Browsing by Subject "Algebra"
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Item A study of abelian groups: a thesis in mathematics(Texas Tech University, 1955-05) Caskey, John HermonNOT AVAILABLEItem Algebraic points of small height with additional arithmetic conditions(2004) Fukshansky, Leonid Eugene; Vaaler, Jeffrey D.Item Algebraic theorems obtained by use of extended analytic geometry(Texas Tech University, 1955-08) Garner, J.H.Te understand the theorems presented here one must first understand the basic principles which have been developed by Dr. Ralph Underwood. The basic process is a method by which equations with three or more variables may be represented on the XY plane. While the process does not represent a true projection many of the basic features of the loci, in the case of three variables, are preserved. The two basic methods which have been used previously are called System A and System B, (9:527) however, an infinity of methods or plotting rules are available. In the first ten theorems presented here a more flexible method is employed. In graphing the locus of the equation a point on the locus is first found, and the equation of a tangent hyperplane is written by the method illustrated below« one then may use a graphing rule so that the locus of the tangent hyperplane is a straight line on the XY plane.Item An evaluation of four remediation methods within computer-based instruction(Texas Tech University, 1999-05) Willis, Ross E.When teaching does not result in complete and error-free learning, remediation is required. Computer-based instruction may address a student's remedial needs along two dimensions: reteaching format and remedial elements. In terms of reteaching format, the computer-based instructional system can reteach the deficient knowledge and skills using the initial instructional screens in the form a review of the original material. Alternatively, the system can present a set of new instructional screens based on an alternative pedagogical strategy. In terms of remedial elements, the instructional system may present a complete set of instructional screens designed to address all elements of the curriculum, as in a complete review of the initial instructional materials. Alternatively, the system may diagnose the student's error and select a set of instructional screens specifically designed to address the student's apparent misconception.Item Item Evaluating the persistence and performance of 'successful' precalculus students in subsequent mathematics courses(Texas Tech University, 2000-05) Jarrett, Ebonee"Each year approximately 600,000 college students take a pre-calculus/college algebra and trigonometry course; yet only about 15-20% of them ever go on to start calculus" (Gordon, 1994, p. 136). Pre-calculus mathematics courses are often the "critical filter" to many college majors (Whitely, 1987). Unfortunately, these same courses are often deterrents, not only for persistence in mathematics and science fields, but in the pursuance of higher education in general. Even though many mathematics and science based fields expect that students begin in calculus, this is not the case for many college freshman. More specifically, introductory or pre-calculus mathematics courses are sometimes the cause of change m major for students considering majors in math or science-based fields (Whitely, 1987). Even more disparaging is the fact that"... students who do go on to calculus... retain little of the material they were taught and do not complete calculus" (Gordon, p. 136). Nationally, only about 40% of students who enroll in a pre-calculus course complete a first semester of calculus (Murray, 1999). Students are not able to apply the skills they have learned in pre-calculus courses toward subsequent courses, like calculus. Because minorities and women seem to compose the majority in many pre-calculus programs, they are often the focus of many studies (Hagedom, 1997; Treisman, 1992). Female and minority students, in particular, are imderrepresented in upper-level mathematics and science courses. Strenta et al. (1993) reported that on a national level, persistence in science, math and engineering majors was approximately 30% for women as opposed to 39% for men. Astin and Astin (1993) reported that approximately one-third of Hispanic students and one-half of black students completed degrees in science, mathematics and engineering fields. Seymour (1997) discovered that 41.2% of students of color attributed leaving a science, mathematics or engineering major to curriculum overload and fast pace of introductory mathematics and science courses. Even though contemporary research has begun to show changes in trends of mathematic achievement, problems in undergraduate mathematics programs still exist. Is pre-calculus study on college level effective? Some studies consider any course before calculus as remedial (Hagedom, 1997). Therefore, some institutions question the need for pre-calculus instruction on a college level. Likewise, some programs for mathematics and science-based majors expect that students begin in calculus. In the early 1990's, a reform movement began to improve the college calculus cumculum. However, for a time nothing was done to update prerequisite pre-calculus courses. As the problem receives more and more publicity with introduction of poticies and standards, such as the Curriculum and Evaluation Standards for School Mathematics by the National Council of Teachers of Mathematics, institutions have been forced to evaluate their pre-calculus programs for effectiveness. Some institutions since have implemented new pre-calculus programs, but some continue to use the same instructional methods. What happens to the students who are not ready for calculus when they enter college? What happens to the students who do not seem to benefit from present precalculus curriculimis?Item Explorations in algebra and topology(2016-05) Gal, Itamar; Hadani, Ronny; Allcock, Daniel; Blumberg, Andrew; Vaaler, JeffreyThree independent investigations are expounded, two in the domain of algebra and one in the domain of topology. We first consider algebraic extensions generated by elements of bounded degree and consider the question of whether or not the finite sub-extensions of such fields can be bounded. We give partial results which will hopefully lead to a full classification in the future. These results are fundamentally group theoretic but have applications to number theory. Next we develop the notational system originated by Conway and Sloane for working with quadratic forms over the 2-adic integers and prove its