On a Generalization of the Frobenius Number

We consider a generalization of the Frobenius Problem where the object of interest is the greatest integer which has exactly $j$ representations by a collection of positive relatively prime integers. We prove an analogue of a theorem of Brauer and Shockley and show how it can be used for computation.Comment: 5 page

Paper 1: Engaging in Lesson Study at Georgia College

A lesson study cycle is a professional development process that integrates research and reflection through collaboration. The cycle allows a group to refine a lesson based on these collaboration efforts such as interaction with students and the post-lesson discussion. Secondary pre-service teachers in a mathematics methods course engaged in a lesson study cycle through collaboration between in-service teachers, Georgia College professors, and students in a local high school classroom. We systematically investigated this process to determine that through preparing, enacting and reflecting on their practice, Pre-service Teachers (PST) developed insight, reasoning, and understanding of the mathematics that they taught

Engaging in Lesson Study at Georgia College

A lesson study cycle is a professional development process that integrates research and reflection through collaboration. The cycle allows a group to refine a lesson based on these collaboration efforts such as interaction with students and the post-lesson discussion. Secondary pre-service teachers in a mathematics methods course engaged in a lesson study cycle through collaboration between in-service teachers, Georgia College professors, and students in a local high school classroom. We systematically investigated this process to determine that through preparing, enacting and reflecting on their practice, Pre-service Teachers (PST) developed insight, reasoning, and understanding of the mathematics that they taught

First cohomology for finite groups of Lie type: simple modules with small dominant weights

Let $k$ be an algebraically closed field of characteristic $p > 0$, and let $G$ be a simple, simply connected algebraic group defined over $\mathbb{F}_p$. Given $r \geq 1$, set $q=p^r$, and let $G(\mathbb{F}_q)$ be the corresponding finite Chevalley group. In this paper we investigate the structure of the first cohomology group $H^1(G(\mathbb{F}_q),L(\lambda))$ where $L(\lambda)$ is the simple $G$-module of highest weight $\lambda$. Under certain very mild conditions on $p$ and $q$, we are able to completely describe the first cohomology group when $\lambda$ is less than or equal to a fundamental dominant weight. In particular, in the cases we consider, we show that the first cohomology group has dimension at most one. Our calculations significantly extend, and provide new proofs for, earlier results of Cline, Parshall, Scott, and Jones, who considered the special case when $\lambda$ is a minimal nonzero dominant weight.Comment: 24 pages, 5 figures, 6 tables. Typos corrected and some proofs streamlined over previous versio

Effectiveness of a Supplemental Instruction Program in a Statistics Course

At most universities, an introductory statistics course is required for the majority of the students before they begin their specific major classes. Of these students, several will fail to retain the information, making future classes more difficult, or fail to successfully pass the course which increases the likelihood a student will not graduate on time. Providing academic support through the implementation of a Supplemental Instruction (SI) Program gives students the opportunity to receive extra help focused on student achievement in this course. Students are able to attend sessions to receive conceptual help while reviewing class material, developing study strategies, and collaborating with classmates. We will be focusing on the effects SI can have on student achievement in statistics classrooms. We will share our data analysis for using SI in a statistics course over a 4-year period, providing participants the opportunity to identify the positive effects SI has on student success

Heading toward Equality, Preservice Teachers’ Interventions to Change Students’ Conceptions of the Equal Sign

Research indicates that a large percentage of students have misconceptions of the equal sign and that this can be detrimental to their mathematical pursuits as they progress through algebra. In order for students to develop a strong understanding of the meaning of the equal sign, algebra teachers must be aware of the viewpoints their students will hold pertaining to the equal sign and strategies to encourage a relational view of the equal sign and a progression of sophistication within the relational spectrum. In an algebra course for preservice teachers, we created a project where they investigated students’ conceptions of the equal sign and determined interventions to foster more productive views. The purpose of our study was to examine our preservice teachers’ ability to differentiate between the various students’ conceptions of the equal sign, to determine which teacher interventions resonated well with the preservice teachers, and to evaluate the overall effectiveness of the project. This article will also outline ideas for other mathematics teacher educators that wish to implement similar practice-based teaching experiences in their content courses and potential paths of future research

K-2 Mathematicians and Writers: Professional Learning Community for Developing Conceptual Understanding

Recent mathematics reform efforts emphasize the importance of conceptual understanding, procedural fluency, problem solving, and subject specific discourse (Boaler, 2015). NCTM has supported the use of children’s literature and writing to improve conceptual understanding of mathematics since 2000. Yet, many math teachers report that it is challenging to integrate math and literacy, often viewing these as mutually exclusive content areas. This notion fragments the curriculum and isolates the resources that literacy yields for thinking and communicating in math. Our session will examine a yearlong professional development initiative designed to provide K-2 teachers in a rural school district with resources for an integrated approach to mathematics instruction involving conceptual understanding through the use of children’s literature, manipulatives, and writing. Teachers participated in a professional learning community and assumed roles of teacher leaders in developing collaborative learning experiences with grade level colleagues and attended workshops throughout the academic year. This approach represents a unique innovation that utilized a teachers teaching teachers model to improve sustainability, embedded professional learning within grade levels, alignment across the district K-2 grades in mathematics, and vertical alignment across K-2 math within four elementary schools. We assert that as children begin school, using resources across the curriculum will support development as mathematicians, readers and writers, and conceptual, metacognitive learners. The following research questions evolved from the analysis of the data. When teachers create lessons integrating literacy and math, how connected are the various components? What misconceptions were revealed in the enactment of the integrated lessons