This dissertation documents a new way of conceptualizing vectors in college mathematics, especially in geometry. First, I will introduce three problems to show the complexity and subtlety of the construct of vectors with the classical vector representations. These highlight the need for a new framework that: (1) differentiates abstraction from a physical embodiment, (2) intertwines the representational perspective and the cognitive perspective on vectors, and (3) reveals cognitive development on geometric representations of vectors. These needs ground the development of the framework that permits a layered view of the construct of vectors. The framework comprises three layers of progressive refinements: (1) a layer that describes a global distinction between physical vectors and mathematical vectors by the difference between a physical embodiment and abstraction, (2) a layer that recounts the difference between the representational perspective and the cognitive perspective on vectors as the difference between ontological perspective and epistemological perspective, and (3) a layer that identifies ontological and epistemological obstacles in terms of transitions towards abstraction. Data was gathered from four empirical studies with ninety-eight total students to find evidence of the three major transition points in the new framework: physical to mathematical coming from the first layer, geometric to symbolic and analytic to synthetic from the second layer, and the prevalence of the analytic approach over the synthetic approach while developing abstraction enlightened by the third layer. Limitations of the framework on the distinction of physical and mathematical vectors, directions of cognitive development, and repetition of transitions suggest further refinements of the framework and the implications for teaching and learning of vectors.
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