Students' construction of fractional knowledge through modification of their Generalized Number Sequence (GNS)
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The purpose of this study was to understand how eight grade students constructed knowledge of fraction multiplication and [measurement] division, and further Rational Numbers of Arithmetic (RNA) based on their abstract whole number sequences [Generalized Number Sequences] in interaction with a teacher-researcher. The second-order models for students’ constructions of fractional knowledge established by Steffe’s and Olive’s (1990) The Fraction Project guided the present study as mathematics of children and played a role as my major theoretical basis in constituting mathematics for children for the two participating students during the teaching experiment of this study. As a teacher-researcher, I taught two eighth graders at a rural middle school in Georgia in a constructivist teaching experiment from October 2008 to May 2009. All teaching episodes were videotaped with two cameras—one to capture the students’ works and one to capture the whole interactions among the students and the teacher-researcher [me]. In retrospective analysis of the videotapes, I constructed second-order models that explained the changes in the students’ mathematical ways of operating and how the students constructed their mathematical knowledge in the context of fraction multiplication, fraction division, and multiplicative transformation between two fractions. The students’ whole number knowledge [GNS] was significant in that the students conducted their partitioning activities by modifications of their GNS such as recursive partitioning operations, distributive partitioning operations and common partitioning operations to cope with the posed tasks throughout the teaching experiment. In addition, the students demonstrated modifications of their unit-segmenting schemes as fraction measurement division situations became complicated. The reported struggles of the two participating students, due to the lack of their interiorized use of a Fractional Connected Number Sequence (FCNS) for further mathematical activities involving fractions, also suggests that the curriculum in school mathematics for students’ fraction learning needs to be revisited and reorganized to take into account the importance of students’ construction of a multiplicative relationship of unit fractions to a referent whole.