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.

Student Transition to College

In this action research study of recent graduates from my district, I investigated their level of readiness for college-level mathematics courses. I discovered that the students have a wide variety of experiences in college. There are many factors that determine success in college mathematics courses. These factors include size of college, private or public, university or community college. Other factors include students’ choice of major, maturity level, and work ethic. As a result of this research, I plan to raise the individual expectations in my classroom. It is our duty as high school educators to prepare the students for a wide variety of experiences in college. We cannot control where the students attend college or what they study. High schools need to prepare the students for all possibilities and ensure that they have a solid knowledge of the baseline mathematics skills.

Holding Students Accountable

This action research study of approximately 90 high school algebra students investigates how frequent quizzing benefits them during the course of a semester. The intent of the research was to see how well students kept up with the material and if frequent quizzing helped them on the chapter tests. It was also designed to help me gain a better understanding of what students know and how I need to adjust daily routines so that all students stay caught up. I discovered that although frequent quizzes are not the students’ favorite activity to take part in, they learn to accept the quizzes and benefit greatly because of the amount of information students learn from them. Holding students accountable with frequent quizzes forces students to stay caught up and pushes them to excel as many found the tests to be much easier because of the practice they received. My research revealed many advantages to holding students accountable through frequent quizzes and although it can be somewhat time consuming, it is definitely a practice that will be continued in my classroom for years to come.

Daily Problem-Solving Warm-Ups: Harboring Mathematical Thinking In The Middle School Classroom

In this action research study of my classroom of 8th grade mathematics, I investigated the use of daily warm-ups written in problem-solving format. Data was collected to determine if use of such warm-ups would have an effect on students’ abilities to problem solve, their overall attitudes regarding problem solving and whether such an activity could also enhance their readiness each day to learn new mathematics concepts. It was also my hope that this practice would have some positive impact on maximizing the amount of time I have with my students for math instruction. I discovered that daily exposure to problem-solving practices did impact the students’ overall abilities and achievement (though sometimes not positively) and similarly the students’ attitudes showed slight changes as well. It certainly seemed to improve their readiness for the day’s lesson as class started in a more timely manner and students were more actively involved in learning mathematics (or perhaps working on mathematics) than other classes not involved in the research. As a result of this study, I plan to continue using daily warm-ups and problem-solving (perhaps on a less formal or regimented level) and continue gathering data to further determine if this methodology can be useful in improving students’ overall mathematical skills, abilities and achievement.

Vanishing of Ext and Tor over complete intersections

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of Tor_{i}(M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and MtensorN force the vanishing of Tor_{i}(M, N) for all i>0. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and MtensorHom(M,R) are torsion-free, then M is free if and only if M has complexity at most one. If R is a hypersurface and Ext^{i}(M, N) has finite length for all i>>0, then the Herbrand difference [18] is defined as length(Ext^{2n}(M, N))-(Ext^{2n-1}(M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of Ext^{i}(M, N) needed to ensure that Ext^{i}(M, N) = 0 for all i>>0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.

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.

Niche Specialization and Conservation Biology of Cicindela nevadica lincolniana

As with many organisms across the globe, Cicindela nevadica lincolniana is threatened with extinction. Understanding ecological factors that contribute to extinction vulnerability and what methods aid in the recovery of those species is essential in developing successful conservation programs. Here we examine behavioral mechanisms for niche partitioning along with improving techniques for captive rearing protocol and increasing public awareness about the conservation of this local insect. Ovipositional selectivity was examined for Cicindela nevadica lincolniana, Cicindela circumpicta, Cicindela togata, Cicindela punctulata, and Cicindela fulgida. Models reflect that these species of co-occurring tiger beetles select different ranges of salinity in which to oviposit thereby reducing the potential for interspecific competition. In a second study, thermoregulatory niche partitioning was examined for the same complex of tiger beetle species. Time spent in the sun, on different substrates, and engaging in various behaviors associated with thermoregulation were significantly different during different parts of the day and between species. I continued along a previous line of study to develop a viable captive rearing program. So far fourteen adult Cicindela nevadica lincolniana have been successfully reared in captivity. Overwintering mortality has been determined as a key factor in the mortality of this species in captivity. Finally, I examined the potential for using the visual arts to promote the conservation of Cicindela nevadica lincolniana and associated saline wetlands. The results from surveys conducted at the exhibit suggest that art exhibits can have a strong positive impact on members of the community.

