Robert S. Siegler

Developing conceptual understanding and procedural skill in mathematics: An iterative process

The authors propose that conceptual and procedural knowledge develop in an iterative fashion and that improved problem representation is 1 mechanism underlying the relations between them. Two experiments were conducted with 5th- and 6th-grade students learning about decimal fractions. In Experiment 1, childrens initial conceptual knowledge predicted gains in procedural knowledge, and gains in procedural knowledge predicted improvements in conceptual knowledge. Correct problem representations mediated the relation between initial conceptual knowledge and improved procedural knowledge. In Experiment 2, amount of support for correct problem representation was experimentally manipulated, and the manipulations led to gains in procedural knowledge. Thus, conceptual and procedural knowledge develop iteratively, and improved problem representation is 1 mechanism in this process.

Three aspects of cognitive development

Abstract An attempt was made to characterize and explain developmental differences in childrens thinking, specifically in their understanding of balance scale problems. Such differences were sought in three domains: existing knowledge about the problems, ability to acquire new information about them, and process-level differences underlying developmental changes in the first two areas. In Experiment 1, four models of rules that might govern childrens performance on balance scale problems were proposed. The rules proved to accurately describe individual performance and also to accurately predict developmental trends on different types of balance scale problems. Experiment 2 examined responsiveness to experience; it was found that older and younger children, equated for initial performance on balance scale problems, derived different benefits from identical experience. Experiment 3 examined a potential cause of this discrepancy, that younger children might be less able than older ones to benefit from experience because their encoding of stimuli was less adequate. Independent assessment procedures revealed that the predicted differences in older and younger childrens encoding were present; it was also found that these differences were not artifactual and that reducing them also reduced the previously observed differences in responsiveness to experience. It was concluded, therefore, that the encoding hypothesis explained a large part of the developmental difference in ability to acquire new information.

Developmental Sequences within and between Concepts.

SIEGLER, ROBERT S. Developmental Sequences Within and Between Concepts. With Commentary by SIDNEY STRAUSS AND IRIS LEVIN; with Reply by the author. Monographs of the Society for Research in Child Development, 1981, 46(2, Serial No. 189). The purpose of this Monograph is to describe and illustrate a new research strategy for studying developmental sequences, the rule-assessment approach. The approach begins with a distinction between two types of developmental sequences: within-concept sequences, involving the series of knowledge states leading to mastery of particular concepts; and between-concept sequences, involving the order in which different concepts are understood. In its assumptions concerning within-concept sequences the approach resembles traditional stage models, but in its assumptions concerning between-concept sequences it differs from them. Specifically, like the stage approaches, it assumes that children progress through a series of discrete knowledge states on their way to full understanding of any particular concept, but, unlike them, it does not assume great consistency of reasoning across different concepts. The rule-assessment strategy also differs in its reliance on patterns of nonverbal responses as the primary index of childrens knowledge. Four experiments were performed to illustrate the utility of the ruleassessment approach for studying developmental sequences across a variety of concepts and over a wide range of ages. In experiments 1 and 2, it was applied to three formal operational proportionality tasks: the balance scale, projection of shadows, and probability tasks. In experiments 3 and 4, it was applied to three concrete operational conservation tasks: conservation of liquid quantity, solid quantity, and number. On all six tasks, the approach was successful in revealing the contents of childrens knowledge at different points in their development. On four of the six, the progression of knowledge states proved to be quite different from the traditional Piagetian accounts. The rule-assessment approach also allowed comparisons of the structure of childrens knowledge across the six tasks. Finally, the data obtained in experiments 3 and 4 were sufficiently different from any collected previously to suggest a new theory of conservation acquisition. This content downloaded from 171.64.40.52 on Sat, 18 May 2013 17:05:02 PM All use subject to JSTOR Terms and Conditions

Developmental and individual differences in pure numerical estimation.

The authors examined developmental and individual differences in pure numerical estimation, the type of estimation that depends solely on knowledge of numbers. Children between kindergarten and 4th grade were asked to solve 4 types of numerical estimation problems: computational, numerosity, measurement, and number line. In Experiment 1, kindergartners and 1st, 2nd, and 3rd graders were presented problems involving the numbers 0-100; in Experiment 2, 2nd and 4th graders were presented problems involving the numbers 0-1,000. Parallel developmental trends, involving increasing reliance on linear representations of numbers and decreasing reliance on logarithmic ones, emerged across different types of estimation. Consistent individual differences across tasks were also apparent, and all types of estimation skill were positively related to math achievement test scores. Implications for understanding of mathematics learning in general are discussed.

An Integrated Theory of Whole Number and Fractions Development.

This article proposes an integrated theory of acquisition of knowledge about whole numbers and fractions. Although whole numbers and fractions differ in many ways that influence their development, an important commonality is the centrality of knowledge of numerical magnitudes in overall understanding. The present findings with 11- and 13-year-olds indicate that, as with whole numbers, accuracy of fraction magnitude representations is closely related to both fractions arithmetic proficiency and overall mathematics achievement test scores, that fraction magnitude representations account for substantial variance in mathematics achievement test scores beyond that explained by fraction arithmetic proficiency, and that developing effective strategies plays a key role in improved knowledge of fractions. Theoretical and instructional implications are discussed.

