Mathematics Commons^™

Modeling And Mathematical Analysis Of Plant Models In Ecology, Eric A. Eager

Department of Mathematics: Dissertations, Theses, and Student Research

Population dynamics tries to explain in a simple mechanistic way the variations of the size and structure of biological populations. In this dissertation we use mathematical modeling and analysis to study the various aspects of the dynamics of plant populations and their seed banks.

In Chapter 2 we investigate the impact of structural model uncertainty by considering different nonlinear recruitment functions in an integral projection model for Cirsium canescens. We show that, while having identical equilibrium populations, these two models can elicit drastically different transient dynamics. We then derive a formula for the sensitivity of the equilibrium population to …

An Analysis Of Nonlocal Boundary Value Problems Of Fractional And Integer Order, Christopher Steven Goodrich

Department of Mathematics: Dissertations, Theses, and Student Research

In this work we provide an analysis of both fractional- and integer-order boundary value problems, certain of which contain explicit nonlocal terms. In the discrete fractional case we consider several different types of boundary value problems including the well-known right-focal problem. Attendant to our analysis of discrete fractional boundary value problems, we also provide an analysis of the continuity properties of solutions to discrete fractional initial value problems. Finally, we conclude by providing new techniques for analyzing integer-order nonlocal boundary value problems.

Adviser: Lynn Erbe and Allan Peterson

Commutative Rings Graded By Abelian Groups, Brian P. Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

Rings graded by Z and Z^d play a central role in algebraic geometry and commutative algebra, and the purpose of this thesis is to consider rings graded by any abelian group. A commutative ring is graded by an abelian group if the ring has a direct sum decomposition by additive subgroups of the ring indexed over the group, with the additional condition that multiplication in the ring is compatible with the group operation. In this thesis, we develop a theory of graded rings by defining analogues of familiar properties---such as chain conditions, dimension, and Cohen-Macaulayness. We then study the …

Prime Ideals In Two-Dimensional Noetherian Domains And Fiber Products And Connected Sums, Ela Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

This thesis concerns three topics in commutative algebra:

1) The projective line over the integers (Chapter 2),

2) Prime ideals in two-dimensional quotients of mixed power series-polynomial rings (Chapter 3),

3) Fiber products and connected sums of local rings (Chapter 4),

In the first chapter we introduce basic terminology used in this thesis for all three topics.

In the second chapter we consider the partially ordered set (poset) of prime ideals of the projective line Proj(Z[h,k]) over the integers Z, and we interpret this poset as Spec(Z[x]) U Spec(Z[1/x]) with an appropriate identification. …

Joanne Growney's Poetry-With-Mathematics Blog -- An Appreciation, Gregory E. Coxson

Journal of Humanistic Mathematics

Now is a good time to work on the boundaries of practice and theory, of art and science. We are seeing a rising tide of interest in these boundaries. Witness the growing Bridges movement, which has been exploring the connections between mathematics and the arts. Similarly, JoAnne Growney's blog, Intersections -- Poetry with Mathematics, explores the connections between mathematics and poetry. Through this review, I aim to give readers a taste of what can be found in Intersections as a way of encouraging others, be they mathematicians, poets, or neither, to visit the blog.

The Weak Discrepancy And Linear Extension Diameter Of Grids And Other Posets, Katherine Victoria Johnson

Department of Mathematics: Dissertations, Theses, and Student Research

A linear extension of a partially ordered set is simply a total ordering of the poset that is consistent with the original ordering. The linear extension diameter is a measure of how different two linear extensions could be, that is, the number of pairs of elements that are ordered differently by the two extensions. In this dissertation, we calculate the linear extension diameter of grids. This also gives us a nice characterization of the linear extensions that are the farthest from each other, and allows us to conclude that grids are diametrally reversing.

A linear extension of a poset might …

Generalized Diamond-Alpha Dynamic Opial Inequalities, Nuriye Atasever, Billûr Kaymakçalan, Goran Lesaja, Kenan Taş

Mathematical Sciences: Faculty Publications

We establish some new dynamic Opial-type diamond alpha inequalities in time scales. Our results in special cases yield some of the recent results on Opial's inequality and also provide new estimates on inequalities of this type. Also, we introduce an example to illustrate our result.

