Learning Mathematics in the Zone of Proximal Development: The Importance of Flexible Use of Knowledge
Ferrara, Roberta Ann
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Permalink
https://hdl.handle.net/2142/69697
Description
Title
Learning Mathematics in the Zone of Proximal Development: The Importance of Flexible Use of Knowledge
Author(s)
Ferrara, Roberta Ann
Issue Date
1987
Doctoral Committee Chair(s)
Brown, Ann L.,
Department of Study
Psychology
Discipline
Psychology
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Education, Mathematics
Education, Educational Psychology
Psychology, Developmental
Abstract
Individual differences in young children's responsiveness to beginning mathematics instruction were assessed, in order to determine the contributions of domain-specific background knowledge, general intellectual ability, and learning and transfer processes.
A preliminary study examined the structure underlying preschoolers' background knowledge of counting--specifically, knowledge expected to facilitate early mathematics learning. Factor analysis indicated that a higher-order factor, and three first-order factors, accounted for over 50% of the variance in performance on a variety of counting tasks. The first-order factors appear to reflect verbal knowledge of the number-word sequence, action knowledge of tagging procedures, and contextual knowledge of the pre-conditions and goals associated with various counting tasks.
The second study had three phases: static pre-assessments of competence, dynamic assessments of learning and transfer abilities, and static post-assessments of mathematics gain. In the initial phase, kindergartener's current development levels were assessed, that is, (a) starting competence on the target problems (single-digit addition items presented in word-problem form using toy props), (b) background knowledge of counting, and (c) IQ. During the following phase, each child learned to solve a subset of the target problems (using a counting-tokens strategy) and was subsequently asked to solve transfer problems that differed systematically from those originally learned. Throughout this dynamic assessment phase, children were given a sequence of hints as needed to solve the problems. The numbers of hints required by individual children were of major interest as inverse measures of learning and transfer efficiency. In the final phase, which was isomorphic to the initial phase, improvement in independent solutions to the target problems was assessed.
Multiple regression indicated that although IQ and background knowledge together accounted for up to 37% of the variance in residual gain scores, transfer efficiency explained an additional 32% of the variance. These results suggest that while general intellectual ability and prior knowledge of a topic may play substantial roles in determining how responsive a child will be to instruction, an additional factor of great importance is the ability to apply prior knowledge flexibly.
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