Modeling Middle Grade Students' Algebraic and Covariational Reasoning using Unit Transformations and Working Memory

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.