Is Competition Making a Comeback? Discovering Methods to Keep Female Adolescents Engaged in STEM: A Phenomenological Approach

The decreasing number of women who are graduating in the Science, Technology, Engineering and Mathematics (STEM) fields continues to be a major concern. Despite national support in the form of grants provided by National Science Foundation, National Center for Information and Technology and legislation passed such as the Deficit Reduction Act of 2005 that encourages women to enter the STEM fields, the number of women actually graduating in these fields is surprisingly low. This research study focuses on a robotics competition and its ability to engage female adolescents in STEM curricula. Data have been collected to help explain why young women are reticent to take technology or engineering type courses in high school and college. Factors that have been described include attitudes, parental support, social aspects, peer pressure, and lack of role models. Often these courses were thought to have masculine and “nerdy” overtones. The courses were usually majority male enrollments and appeared to be very competitive. With more female adolescents engaging in this type of competitive atmosphere, this study gathered information to discover what about the competition appealed to these young women. Focus groups were used to gather information from adolescent females who were participating in the First Lego League (FLL) and CEENBoT competitions. What enticed them to participate in a curriculum that data demonstrated many of their peers avoided? FLL and CEENBoT are robotics programs based on curricula that are taught in afterschool programs in non-formal environments. These programs culminate in a very large robotics competition. My research questions included: What are the factors that encouraged participants to participate in the robotics competition? What was the original enticement to the FLL and CEENBoT programs? What will make participants want to come back and what are the participants’ plans for the future? My research mirrored data of previous findings such as lack of role models, the need for parental support, social stigmatisms and peer pressure are still major factors that determine whether adolescent females seek out STEM activities. An interesting finding, which was an exception to previous findings, was these female adolescents enjoyed the challenge of the competition. The informal learning environments encouraged an atmosphere of social engagement and cooperative learning. Many volunteers that led the afterschool programs were women (role models) and a majority of parents showed support by accommodating an afterschool situation. The young women that were engaged in the competition noted it was a friendly competition, but they were all there to win. All who participated in the competition had a similar learning environment: competitive but cooperative. Further research is needed to determine if it is the learning environment that lures adolescent females to the program and entices them to continue in the STEM fields or if it is the competitive aspect of the culminating activity. Advisors: James King and Allen Steckelberg

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Class Notes for Math 921/922: Real Analysis, Instructor Mikil Foss

Topics include: Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon- Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration, Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon-Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration.

Class Notes for Math 905: Commutative Algebra, Instructor Sylvia Wiegand

Topics include: Rings, ideals, algebraic sets and affine varieties, modules, localizations, tensor products, intersection multiplicities, primary decomposition, the Nullstellensatz

Class Notes for Math 918: Homological Conjectures, Instructor Tom Marley

This course was an overview of what are known as the “Homological Conjectures,” in particular, the Zero Divisor Conjecture, the Rigidity Conjecture, the Intersection Conjectures, Bass’ Conjecture, the Superheight Conjecture, the Direct Summand Conjecture, the Monomial Conjecture, the Syzygy Conjecture, and the big and small Cohen Macaulay Conjectures. Many of these are shown to imply others. This document contains notes for a course taught by Tom Marley during the 2009 spring semester at the University of Nebraska-Lincoln. The notes loosely follow the treatment given in Chapters 8 and 9 of Cohen-Macaulay Rings, by W. Bruns and J. Herzog, although many other sources, including articles and monographs by Peskine, Szpiro, Hochster, Huneke, Grith, Evans, Lyubeznik, and Roberts (to name a few), were used. Special thanks to Laura Lynch for putting these notes into LaTeX.

Class Notes for Math 918: Cohen Macaulay Modules, Instructor Roger Wiegand

Topics covered are: Cohen Macaulay modules, zero-dimensional rings, one-dimensional rings, hypersurfaces of finite Cohen-Macaulay type, complete and henselian rings, Krull-Remak-Schmidt, Canonical modules and duality, AR sequences and quivers, two-dimensional rings, ascent and descent of finite Cohen Macaulay type, bounded Cohen Macaulay type.

Class Notes for Math 918: Local Cohomology, Instructor Tom Marley

Topics include: Injective Module, Basic Properties of Local Cohomology Modules, Local Cohomology as a Cech Complex, Long exact sequences on Local Cohomology, Arithmetic Rank, Change of Rings Principle, Local Cohomology as a direct limit of Ext modules, Local Duality, Chevelley’s Theorem, Hartshorne- Lichtenbaum Vanishing Theorem, Falting’s Theorem.

Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe

Topics include: Topological space and continuous functions (bases, the product topology, the box topology, the subspace topology, the quotient topology, the metric topology), connectedness (path connected, locally connected), compactness, completeness, countability, filters, and the fundamental group.