Additive relationships and signed quantities
Ulrich, Catherine Louise
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This study investigates the cognitive and social factors that influence how a student constructs quantities that can take on positive and negative values and how the student constructs sums and differences of these signed values. Four sixth-grade students worked in pairs during this five-month teaching experiment. We first determined the nature of the students’ number sequences, additive reasoning, and their ability to reason about multiple levels of multiplicatively related units. The students then worked in three signed contexts: a game in which they kept track of how many points they were winning or losing by, changes in savings after combinations of withdrawals and deposits, and displacement on a vertical pole after combinations of trips up and down the pole. My findings were consistent with past research (e.g., Peled, 1991) that found that students start off constructing positive values as comprising one quantity and negative values comprising a different quantity. The students gradually move towards a unitary signed quantity that can take on both positive and negative values. My other findings included the fact that utilizing the idea of additive inverses in adding or subtracting signed quantities is facilitated by the ability to assimilate mathematical situations using three levels of units and that forming a unitary signed quantity requires the construction of an explicitly nested number sequence (Steffe, 2010b). I also discuss how gender, personal interaction style, and disparities in zones of potential construction may have affected the quality of my data and the amount of learning opportunities for the students.