Balance With Unbounded Complexes, Edgar E. Enochs, Sergio Estrada, Alina Iacob

Mathematical Sciences: Faculty Publications

Given a double complex X there are spectral sequences with the E₂ terms being either H_I (H_II(X)) or H_II(H_I(X)). But if H_I(X)=H_II(X)=0, then both spectral sequences have all their terms 0. This can happen even though there is nonzero (co)homology of interest associated with X. This is frequently the case when dealing with Tate (co)homology. So, in this situation the spectral sequences may not give any information about the (co)homology of interest. In this article, we …

How Ipads Can Be Used In The Math Classroom To Improve Student Learning, Katie Ann Garrity Smieja

Mathematics Graduate Theses

This study explores how iPads can be used to enhance student learning in the math classroom. The author reviews current literature to determine how iPads are being used to facilitate communication, formative assessment, and promising practices in math education such as discovery learning and use of spreadsheets, graphing calculators, and manipulatives. Additionally, challenges and drawbacks to using iPads in the classroom are examined. Research supports the use of iPads to provide students access to digital content including online discussion boards and practice activities with instant feedback. iPads can also be used as student response systems to engage students in instruction …

Mathematics Classrooms At The Middle School Level: Does Grouping Make A Difference In Achievement, Adam Smieja

Mathematics Graduate Theses

This paper reviewed related research on student achievement based on how students were grouped, homogeneously or heterogeneously. Questions examined were whether students performed better in heterogeneous or homogenous groupings and to what extent did high-, average-, and low-ability students perform better in a particular group. Results showed that high-ability students in homogeneous groupings had higher gains in achievement when compared to their counterparts in heterogeneous groupings. However, gains for these high-ability students were not statistically significant. On the contrary, average-, and low-ability students did show statistically significant gains in achievement in the heterogeneous groupings. Yet, despite their effectiveness, implementation of …

Formulæ For The Number Of Partitions Of N Into At Most M Parts (Using The Quazi-Polynomial Ansatz), Andrew Sills, Doron Zeilberger

Mathematical Sciences: Faculty Publications

The purpose of this short article is to announce, and briefly describe, a Maple package, PARTITIONS, that (inter alia) completely automatically discovers, and then proves, explicit expressions (as sums of quasi-polynomials) for p_m(n) for any desired m. We do this to demonstrate the power of “rigorous guessing” as facilitated by the quasi-polynomial ansatz.

American Mathematical Association Of Two Year Colleges Reform Policies In Practice : Implementing Standards In Classroom Instruction For Basic Skills Mathematics At One Four-Year College In New Jersey, Patricia J. Garruto

Theses, Dissertations and Culminating Projects

High school graduates continue to enter post-secondary education lacking in basic mathematical skills and thus not academically prepared to enroll in college-level mathematics courses (ACT, 2010). Although it can be argued that those mathematical concepts should have been mastered in grades K-12, educating those students in basic skills mathematics has become the responsibility of universities and colleges. Two publications of the American Mathematical Association of Two Year Colleges (AMATYC), Crossroads (1995) and Beyond Crossroads (2006) set forth standards for mathematics programs and courses offered to students during their first two years of postsecondary education, which includes basic skills programs. Those …

Serving Two Masters : A Study Of Quantitative Literacy At Small Colleges And Universities, Jodie Ann Miller

Theses, Dissertations and Culminating Projects

The past twenty years have seen a growing interest in promoting quantitative literacy (QL) courses at the college level. At small institutions, financial realities impose limitations on faculty size and therefore the variety of courses that may be offered. This study examined course offerings below calculus at four hundred twenty-eight small colleges to gain a thorough understanding of the approaches to developing QL among the general population of undergraduate students. Using a three-phase model of examining progressively narrower subsets of QL programs at small institutions, document-based data from college catalogs and communication with mathematics program chairs were studied to summarize …

K-8 Preservice Teachers’ Inductive Reasoning In The Problem-Solving Contexts, Marta Magiera

Mathematics, Statistics and Computer Science Faculty Research and Publications

This paper reports the results from an exploratory study of K-8 pre-service teachers’ inductive reasoning. The analysis of 130 written solutions to seven tasks and 77 reflective journals completed by 20 pre-service teachers lead to descriptions of inductive reasoning processes, i.e. specializing, conjecturing, generalizing, and justifying, in the problem-solving contexts. The uncovered characterizations of the four inductive reasoning processes were further used to describe pathways of successful generalizations. The results highlight the importance of specializing and justifying in constructing powerful generalizations. Implications for teacher education are discussed.

