Quantitative reasoning permeates mathematical thinking, and mathematics education researchers have taken a quantitative reasoning approach to examining and modeling students' mathematical thinking and development in various domains. From this approach, secondary and post-secondary researchers have focused on students' ability to reason about how two quantities vary together (covariational reasoning). However, little is known about how covariational reasoning develops from, or connects with, arithmetic and algebraic reasoning. This study begins to bridge the gap in this knowledge. Originally this study was designed to examine middle grade students' units coordination in covariational reasoning across stages and consider the cognitive limiting factor of working memory. In this case study of Daniel, an advanced Stage 2 middle-grade algebra student, I examined the role his units coordinating structures played in his covariational reasoning in non-graphing and algebra tasks. I considered three main components in covariational reasoning (type of quantity, modality of change, and role of time) when analyzing covariational reasoning and capturing the underlying mental units and actions. I found type of quantity and time were the two biggest factors when determining Daniel's covariational reasoning. Daniel also used his units coordinating structures in various ways in the different covariation tasks, generating three different types of change units that were cognitively structurally different. These findings suggest cognitive connections between the types of units a student assimilates with, and the types of covariational reasoning they engage in, are interconnected and warrant future study. / Doctor of Philosophy / This study examines connections between middle-grade students' arithmetic reasoning and algebraic reasoning in their conceptualization of how two quantities vary together (covariation). I interviewed 6 cognitively diverse middle-grade students to investigate these connections and determine at the level of mental action level the types of quantities and actions students use in covariation. After collecting data on the 6 students and reflecting on the richness of each case, I elected to focus on one student for a fine-grain analysis. From this case study of Daniel, an algebra student, I found he used his arithmetic unit structures in unique ways depending on what quantities a task asked him to work with. I also found that Daniel's use of time as a measured quantity in his covariational reasoning influenced how he conceptualized two quantities changing together.

Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical domains, including counting sequences, whole number multiplicative conceptions, and fractions schemes. I describe how consideration of units coordination as a Piagetian (1970b) structure is useful for modeling units coordination across contexts. In this study, I extend teaching experiment methodology (Steffe and Thompson, 2000) to model the dynamics of students' units coordinating activity across contexts within a teaching experiment, using the construct of propensity to coordinate units. Two video-recorded teaching experiments involving pairs of sixth-grade students were analyzed to form a model of the dynamics of students' units coordinating activity. The modeling involved separation of transcriptions into chunks that were coded dichotomously for the units coordinating activity of a single student in each dyad. The two teaching experiments were used to form 5 conjectures about the output of the model that were then tested with a third teaching experiment. The results suggest that modeling units coordination activity via the construct of propensity to coordinate units was useful for describing patterns in the students' perturbations during the teaching sessions. The model was moderately useful for identifying sequences of interactions that support growth in units coordination. Extensions, modifications, and implications of the modeling approach are discussed. / Ph. D.