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Connecting the Multiplicative Field with Student Mathematical Thinking


Even though algebraic ideas are addressed across a number of grades, algebra continues to serve as a gatekeeper to upper mathematics and degree attainment because of the high percentage of students that fail algebra classes and become halted in their educational progress. One reason for this is students not having the opportunity to build on their own thinking to connect mathematical ideas from elementary through middle school. The multiplicative field consists of major components of the mathematical ideas learned in these important years, but research within the multiplicative field has focused primarily on the mathematics and not enough on connections within student mathematical thinking. This study focuses on examining the connections in student strategies between whole number, fraction, and two-step rate problems, as well as, how students’ ideas of grouping connect to graphing their strategies. Findings add to previous research of student strategies with multiplication and division by detailing some of the nuance in students’ use of grouping. A focus on grouping strategies reveals students’ progression in understanding the mathematical properties from implicit to explicit to purposeful planning of use of the distributive and associative properties of multiplication. Progression within strategies occurs not as a trajectory, but as part of a constellation of ideas. Additionally, student grouping strategies provided a context to begin to connect solutions to graphing. Implications from this research indicate the need for researchers and teachers to uncover what students know and examine use of grouping to support connections across mathematical concepts within the multiplicative field.

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