: 4 years, 4 months) and kindergarten (mean age: 5 years, 10 months) children sorted pictures according to their color or category membership, and then received either a recall test (Experiment 1) or a recognition test (Experiment 2). Better retention of category- than color-sorted items was observed for kindergarten children in free recall, preschool and kindergarten children in cued recall, and neither group in recognition. These results were interpreted in terms of the retrieval strategies used by children in each of the memory tasks. The importance of distinguishing between voluntary and involuntary memory processes, and between acquisition and retrieval, in studies of depth-of-processing was emphasized. Developmental differences in performance appear to derive primarily from the role of voluntary search strategies in retrieval, rather than from age differences in involuntary encoding processes.

Early developments in logical reasoning: Considering alternative possibilities

Abstract Three experiments showed that starting at about 4 years of age children take into account alternative possibilities in reasoning about the location of a hidden object. The children had to infer from event sequences they observed which of several locations might contain a hidden toy and which locations definitely could not. In Experiment 1, there were always two possible places (in an array of four), and 4- and 6-year-old children were quite successful in identifying the two possible locations. In Experiment 2, the number of possibilities varied from one to three (in an array of six), and 4- and 6-year-olds appropriately adjusted the number of locations they chose as the number of possibilities varied, indicating that they could determine how many possibilities were compatible with information they had. In addition, they chose the correct locations (which were always ones the experimenter had visited) over incorrect locations that the experimenter had also visited and over incorrect locations that she had not visited at all. Experiment 3 extended the research to 2 1 2 -, 3-, and 4-year-old children, using an analog of Sperlings partial report technique to evaluate knowledge of alternative possibilities from single searches. Even the 2 1 2 - year-olds showed some logical reasoning, but only the 4-year-olds gave evidence of considering more than one possibility.

Proportional reasoning in young children: The parts and the whole of it.

Evidence concerning intuitive foundations for fraction learning was obtained in a study of early developments in proportional reasoning. Children aged 5 to 7 years (kindergarten to 2nd grade) were given problems constructed so as to differentiate between reasoning based on the relations of a part to the whole versus reasoning based on relations between one part and another. The participants were able to use part-whole relations to compare proportions by 7 years of age. In addition, a developmental shift toward increasing reliance on part-whole reasoning was observed in childrens responses to conflict problems that pitted part-whole and part-part matches against each other.

Units of counting: Developmental changes

Abstract Insofar as counting is directed toward a definite quantification goal, only items that qualify as valid instances of what is being quantified should be included in the count. Thus, the choice of what to treat as a unit to be counted depends upon ones quantification goals. The present research examined developmental changes in the way children define units for counting. In the first experiment, children were shown arrays of toy animals and asked to count either the number of families or the number of individuals within a family. In the second and third experiments, the stimuli were objects that came apart into two pieces. Children were shown arrays composed of some intact objects and some objects that were separated into their parts, and they were asked either to count the number of wholes or the number of pieces in the entire array. Virtually all the counts children generated were based on some type of common unit, even if it was only defined by physical discreteness. However, marked age differences emerged in childrens adaptation of their counting units to what they were asked to count.

Does Recognition Memory Improve with Age

Abstract In order to test the hypothesis that recognition is a developmentally stable component of the memory system, age differences in recognition of faces were examined while controlling for nonmemory factors that might contribute to differences between the groups. Three groups of children (mean ages: 3 years, 4 months; 4 years, 9 months; and 6 years, 11 months) and a group of college students were tested on a recognition task and a similar matching task. The results indicated no change in recognition across the preschool years but an improvement from the later preschool period to the first grade. Further analyses indicated that this improvement was not due to changes in decision criteria or perceptual skills. These findings call into question the view that recognition is a developmentally invariant component of the memory system.

Learning about What Fits: Preschool Children's Reasoning about Effects of Object Size.

Preschool childrens understanding of the mathematical significance of unit size was examined through problems that involved making judgments about the number of larger objects versus the number of smaller objects that would fit in a container or bounded space. Childrens judgments about effects of object size were elicited both before and after the presentation of a series of demonstration trials. The accuracy of childrens judgments about the effects of object size improved significantly from pretest to posttest, and the degree of improvement was similar for 3-year-olds versus 4-year-olds and for Head Start children versus children who attended private preschools. The Head Start children, however, were much less likely than their peers attending private preschools to be able to articulate relevant quantitative features when asked to explain the outcomes that they observed during the training trials.