In this work, we delve into geometric analysis, particularly examining the interplay between lower bounds on Ricci curvature and specific functionals. Our exploration begins with an investigation into the implications of Yamabe invariants for asymptotically Poincare-Einstein manifolds and their conformal boundaries under conditions of…

Mathematical justifications offer an approach to which students deepen and extend their mathematical thinking. This process has the potential to enhance students' conceptual understanding. Mathematical practices and process standards ask students to use critical thinking skills to justify their reasoning, but this is usually seen as proofs in high…

Geometric proof-writing is a widely known cause of stress for secondary school students and teachers alike. As the textbook is the primary curricular tool utilized by novice teachers, a two-part qualitative study was conducted to determine (a) the types of proofs presented in a typical high school geometry textbook and (b) teachers' preparedness…

It is widely accepted by the field of mathematics education that proof and reasoning should be integrated into students' mathematical experiences across all grades and across a breadth of content areas. However, studies have shown that both secondary students and teachers have encountered difficulties with learning and teaching geometry proofs. By…

In mathematics, engaging in reasoning-and-proving (RP) includes investigating mathematical relationships, formulating conjectures, evaluating others' conjectures or arguments, generating arguments, and communicating mathematical knowledge. These mathematical processes are key habits of mind that help learners think critically within and…

This study concentrated on the characteristics of Grade 2 students' writing in mathematics class who used the Project M[superscript 2] units to learn geometry and measurement. Included in the study were students with high and low levels of content knowledge. Archived data from the Project M[superscript 2] study were analyzed to determine the…

We establish an isoperimetric inequality for the fundamental tone (first nonzero eigenvalue) of the free plate of a given area, proving the ball is maximal. Given tau greater than 0, the free plate eigenvalues omega and eigenfunctions upsilon are determined by the equation Delta Delta upsilon - tau Delta upsilon = omega upsilon together with…

This dissertation was conducted in a gendered academy in a low-income community in northern Indiana. The study examined how involvement with inquiry-based mathematical tasks embedded in a curricular unit impacted the geometric reasoning of elementary aged African-American males. The curriculum included activities focused on angles, lines and…

Prior research on students' uses of technology has suggested it can be used to support students' development of formal justifications and proofs. The ways in which these technologies influence the construction of arguments and proofs remain uncertain. Furthermore, research has not been conducted that compares the arguments students develop while…

Current reforms in undergraduate mathematics education devoted to proof and counterexample highlight its importance in teaching and learning mathematics. While undergraduate students who have taken a number of upper-level mathematics courses are expected to have mastered the skills required to verify statements as well as to read and write proofs…

This study describes how two 10th grade teachers designed a cross-disciplinary unit, using PBL instruction as a tool for helping students create a valid and convincing argument in their math class. The five week PBL unit focused around American Studies/English and geometry content. In order to measure the rigor and relevance of the unit, multiple…

The purpose of this study is to determine whether educators believe mathematics should be taught to liberal arts students, what topics are taught in liberal arts mathematics courses, and what the motivation is for teaching such topics. The study consists of two parts, a review of the opinions of educators regarding liberal arts mathematics, and a…

How can the secondary Geometry course serve as an opportunity for students to learn to "be like" a mathematician--that is, to acquire a mathematical sensibility? In the first part of this dissertation, I investigate what might be meant by "mathematical sensibility". By analyzing narratives of mathematicians and their work, I identify a collection…