Modeling Middle Grade Students' Algebraic and Covariational Reasoning using Unit Transformations and Working Memory
Kerrigan, Sarah Therese
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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.
General Audience Abstract
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.
- Doctoral Dissertations