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Abstract
Intensive quantities, those quantities which characterize a multiplicative relationship between two quantities, represent a critical component of students mathematical learning. Examples of reasoning with intensive quantities include making proportional comparisons, reasoning about linear functions that have constant rates of change, considering the densities of various materials, and analyzing the rates at which quantities covary. In this study, I investigated the mental schemes and operations that students used to construct and reason with intensive quantities and the covariational relationships those quantities described. This dissertation reports the findings from a constructivist teaching experiment I conducted with two tenth-grade students from October 2013 to March 2014. As the students primary teacher, I posed a variety of tasks designed to investigate how the students would construct and reason with constant multiplicative relationships between covarying quantities. After completing the teaching experiment, I conducted a retrospective analysis of the teaching session interactions in order to construct second-order models that accounted for the students mathematical activity and changes they made to their quantitative reasoning over the course of the study. Findings include the identification of seven constructive resources that facilitated the students ability to construct intensive quantities and to make sense of constant covariational relationships: a) reasoning with three levels of units; b) incorporating a strategy of coordinated partitioning/iterating; c) the construction of a splitting scheme; d) the construction of iterable composite units; e) the construction of a process for quantifying a unit ratio; f) the construction of a simultaneous awareness of a measured quantity as a single composite whole and as a sequence of individual units; and g) the ability to use ones operations recursively in order to flexibly change the measurement units of both quantities in a given ratio. In addition, these conceptual resources were involved in the construction of a reversible distributive partitioning scheme that enabled the construction of distributive reasoning. These results have implications for how researchers and teachers conceptualize goals for students mathematical learning in schools.