Occasional white boarding : examining the effects of physics students' understanding of motion graphs

The Modeling method of teaching has demonstrated well--‐documented success in the improvement of student learning. The teacher/researcher in this study was introduced to Modeling through the use of a technique called White Boarding. Without formal training, the researcher began using the White Boarding technique for a limited number of laboratory experiences with his high school physics classes. The question that arose and was investigated in this study is “What specific aspects of the White Boarding process support student understanding?” For the purposes of this study, the White Boarding process was broken down into three aspects – the Analysis of data through the use of Logger Pro software, the Preparation of White Boards, and the Presentations each group gave about their specific lab data. The lab used in this study, an Acceleration of Gravity Lab, was chosen because of the documented difficulties students experience in the graphing of motion. In the lab, students filmed a given motion, utilized Logger Pro software to analyze the motion, prepared a White Board that described the motion with position--‐time and velocity--‐time graphs, and then presented their findings to the rest of the class. The Presentation included a class discussion with minimal contribution from the teacher. The three different aspects of the White Boarding experience – Analysis, Preparation, and Presentation – were compared through the use of student learning logs, video analysis of the Presentations, and follow--‐up interviews with participants. The information and observations gathered were used to determine the level of understanding of each participant during each phase of the lab. The researcher then looked for improvement in the level of student understanding, the number of “aha” moments students had, and the students’ perceptions about which phase was most important to their learning. The results suggest that while all three phases of the White Boarding experience play a part in the learning process for students, the Presentations provided the most significant changes. The implications for instruction are discussed.

Fixed block configuration GDDs with block size 6 and (3, r)-regular graphs

Chapter 1 is used to introduce the basic tools and mechanics used within this thesis. Most of the definitions used in the thesis will be defined, and we provide a basic survey of topics in graph theory and design theory pertinent to the topics studied in this thesis. In Chapter 2, we are concerned with the study of fixed block configuration group divisible designs, GDD(n; m; k; λ1; λ2). We study those GDDs in which each block has configuration (s; t), that is, GDDs in which each block has exactly s points from one of the two groups and t points from the other. Chapter 2 begins with an overview of previous results and constructions for small group size and block sizes 3, 4 and 5. Chapter 2 is largely devoted to presenting constructions and results about GDDs with two groups and block size 6. We show the necessary conditions are sufficient for the existence of GDD(n, 2, 6; λ1, λ2) with fixed block configuration (3; 3). For configuration (1; 5), we give minimal or nearminimal index constructions for all group sizes n ≥ 5 except n = 10, 15, 160, or 190. For configuration (2, 4), we provide constructions for several families ofGDD(n, 2, 6; λ1, λ2)s. Chapter 3 addresses characterizing (3, r)-regular graphs. We begin with providing previous results on the well studied class of (2, r)-regular graphs and some results on the structure of large (t; r)-regular graphs. In Chapter 3, we completely characterize all (3, 1)-regular and (3, 2)-regular graphs, as well has sharpen existing bounds on the order of large (3, r)- regular graphs of a certain form for r ≥ 3. Finally, the appendix gives computational data resulting from Sage and C programs used to generate (3, 3)-regular graphs on less than 10 vertices.

Hamilton decompositions of 6-regular abelian Cayley graphs

In 1969, Lovasz asked whether every connected, vertex-transitive graph has a Hamilton path. This question has generated a considerable amount of interest, yet remains vastly open. To date, there exist no known connected, vertex-transitive graph that does not possess a Hamilton path. For the Cayley graphs, a subclass of vertex-transitive graphs, the following conjecture was made: Weak Lovász Conjecture: Every nontrivial, finite, connected Cayley graph is hamiltonian. The Chen-Quimpo Theorem proves that Cayley graphs on abelian groups flourish with Hamilton cycles, thus prompting Alspach to make the following conjecture: Alspach Conjecture: Every 2k-regular, connected Cayley graph on a finite abelian group has a Hamilton decomposition. Alspach’s conjecture is true for k = 1 and 2, but even the case k = 3 is still open. It is this case that this thesis addresses. Chapters 1–3 give introductory material and past work on the conjecture. Chapter 3 investigates the relationship between 6-regular Cayley graphs and associated quotient graphs. A proof of Alspach’s conjecture is given for the odd order case when k = 3. Chapter 4 provides a proof of the conjecture for even order graphs with 3-element connection sets that have an element generating a subgroup of index 2, and having a linear dependency among the other generators. Chapter 5 shows that if Γ = Cay(A, {s1, s2, s3}) is a connected, 6-regular, abelian Cayley graph of even order, and for some1 ≤ i ≤ 3, Δi = Cay(A/(si), {sj1 , sj2}) is 4-regular, and Δi ≄ Cay(ℤ3, {1, 1}), then Γ has a Hamilton decomposition. Alternatively stated, if Γ = Cay(A, S) is a connected, 6-regular, abelian Cayley graph of even order, then Γ has a Hamilton decomposition if S has no involutions, and for some s ∈ S, Cay(A/(s), S) is 4-regular, and of order at least 4. Finally, the Appendices give computational data resulting from C and MAGMA programs used to generate Hamilton decompositions of certain non-isomorphic Cayley graphs on low order abelian groups.

A STUDY ON THE FEASIBILITY OF UNIVERSAL CHIP CONTROL IN MACHINING

A novel solution to the long standing issue of chip entanglement and breakage in metal cutting is presented in this dissertation. Through this work, an attempt is made to achieve universal chip control in machining by using chip guidance and subsequent breakage by backward bending (tensile loading of the chip's rough top surface) to effectively control long continuous chips into small segments. One big limitation of using chip breaker geometries in disposable carbide inserts is that the application range is limited to a narrow band depending on cutting conditions. Even within a recommended operating range, chip breakers do not function effectively as designed due to the inherent variations of the cutting process. Moreover, for a particular process, matching the chip breaker geometry with the right cutting conditions to achieve effective chip control is a very iterative process. The existence of a large variety of proprietary chip breaker designs further exacerbates the problem of easily implementing a robust and comprehensive chip control technique. To address the need for a robust and universal chip control technique, a new method is proposed in this work. By using a single tool top form geometry coupled with a tooling system for inducing chip breaking by backward bending, the proposed method achieves comprehensive chip control over a wide range of cutting conditions. A geometry based model is developed to predict a variable edge inclination angle that guides the chip flow to a predetermined target location. Chip kinematics for the new tool geometry is examined via photographic evidence from experimental cutting trials. Both qualitative and quantitative methods are used to characterize the chip kinematics. Results from the chip characterization studies indicate that the chip flow and final form show a remarkable consistency across multiple levels of workpiece and tool configurations as well as cutting conditions. A new tooling system is then designed to comprehensively break the chip by backward bending. Test results with the new tooling system prove that by utilizing the chip guidance and backward bending mechanism, long continuous chips can be more consistently broken into smaller segments that are generally deemed acceptable or good chips. It is found that the proposed tool can be applied effectively over a wider range of cutting conditions than present chip breakers thus taking possibly the first step towards achieving universal chip control in machining.