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Abstract
In this dissertation study, I report on six middle school students’ construction and interpretation of graphs and associated dynamic situations. Constructing and interpreting graphs represents a critical moment in middle school mathematics due to its opportunity to provide a powerful foundation for learning. Nevertheless, researchers have frequently reported on the challenges students experience in interpreting and making sense of graphs that ultimately affect their learning of many topics in algebra and calculus.
I explore the ways in which middle school students’ graphing meanings involve quantitative and covariational reasoning. In particular, I conducted teaching experiments with each participant to engage them in activities intended to leverage quantitative and covariational reasoning with the goal of understanding and developing models of their thinking. In light of the essential role of quantitative and covariational reasoning in mathematical development, I designed tasks in order to understand opportunities for students to construct and reason with quantities’ magnitudes in dynamic real-world situations. Based on my findings, I describe mental operations that constitute productive meanings for graphs, as well as those mental actions that constrain the students’ graphing activity.
In my analysis, I identified various ways students organize one- and two-dimensional space to construct or make sense graphs within those spaces. I situate these different meanings in terms of representing a multiplicative object. Those meanings include representing (i) non-multiplicative object (iconic and transformed iconic translation), (ii) spatial-quantitative multiplicative object, and (iii) quantitative multiplicative object (Type 1 and Type 2). As part of my analysis of the teaching experiments, I also identify various ways of reasoning the students exhibited in dynamic situations including (i) quantitative covariational reasoning, (ii) spatial proximity reasoning, and (iii) matching the perceptual features of motion in two different spaces. Outlining those meanings and ways of reasoning can enable researchers and teachers to be more attentive to those meanings students might hold for their representational activity and to those types of reasoning students might demonstrate in dynamic situations. Based on my analysis, I also outline critical cognitive resources involved in developing a meaning for graphs as an emergent representation of two covarying quantities. The findings have important implications for research, teaching, and curriculum in terms of students’ representational practices.
I explore the ways in which middle school students’ graphing meanings involve quantitative and covariational reasoning. In particular, I conducted teaching experiments with each participant to engage them in activities intended to leverage quantitative and covariational reasoning with the goal of understanding and developing models of their thinking. In light of the essential role of quantitative and covariational reasoning in mathematical development, I designed tasks in order to understand opportunities for students to construct and reason with quantities’ magnitudes in dynamic real-world situations. Based on my findings, I describe mental operations that constitute productive meanings for graphs, as well as those mental actions that constrain the students’ graphing activity.
In my analysis, I identified various ways students organize one- and two-dimensional space to construct or make sense graphs within those spaces. I situate these different meanings in terms of representing a multiplicative object. Those meanings include representing (i) non-multiplicative object (iconic and transformed iconic translation), (ii) spatial-quantitative multiplicative object, and (iii) quantitative multiplicative object (Type 1 and Type 2). As part of my analysis of the teaching experiments, I also identify various ways of reasoning the students exhibited in dynamic situations including (i) quantitative covariational reasoning, (ii) spatial proximity reasoning, and (iii) matching the perceptual features of motion in two different spaces. Outlining those meanings and ways of reasoning can enable researchers and teachers to be more attentive to those meanings students might hold for their representational activity and to those types of reasoning students might demonstrate in dynamic situations. Based on my analysis, I also outline critical cognitive resources involved in developing a meaning for graphs as an emergent representation of two covarying quantities. The findings have important implications for research, teaching, and curriculum in terms of students’ representational practices.