Building Leadership: The Knowledge of Principals in Creating Collaborative Communities of Professional Learning

Research literature is replete with the importance of collaboration in schools, the lack of its implementation, the centrality of the role of the principal, and the existence of a gap between knowledge and practice--or a "Knowing-Doing Gap." In other words, there is a set of knowledge that principals must know in order to create a collaborative workplace environment for teachers. This study sought to describe what high school principals know about creating such a culture of collaboration. The researcher combed journal articles, studies and professional literature in order to identify what principals must know in order to create a culture of collaboration. The result was ten elements of principal knowledge: Staff involvement in important decisions, Charismatic leadership not being necessary for success, Effective elements of teacher teams, Administrator‘s modeling professional learning, The allocation of resources, Staff meetings focused on student learning, Elements of continuous improvement, and Principles of Adult Learning, Student Learning and Change. From these ten elements, the researcher developed a web-based survey intended to measure nine of those elements (Charismatic leadership was excluded). Principals of accredited high schools in the state of Nebraska were invited to participate in this survey, as high schools are well-known for the isolation that teachers experience--particularly as a result of departmentalization. The results indicate that principals have knowledge of eight of the nine measured elements. The one that they lacked an understanding of was Principles of Student Learning. Given these two findings of what principals do and do not know, the researcher recommends that professional organizations, intermediate service agencies and district-level support staff engage in systematic and systemic initiatives to increase the knowledge of principals in the element of lacking knowledge. Further, given that eight of the nine elements are understood by principals, it would be wise to examine reasons for the implementation gap (Knowing-Doing Gap) and how to overcome it.

Improving the Effectiveness of Independent Practice with Corrective Feedback

In this action research study of my 8th grade Algebra class, I investigated the effects of teacher-to-student written corrective feedback on student performance and attitude toward mathematics. The corrective feedback was given on solutions for selected independent practice problems assigned as homework. Each problem being assessed was given a score based on a 3- point rubric and additional comments were written. I discovered that providing teacher-to-student written corrective feedback for independent practice problems was beneficial for both students and teachers. The feedback positively affected the attitudes of students and teacher toward independent practice work resulting in an improved quality of solutions produced by students. I plan to extend my research to explore ways to provide corrective feedback to students in all of my mathematics classes.

Reasonable or Not? A Study of the Use of Teacher Questioning to Promote Reasonable Mathematical Answers from Sixth Grade Students

In this action research study of my sixth grade mathematics class, I investigated the influence a change in my questioning tactics would have on students’ ability to determine answer reasonability to mathematics problems. During the course of my research, students were asked to explain their problem solving and solutions. Students, amongst themselves, discussed solutions given by their peers and the reasonability of those solutions. They also completed daily questionnaires that inquired about my questioning practices, and 10 students were randomly chosen to be interviewed regarding their problem solving strategies. I discovered that by placing more emphasis on the process rather than the product, students became used to questioning problem solving strategies and explaining their reasoning. I plan to maintain this practice in the future while incorporating more visual and textual explanations to support verbal explanations.

Applications of Linear Programming to Coding Theory

Maximum-likelihood decoding is often the optimal decoding rule one can use, but it is very costly to implement in a general setting. Much effort has therefore been dedicated to find efficient decoding algorithms that either achieve or approximate the error-correcting performance of the maximum-likelihood decoder. This dissertation examines two approaches to this problem. In 2003 Feldman and his collaborators defined the linear programming decoder, which operates by solving a linear programming relaxation of the maximum-likelihood decoding problem. As with many modern decoding algorithms, is possible for the linear programming decoder to output vectors that do not correspond to codewords; such vectors are known as pseudocodewords. In this work, we completely classify the set of linear programming pseudocodewords for the family of cycle codes. For the case of the binary symmetric channel, another approximation of maximum-likelihood decoding was introduced by Omura in 1972. This decoder employs an iterative algorithm whose behavior closely mimics that of the simplex algorithm. We generalize Omura's decoder to operate on any binary-input memoryless channel, thus obtaining a soft-decision decoding algorithm. Further, we prove that the probability of the generalized algorithm returning the maximum-likelihood codeword approaches 1 as the number of iterations goes to infinity.