Team-Teaching Experiences Of A Mathematician And A Mathematics Teacher Educator: An Interpretative Phenomenological Case Study, Sarah K. Bleiler

USF Tampa Graduate Theses and Dissertations

In recent years, experts and organizations involved in mathematics education have emphasized the importance of collaboration between mathematicians and mathematics teacher educators as a means of improving the professional preparation of mathematics teachers. While several such collaborative endeavors have been documented in the extant literature, most research reports have focused on the products, rather than the process, of collaboration. The purpose of this interpretative phenomenological case study is to gain an understanding of the lived experiences of a mathematician and a mathematics teacher educator as they engaged in a team-teaching collaboration within the context of prospective secondary mathematics teacher preparation. …

The Mathematics Portfolio: An Alternative Tool To Evaluate Students’ Progress, Marla A. Sole

Publications and Research

This article describes the need for more thorough and varied forms of assessment to evaluate students’ level of understanding in mathematics. Portfolios are one type of assessment tool that, when added to a teacher’s repertoire can improve students’ comprehension and retention and enable students to monitor their own progress and to take more responsibility for their own learning. Portfolio assignments can also help students and teachers to detect and remedy weaknesses and misunderstandings and can increase students’ self-confidence in mathematics. This article discusses what a portfolio is, gives an example of a unit portfolio used in an undergraduate Finite Mathematics …

How To Reach All Types Of Learners Through The Use Of Manupulatives In Grades Three, Four, And Five, Cadie Fairchild

Mathematics Graduate Theses

This paper is an examination of research concerning the use of manipulatives with third, fourth, and fifth grade mathematics instruction. The research discusses the power of child discovery, stages of representation, using manipulatives to support learning, eliminating manipulatives prematurely, and eliminating negative stereotypes linked to manipulative use. A major concern of teaching with manipulatives is the knowledge of the teacher. An educator’s role is to provide the best learning opportunities for students. Research reviewed for this paper indicates frequent hands on activities using manipulatives. They need to be aware of the many learning stages and how to address each individual …

Generalized Borcea-Voisin Construction, Jimmy Dillies

Mathematical Sciences: Faculty Publications

C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi-Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi-Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the …

Combinatorics Using Computational Methods, Derrick Stolee

Department of Mathematics: Dissertations, Theses, and Student Research

Computational combinatorics involves combining pure mathematics, algorithms, and computational resources to solve problems in pure combinatorics. This thesis provides a theoretical framework for combinatorial search, which is then applied to several problems in combinatorics. Some results in space-bounded computational complexity are also presented.

Lecture Hall Sequences, Q-Series, And Asymmetric Partition Identities, Sylvie Corteel, Carla D. Savage, Andrew Sills

Mathematical Sciences: Faculty Publications

We use generalized lecture hall partitions to discover a new pair of q-series identities. These identities are unusual in that they involve partitions into parts from asymmetric residue classes, much like the little Göllnitz partition theorems. We derive a two-parameter generalization of our identities that, surprisingly, gives new analytic counterparts of the little Göllnitz theorems. Finally, we show that the little Göllnitz theorems also involve “lecture hall sequences,” that is, sequences constrained by the ratio of consecutive parts.

Session D-3: Discrete Mathematics: A Great Curriculum Connector, Donald Porzio

Professional Learning Day

Many topics that fall under the umbrella of Discrete Mathematics cut across the traditional high school curriculum areas of algebra, geometry, and pre-calculus. Come try some classroom-ready hands-on Discrete Mathematics activities that illustrate the true interconnectedness of mathematics.

Undergraduate Students' Self-Reported Use Of Mathematics Textbooks, Aaron Weinberg, Emilie Wiesner, Bret Benesh, Timothy Boester

Mathematics Faculty Publications

Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed, and asked to describe how they used their textbook. The results indicate that students tend to use examples, instead of the expository text, to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself, as well as students' beliefs about reading and the …

Professional Learning Community Impacts On Student Achievement In Middle School Mathematics, Jessica Stuewe

Mathematics Graduate Theses

The development of professional learning communities in middle level education is a relatively new and trendy concept sweeping the nation. School professional development committees are training large numbers of teachers through conferences, book studies and online webinars on how to collaborate to improve student learning. This research will explore the impact of the first year of implementation of a departmental professional learning community (PLC) on student achievement at the middle school level in the content area of mathematics. Student standardized test scores from 2010 and 2011 will be compared as a measure of success of the PLC. Teacher surveys are …

What Mathematics Do Elementary Education Teachers Need To Know?, Bret Benesh

Forum Lectures

Almost no one is happy with the state of America's mathematics education. I examine the mathematics textbooks elementary education majors commonly use in college to determine what effect this might be having on their future elementary school students. In this Thursday Forum, I report on what I found in these textbooks---and why I do not like them. I then supply an alternative vision that would better serve our elementary school students.