correctness. This provides a proof which was missing from the literature. Finally we study distributions of persistent homology barcodes obtained by sampling finite point sets from metric measure spaces. The main result here is the derivation of robust statistics for topological data analysis.Item The height in terms of the normalizer of a stabilizer(2008-05) Garza, John Matthew, 1975-; Vaaler, Jeffrey D.This dissertation is about the Weil height of algebraic numbers and the Mahler measure of polynomials in one variable. We investigate connections between the normalizer of a stabilizer and lower bounds for the Weil height of algebraic numbers. In the Archimedean case we extend a result of Schinzel [Sch73] and in the non-archimedean case we establish a result related to work of Amoroso and Dvornicich [Am00a]. We establish that amongst all polynomials in Z[x] whose splitting fields are contained in dihedral Galois extensions of the rationals, x³-x-1, attains the lowest Mahler measure different from 1.Item Integrating topology into the standard high school geometry curriculum(2012-08) Kiker, William George; Odell, E. (Edward); Daniels, MarkThis report conveys some of the modern investigations surrounding the use of topology in a contextual setting. Topics discussed include applications of topology relating to the modeling of biological structures and common objects like sunshades, elementary knot theory, and the connection between the fields of topology and algebra. A brief overview and discussion of the incorporation of elementary topology into the standard Geometry curriculum of secondary schools is also examined.Item Loci of second degree equations of the 3-axes plane(Texas Tech University, 1949-06) Eaves, Leslie LevonThis paper is concerned with the loci of second degree equations on the plane In the coordinate system discussed by Dr. R.S. Underwood in the American Mathematical Monthly (May, 1945 and March 1949).Item Maintenance effects of strategy instruction for algebra skills with students with challenging behavior(2011-12) Roundhill, Marie Colleen; Flower, Andrea; Garcia, Shernaz; Pazey, BarbaraThis thesis consists of a single subject multiple baseline study of a math intervention for students with behavioral challenges. Students with behavioral challenges were given instruction using a concrete, representational, abstract (CRA) sequence in Algebra problems requiring transformations on both sides of the equation. This study examined maintenance of those skills. Results indicate that while accuracy decreased from the post-intervention to maintenance phases, scores remained well-above baseline levels indicating that the students retained understanding of the concepts taught. In a social validity survey, participants indicated that they liked the intervention, found it beneficial, and sometimes use it in their classes.Item Multiplicative and dynamical analysis on idèles and idèle class groups(2016-05) Hughes, Adam Miles; Vaaler, Jeffrey D.; Ciperiani, Mirela, 1976-; Mohammadi, Amir; Allcock, Daniel; Sinclair, Christopher; Widmer, MartinWe prove an extension of a result due to Allcock and Vaaler from 2009. In the main theorem we show that an idèle group associated to Q-bar is naturally dense in a Banach algebra normed by the Weil height. We establish bounds for the dynamics of generic idèlic points of a field modulo the diagonally-embedded multiplicative groups of the associated fields.Item 'Playing the game' of story problems : situated cognition in algebra problem solving(2010-12) Walkington, Candace Ann; Petrosino, Anthony J. (Anthony Joseph), 1961-; Carmona-Dominguez, Guadalupe; Marshall, Jill; Walker, Mary; Greeno, JimThe importance of mathematics instruction including "real life" contexts relevant to students’ lives and experiences is widely acknowledged (Common Core State Standards Initiative, 2010; National Council of Teachers of Mathematics, 2000; 2006; 2009), however questions about why contextualized mathematics is beneficial and how different types of contextualization impact problem solving have yet to be fully addressed by research. Common justifications for contextualized mathematics include the idea that relevant contexts may help students to apply what they learn in school to out-of-school situations, and that relevant contexts may scaffold learning by providing a bridge between what students understand and the content they are trying to learn. The present study investigates these justifications, as well as students' beliefs and problem-solving methods, using story problems on linear functions. A situated cognition theoretical framework (Greeno, 2006) is used to interpret student behavior in the complex, social system of "school mathematics." In a series of interviews, students from a low-performing urban school were presented with algebra problems. Some problems were personalized to the ways in which they described using mathematics in their everyday lives, while others were normal story problems, story problems with equations, or abstract symbolic equations. Results showed that students rarely explicitly used situational knowledge when solving story problems, had consistent issues with verbal interpretation of stories, and engaged in non-coordinative reasoning where they bypassed the intermediate step of understanding the given situation before trying to solve the problem. After completing most of Algebra I, students still had considerable difficulty with symbolic representations, and struggled to coordinate formal and informal mathematical reasoning. Problems with the same mathematical structure with different amounts of verbal and symbolic support elicited different strategies from students, with personalized problems having high response rates and high use of informal strategies. This suggests that students can use sophisticated, situation-based reasoning on contextualized problems, and that different problem framings may scaffold learning. However, results also demonstrated that the culture of schooling, and story problems as an artifact of this culture, undermines many of the justifications for contextualizing mathematics, and that students need more authentic ways to develop their mathematical reasoning.Item Two theorems related to group schemes(2010-12) Jones, James Hunter, 1982-; Voloch, José Felipe; Helm, DavidAfter presenting some preliminary information, this paper presents two proofs regarding group schemes. The first relates the category of affine group schemes to the category of commutative Hopf algebras. The second shows that a commutative group scheme of finite order is in fact killed by its order.