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Indicators of Multiplicative Reasoning among Fourth Grade Students

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Title: Indicators of Multiplicative Reasoning among Fourth Grade Students
Language: English
Authors: Carrier, James A.
Source: ProQuest LLC. 2010Ph.D. Dissertation, The University of North Carolina at Greensboro.
Availability: ProQuest LLC. 789 East Eisenhower Parkway, P.O. Box 1346, Ann Arbor, MI 48106. Tel: 800-521-0600; Web site: http://www.proquest.com/en-US/products/dissertations/individuals.shtml
Peer Reviewed: N
Physical Description: PDF
Page Count: 160
Publication Date: 2010
Document Type: Dissertations/Theses - Doctoral Dissertations
Education Level: Elementary Education
Elementary Secondary Education
Grade 4
Descriptors: Formal Operations, Test Items, Number Systems, Grade 4, Mathematics Instruction, Teaching Methods, Mathematics Skills, Thinking Skills, Case Studies, Mathematics Tests, Learning Processes, Multiplication, Child Development
ISBN: 978-1-109-77351-4
Abstract: Many students encounter difficulty in their transition to advanced mathematical thinking. Such difficulty may be explained by a lack of understanding of many concepts taught in early school years, especially multiplicative reasoning. Advanced mathematical thinking depends on the development of multiplicative reasoning. The purpose of this study was to identify indicators of multiplicative reasoning among fourth grade students. Inhelder and Piaget (1958) suggested that children circa age eleven are transitioning from the Concrete Operational Stage to the Formal Operations Stage and that it is not likely for children to demonstrate multiplicative reasoning without the structures of development supporting logical and abstract thinking. By employing a cross-case analysis, this study explores the thinking of fourteen math students from a low socioeconomic school. Through cross-case analysis, the researcher probed for patterns of multiplicative reasoning as students progressed through a test instrument which invoked varying levels of multiplicative reasoning. Section one did not distinguish between multiplicative algorithms and multiplicative reasoning. Section two discriminated with respect to multiplicative scheme extension. Section three discriminated with respect to unequal group identification and manipulation. Section 4 discriminated with respect to proportional reasoning but not with respect to multiplicative reasoning. The fourth grade subjects fell into three categories: pre-multiplicative, emergent, and multipliers. Those subjects who utilized multiplicative reasoning on less than four questions were considered pre-multiplicative, whereas those subjects who utilized multiplicative reasoning on six or more questions were considered multipliers. The remaining seven were those subjects who changed their approach from test item to test item, sometimes demonstrating multiplicative reasoning strategies and at other times demonstrating additive reasoning strategies. These subjects were considered emergent in the development of multiplicative reasoning. This study developed twelve new sub-levels that describe in more detail the multiplicative thinking of these fourth graders. These new sub-levels are Level 1 Non-quantifier, Level 1 Spontaneous Guesser, Level 2 Keyword Finder, Level 2 Counter, Level 2 Adder, Level 2 Quantifier, Level 2 Measurer, Level 3 Repeated Adder, Level 3 Coordinator, Level 4 Multiplier, Level 4 Splitter and Level 5 Predictor. This paper suggests that when teachers understand a child's method of deriving multiplying schemes and multiplicative reasoning strategies, they are in a better position to provide the appropriate learning environment for the child. Such interaction allows the listening teacher to build on the child's current level of mathematical understanding. Students should be encouraged to discover for themselves the needed theorems, definitions, and mechanics of the number system, and to personally develop any "short cutting" algorithms, rather than simply being handed the algorithms by the instructor with little or no understanding. [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: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]
Abstractor: As Provided
Entry Date: 2011
Access URL: https://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3403677
Accession Number: ED516672
Database: ERIC
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  Data: Many students encounter difficulty in their transition to advanced mathematical thinking. Such difficulty may be explained by a lack of understanding of many concepts taught in early school years, especially multiplicative reasoning. Advanced mathematical thinking depends on the development of multiplicative reasoning. The purpose of this study was to identify indicators of multiplicative reasoning among fourth grade students. Inhelder and Piaget (1958) suggested that children circa age eleven are transitioning from the Concrete Operational Stage to the Formal Operations Stage and that it is not likely for children to demonstrate multiplicative reasoning without the structures of development supporting logical and abstract thinking. By employing a cross-case analysis, this study explores the thinking of fourteen math students from a low socioeconomic school. Through cross-case analysis, the researcher probed for patterns of multiplicative reasoning as students progressed through a test instrument which invoked varying levels of multiplicative reasoning. Section one did not distinguish between multiplicative algorithms and multiplicative reasoning. Section two discriminated with respect to multiplicative scheme extension. Section three discriminated with respect to unequal group identification and manipulation. Section 4 discriminated with respect to proportional reasoning but not with respect to multiplicative reasoning. The fourth grade subjects fell into three categories: pre-multiplicative, emergent, and multipliers. Those subjects who utilized multiplicative reasoning on less than four questions were considered pre-multiplicative, whereas those subjects who utilized multiplicative reasoning on six or more questions were considered multipliers. The remaining seven were those subjects who changed their approach from test item to test item, sometimes demonstrating multiplicative reasoning strategies and at other times demonstrating additive reasoning strategies. These subjects were considered emergent in the development of multiplicative reasoning. This study developed twelve new sub-levels that describe in more detail the multiplicative thinking of these fourth graders. These new sub-levels are Level 1 Non-quantifier, Level 1 Spontaneous Guesser, Level 2 Keyword Finder, Level 2 Counter, Level 2 Adder, Level 2 Quantifier, Level 2 Measurer, Level 3 Repeated Adder, Level 3 Coordinator, Level 4 Multiplier, Level 4 Splitter and Level 5 Predictor. This paper suggests that when teachers understand a child's method of deriving multiplying schemes and multiplicative reasoning strategies, they are in a better position to provide the appropriate learning environment for the child. Such interaction allows the listening teacher to build on the child's current level of mathematical understanding. Students should be encouraged to discover for themselves the needed theorems, definitions, and mechanics of the number system, and to personally develop any "short cutting" algorithms, rather than simply being handed the algorithms by the instructor with little or no understanding. [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: http://www.proquest.com/en-US/products/dissertations/individuals.shtml.]
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RecordInfo BibRecord:
  BibEntity:
    Languages:
      – Text: English
    PhysicalDescription:
      Pagination:
        PageCount: 160
    Subjects:
      – SubjectFull: Formal Operations
        Type: general
      – SubjectFull: Test Items
        Type: general
      – SubjectFull: Number Systems
        Type: general
      – SubjectFull: Grade 4
        Type: general
      – SubjectFull: Mathematics Instruction
        Type: general
      – SubjectFull: Teaching Methods
        Type: general
      – SubjectFull: Mathematics Skills
        Type: general
      – SubjectFull: Thinking Skills
        Type: general
      – SubjectFull: Case Studies
        Type: general
      – SubjectFull: Mathematics Tests
        Type: general
      – SubjectFull: Learning Processes
        Type: general
      – SubjectFull: Multiplication
        Type: general
      – SubjectFull: Child Development
        Type: general
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      – TitleFull: Indicators of Multiplicative Reasoning among Fourth Grade Students
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