Incorporating Quantitative Reasoning In Common Core Courses: Mathematics For The Ghost Map, John R. Jungck

Numeracy

How can mathematics be integrated into multi-section interdisciplinary courses to enhance thematic understandings and shared common readings? As an example, four forms of quantitative reasoning are used to understand and critique one such common reading: Steven Berlin Johnson’s "The Ghost Map: The Story of London's Most Terrifying Epidemic - and How it Changed Science, Cities and the Modern World" (Riverhead Books, 2006). Geometry, statistics, modeling, and networks are featured in this essay as the means of depicting, understanding, elaborating, and critiquing the public health issues raised in Johnson’s book. Specific pedagogical examples and resources are included to illustrate applications and …

Parts Of The Whole: Learn More, Learn Better, Dorothy Wallace

Numeracy

Building on previous columns in Numeracy, this column analyzes various teaching techniques in terms of their ability to build cognitive schema, extend existing schema, reinforce learning, move mean understanding of a group of students, and reduce variance in understanding of a group. We offer a pedagogical cycle as an example of how to address multiple learning goals using common teaching methods.

If Only Math Majors Could Write..., Bernard L. Madison

Numeracy

This text of the opening plenary address to the 2011 Summit of the Appalachian College Association and the meeting of the National Numeracy Network makes an argument that quantitative reasoning and writing should be taught together. The argument is set up by noting that humanists have historically banished quantitative issues from their study of the liberal arts and that science, engineering, and mathematics education suffers from lack of approaches to learning that promote complex, deeper understanding, most notably integrative and reflective learning. Therefore, everyone would profit from combining writing and quantitative reasoning. Five more specific reasons are discussed, drawing evidence …

Period Life Tables: A Resource For Quantitative Literacy, Thomas J. Pfaff, Stanley Seltzer

Numeracy

A period life table provides an estimate of the probability that a person will die at a particular age. Using data available online, we examine tables of expected years to live for males and females against age for three populations: the United States in 2007, the U.S. at the turn of the twentieth century, and the Roman Empire. Scatter plots of males and females for each population show how life expectancy increases with age (e.g., U.S. 2007: 50 year-old female > 40 year-old female > 45 year-old male). The three data sets allow historical comparisons (e.g., of gender disparity, larger now; of …

Motivation For Achievement And Attitudes Toward Mathematics Instruction In A Required Calculus Course At The Norwegian University Of Science And Technology, Donna Sundre, Carol Barry, Vidar Gynnild, Erin Tangen Ostgard

Numeracy

This study from the Norwegian University of Science and Technology (NTNU) examines students’ learning goals and attitudes toward mathematics in a first-year calculus course in undergraduate engineering education. Achievement motivation research using the Achievement Goal Questionnaire (AGQ) is advanced from current literature with two additions: (1) a course specific context using introductory college calculus students, and (2) participation of Norwegian students.

Pre- and posttest measures of attitudes indicate that students do change learning goals over time, unfortunately opposite to the instructors’ aspirations. A significant increase in “Mastery Avoidance” and “Work Avoidance” was accompanied with a drop in “Mastery Approach” and …

Quantitative Literacy Interventions At University Of Cape Town: Effects Of Separation From Academic Disciplines, Vera Frith

Numeracy

The aim of the Numeracy Centre at the University of Cape Town is to develop students’ quantitative literacy (QL) in a manner consistent with their programmes of study and intended roles in the community. Our theoretical perspective on the nature of QL is in line with that of the New Literacies Studies and sees academic QL as practices in different academic disciplinary contexts. This means that for us the ideal curriculum structure for developing QL would fully integrate it into the teaching of the disciplines. This is in practice not achievable in most cases, especially since many students do not …