Early Predictors of High School Mathematics Achievement

Identifying the types of mathematics content knowledge that are most predictive of students’ long-term learning is essential for improving both theories of mathematical development and mathematics education. To identify these types of knowledge, we examined long-term predictors of high school students’ knowledge of algebra and overall mathematics achievement. Analyses of large, nationally representative, longitudinal data sets from the United States and the United Kingdom revealed that elementary school students’ knowledge of fractions and of division uniquely predicts those students’ knowledge of algebra and overall mathematics achievement in high school, 5 or 6 years later, even after statistically controlling for other types of mathematical knowledge, general intellectual ability, working memory, and family income and education. Implications of these findings for understanding and improving mathematics learning are discussed.

Cognitive Variability: A Key to Understanding Cognitive Development

Among the most remarkable characteristics of human beings is how much our thinking changes with age. When we compare the thinking of an infant, a toddler, an elementary school student, and an adolescent, the magnitude of the change is immediately apparent. Ac counting for how these changes oc cur is perhaps the central goal of re searchers who study cognitive development. Alongside this agreement about the importance of the goal of deter mining how change occurs, how ever, is agreement that we tradition ally have not done very well in meeting it. In most models of cogni tive development, children are de picted as thinking or acting in a cer tain way for a prolonged period of time, then undergoing a brief, rather mysterious, transition, and then thinking or acting in a different way for another prolonged period. For example, on the classic conserva tion-of-liquid quantity problem, children are depicted as believing for several years that pouring water into a taller, thinner beaker changes the amount of water; then undergo ing a short period of cognitive con flict, in which they are not sure about the effects of pouring the wa ter; and then realizing that pouring does not affect the amount of liquid. How children get from the earlier to the later understanding is described only superficially. Critiques of the inadequacy of such accounts have been leveled most often at stage models such as Piagets. The problem, however, is far more pervasive. Regardless of whether the particular approach de scribes development in terms of stages, rules, strategies, or theories; regardless of whether the focus is on reasoning about the physical or the social world; regardless of the age group of central interest, most theo ries place static states at center stage and change processes either in the wings or offstage altogether. Thus, 3-year-olds are said to have nonrep resentational theories of mind and 5-year-olds representational ones; 5-year-olds to have absolute views about justice and 10-year-olds rela tivistic ones; 10-year-olds to be in capable and 15-year-olds capable of true scientific reasoning. The em phasis in almost all cognitive developmental theories has been on identifying sequences of one-to-one correspondences between ages and ways of thinking or acting, rather than on specifying how the changes occur.

SCADS: A Model of Children's Strategy Choices and Strategy Discoveries:

Preschoolers show surprising competence in choosing adaptively among alternative strategies and in discovering new approaches. The SCADS computer simulation illustrates how simple processes can generate this impressive competence. The models behavior parallels data on childrens addition in at least eight ways: It uses diverse strategies over prolonged periods of time, makes adaptive choices among strategies, discovers the same strategies as children, discovers strategies in the same sequence as children, makes discoveries without trial and error, makes discoveries without having experienced failure, narrowly generalizes new approaches, and generalizes more broadly following challenging problems. SCADS thus indicates plausible sources of young childrens surprising competence at strategy choice and strategy discovery.

Playing linear numerical board games promotes low-income children's numerical development.

The numerical knowledge of children from low-income backgrounds trails behind that of peers from middle-income backgrounds even before the children enter school. This gap may reflect differing prior experience with informal numerical activities, such as numerical board games. Experiment 1 indicated that the numerical magnitude knowledge of preschoolers from low-income families lagged behind that of peers from more affluent backgrounds. Experiment 2 indicated that playing a simple numerical board game for four 15-minute sessions eliminated the differences in numerical estimation proficiency. Playing games that substituted colors for numbers did not have this effect. Thus, playing numerical board games offers an inexpensive means for reducing the gap in numerical knowledge that separates less and more affluent children when they begin school.

Representational change and children's numerical estimation

We applied overlapping waves theory and microgenetic methods to examine how children improve their estimation proficiency, and in particular how they shift from reliance on immature to mature representations of numerical magnitude. We also tested the theoretical prediction that feedback on problems on which the discrepancy between two representations is greatest will cause the greatest representational change. Second graders who initially were assessed as relying on an immature representation were presented feedback that varied in degree of discrepancy between the predictions of the mature and immature representations. The most discrepant feedback produced the greatest representational change. The change was strikingly abrupt, often occurring after a single feedback trial, and impressively broad, affecting estimates over the entire range of numbers from 0 to 1000. The findings indicated that cognitive change can occur at the level of an entire representation, rather than always involving a sequence of local repairs.