A Study of the Development of the Number Concept by Scalogram Analysis

Wohlwill, Joachim F.

doi:10.1080/00221325.1960.10534340

Cited by 132 publications

(40 citation statements)

References 22 publications

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“…Because the higher types of moral thought replace, rather than add to, the lower modes of thought, the Guttman (1950) scaling technique used by other investigators to establish certain cognitive developmental sequences (Schuessler and Strauss, 1950;Wohlwill, 1960) is not appropriate for our material. A more appropriate statistical model is derived from Guttman's (1954) quasi-simplex correlation matrix.…”

Section: Level II Morality Of Conventional Role-conformitymentioning

confidence: 95%

The Development of Children’s Orientations Toward a Moral Order

Kohlberg¹

1963

Human Development

290

148

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Section: Level II Morality Of Conventional Role-conformitymentioning

confidence: 95%

The Development of Children’s Orientations Toward a Moral Order

Kohlberg¹

1963

Human Development

290

148

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“…Traditional arithmetic has stressed the learning of such skills as counting objects, using written numerals, and, later, calculating. Both Piagetoriented researchers in mathematics learning (e.g., Dienes, 1966Dienes, , 1967Lovell, 1966) and developmental psychologists (e.g., Flavell, 1963; Kohlberg, 1968;Wohlwill, 1960) focus instead on processes that reflect more directly the mathematical definition of the number concept. Mathematicians stress logical relations among ordered sets, and particularly the notion of one-to-one correspondence among sets.…”

Section: Content Of An Introductory Mathematics Curriculummentioning

confidence: 99%

TASK ANALYSIS IN CURRICULUM DESIGN: A HIERARCHICALLY SEQUENCED INTRODUCTORY MATHEMATICS CURRICULUM¹

Resnick

Wang

Kaplan

1973

J of App Behav Analysis

116

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A method of systematic task analysis is applied to the problem of designing a sequence of learning objectives that will provide an optimal match for the child's natural sequence of acquisition of mathematical skills and concepts. The authors begin by proposing an operational definition of the number concept in the form of a set of behaviors which, taken together, permit the inference that the child has an abstract concept of "number". These are the "objectives" of the curriculum. Each behavior in the defining set is then subjected to an analysis that identifies hypothesized components of skilled performance and prerequisites for learning these components. On the basis of these analyses, specific sequences of learning objectives are proposed. The proposed sequences are hypothesized to be those that will best facilitate learning, by maximizing transfer from earlier to later objectives. Relevant literature on early learning and cognitive development is considered in conjunction with the analyses and the resulting sequences. The paper concludes with a discussion of the ways in which the curriculum can be implemented and studied in schools. Examples of data on individual children are presented, and the use of such data for improving the curriculum itself, as well as for examining the effects of other treatment variables, is considered.The curriculum presented in this paper is an intermediate result of a research program exploring application of detailed task-analysis procedures to the problem of designing sequences of learning objectives. The aim of this research program is to develop a systematic method of specifying and validating learning hierarchies so that instructional programs can be designed that provide an optimal match for a child's natural sequence of acquisition. It is assumed that curricula that closely parallel this sequence will facilitate learning under a wide variety of specific teaching methods.The basic rationale for the methods employed here has been presented in papers by Resnick (1967) and by Resnick and Wang (1969).Briefly, the strategy is to develop hierarchies of learning objectives such that mastery of objectives lower in the hierarchy (simpler tasks) 'Preparation of this paper was supported by a grant from the Ford Foundation. Reprints may be obtained from Lauren B. Resnick,

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“…Invariant sequences of cognitive development have also been proposed for much more finely scaled units of behavior, however. For example, Wohlwill (1960) studied the sequence of a set of closely related number behaviors leading to conservation of number; and Smedslurid (1964) studied the sequence of a number of specific classification, seriation, and number behaviors all of which typically appear between 6 and 8 years of age.…”

Section: Definition Of a Hierarchymentioning

confidence: 99%

“…They have also been used by developmental psychologists in studies of the sequence of acquisition of early cognitive skills (e.g., Kofsky, 1963;Wohlwill, 1960). In such studies the sequence of acquisition is inferred from the fact that any child who can perform higher level behaviors can also perform all lower level behaviors in the hierarchical sequence.…”

Section: Validation Of Hypothesized Hierarchiesmentioning

confidence: 99%

Hierarchies in children's learning: A symposium

Resnick¹

1973

Instr Sci

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There has been a continuing increase in the application of hierarchy theory to the problems of instruction and evaluation, since Gagn6 first used the term "hierarchy" in his theory of how human beings acquire complex skills and knowledge. This article, based on a 1970 symposium of the American Educational Research Association, presents an overview and analysis of the use of learning hierarchies in instruction and research, examines some current research on learning hierarchies, and considers the implications of this research for instructional psychology and for theories of cognitive development.Since Gagn6 first used the term "hierarchy" in his theory of how human beings acquired complex skills and knowledge (Gagn6, 1962), there has been a continuing increase in the application of hierarchy theory to problems of instruction and evaluation. Simultaneously, but apparently independently, developmental psychologists have begun to use concepts of hierarchical dependency in studying sequences of cognitive and psychosocial development. This article examines some current research on learning hierarchies and considers the implications of this research for instructional psychology and for theories of cognitive development.This article is based on the proceedings of a symposium held at the 1970 meetings of the American Educational Research Association. The opening section presents an overview and analysis of the key issues in the study and use of learning hierarchies in instruction and in psychological research. This is followed by summaries of four empirical studies, each of which was presented in detail at the symposium, as an example of a particular approach to work in the field. The concluding section contains commentary of three discussants, representing learning, psychometric and developmental psychologists' points of view. 311

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A Study of the Development of the Number Concept by Scalogram Analysis

Cited by 132 publications

References 22 publications

The Development of Children’s Orientations Toward a Moral Order

The Development of Children’s Orientations Toward a Moral Order

TASK ANALYSIS IN CURRICULUM DESIGN: A HIERARCHICALLY SEQUENCED INTRODUCTORY MATHEMATICS CURRICULUM¹

Hierarchies in children's learning: A symposium

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A Study of the Development of the Number Concept by Scalogram Analysis

Cited by 132 publications

References 22 publications

The Development of Children’s Orientations Toward a Moral Order

The Development of Children’s Orientations Toward a Moral Order

TASK ANALYSIS IN CURRICULUM DESIGN: A HIERARCHICALLY SEQUENCED INTRODUCTORY MATHEMATICS CURRICULUM1

Hierarchies in children's learning: A symposium

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TASK ANALYSIS IN CURRICULUM DESIGN: A HIERARCHICALLY SEQUENCED INTRODUCTORY MATHEMATICS CURRICULUM¹