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ERIC Number: ED535577
Record Type: Non-Journal
Publication Date: 2009
Pages: 400
Abstractor: As Provided
Reference Count: N/A
ISBN: ISBN-978-1-1095-7455-5
The Development of Four Fifth Grade Students' Understanding and Skill Representing Fractions as Quotients
Kim, Ahyoung
ProQuest LLC, Ed.D. Dissertation, Arizona State University
This dissertation investigated the conceptual schemes children constructed as they related division number sentences to various types of fractions: Proper fractions, improper fractions, and mixed numbers in both contextual and abstract symbolic forms. It was hypothesized that student's understanding depends heavily on the role played by factors and multiples in numerator/denominator and dividend/divisor, how these quantities are partitioned, and the forms of inscription in which problems are presented and represented by children. Methods followed those of the constructivist teaching experiment. Four fifth-grade students from an inner city school in the southwest United States were interviewed eight times: Pre-test clinical interview, six teaching/semi-structured interviews, and a final post-test clinical interview. Four linked case studies are presented. Results showed that for equal sharing situations, children first conceptualized division in two ways: For mixed fractions, division generated a whole number quotient and a fractional quotient. This provided the conceptual basis to see improper fractions as quotients. For proper fractions, they tended to see the quotient as an instance of the multiplicative structure: a x b = c; a = c/b; b = c/a. The predominance of long division greatly hampered students' ability to recognize this multiplicative structure. Children's partitioning strategies developed from a repeated halving stage [right arrow] consuming all quantity stage [right arrow] whole number objects leftover stage [right arrow] singleton object analysis/multiple objects analysis [right arrow] direct mapping stage. When children connected singleton object analyses and multiple object analyses, they finally became able to conceptualize both division as fractions and fractions as division. Results suggest that children may need to use both singleton object analysis and multiple objects analysis through well designed equal sharing word problems. Second, facility in recall of multiplication and division fact families and understanding the multiplicative structure must be emphasized before learning fraction division. Third, to facilitate understanding of the multiplicative structure children must be fluent in representing division in the form of number sentences for equal sharing word problems or their reliance on long division hampers their use of syntax and their understanding of divisor and dividend and their relation to the concepts of numerator and denominator. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page:]
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Publication Type: Dissertations/Theses - Doctoral Dissertations
Education Level: Elementary Education; Grade 5
Audience: N/A
Language: English
Sponsor: N/A
Authoring Institution: N/A