This study sought to explore whether access to definitions and general representations influences the construction of general direct arguments. Data was collected in college mathematics courses for prospective elementary school teachers. Participant arguments were analyzed along two variables: the generality of the representations and the…

Algebraic thinking is an essential component of the K-12 mathematics curriculum. Early algebra focuses on the overall numeric structure (Kaput, 2008; Kieran,2018; Kieran et al., 2016) which focuses on the noticing of arithmetic patterns leading to generalizations supported by justification, reasoning and justifications (Blanton, 2008; Blanton et…

This dissertation study aims to understand the instructional interactions that foster students' generalizing. By analyzing video recorded lessons from 13 different Grade 3 classrooms in which an early algebra intervention is being implemented, activities related to generalizing and students' generalizing activities were identified, and the nature…

In this thesis, we apply model theory to Lie theory and geometric group theory. These applications of model theory come via nonstandard analysis. In Lie theory, we use nonstandard methods to prove two results. First, we give a positive solution to the local form of Hilbert's Fifth Problem, which asks whether every locally euclidean local…

There is ample literature documenting that, for many decades, high school students view algebra as difficult and do not demonstrate understanding of algebraic concepts. Algebraic reasoning in elementary school aims at meaningfully introducing algebra to elementary school students in preparation for higher-level mathematics. While there is research…

In this thesis, our aim is the study the Vanishing of Maps of Tor Conjecture of Hochster and Huneke. We mainly focus on an equivalent characterization called the Strong Direct Summand Conjecture, due to N. Ranganathan. Our results are separated into three chapters. In Chapter 3, we prove special cases of the Strong Direct Summand Conjecture in…

Scope and method of study: The purpose of this study was to unpack the understandings pre-service elementary teachers have pertaining to the ideas of pattern and function. The intent was to bring insight into how mathematics teacher educators can use patterning activities to prepare pre-service elementary teachers to support the development of…