Mathematics Commons^™

Extending K-8 Mathematics Concepts In Alternate Bases, Dianna J. Spence

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

When learning to represent mathematics with manipulatives, many pre-service K-8 teachers rely on memorized rote procedures to perform the associated mathematical tasks; then they arrange the manipulatives to match their result, often with minimal understanding of underlying mathematical connections. In a Number and Operations course for K-8 pre-service teachers, a portion of the class was conducted in alternate bases: Base 6 and Base 8 Blocks were used to model operations with integers to facilitate deeper understanding of the number systems and arithmetic processes being represented. Fractions and decimals were later covered only in Base 10. On midterm and final exams, …

Proceedings Of The Fourth Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Introductory Texts

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 1st Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents

Investigating Mathematical Literacy Through Teacher Language, Alyson Lischka

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Communication about mathematical concepts using appropriate terminology is a standard established by the National Council of Teachers of Mathematics. However, international test results show that United States’ students are lagging in mathematical literacy. This case study analyzes the ways in which instructors use language to help students move toward conceptual understanding of mathematical vocabulary. Three mathematics education professors at a mid-size four year institution were observed teaching math classes to students enrolled in elementary or secondary certification programs. Collected data included: audio-recorded observations and field notes, lesson artifacts such as quizzes and handouts, and audio-recorded interviews with each participant. Findings …

Generalized Complex Hamiltonian Torus Actions: Examples And Constraints, Thomas Baird, Yi Lin

Mathematical Sciences: Faculty Publications

Consider an effective Hamiltonian torus action T×M→M on a topologically twisted,generalized complex manifold M of dimension 2n. We prove that the rank(T)≤n−2 and that the topological twisting survives Hamiltonian reduction. We then construct a large new class of such actions satisfying rank(T)=n−2, using a surgery procedure on toric manifolds.

Rademacher-Type Formulas For Partitions And Overpartitions, Andrew Sills

Mathematical Sciences: Faculty Publications

A Rademacher-type convergent series formula which generalizes the Hardy-Ramanujan-Rademacher formula for the number of partitions of n and the Zuckerman formula for the Fourier coefficients of ϑ4_0 | τ_−1 is presented.

Some Implications Of Chu's 10Ψ10 Generalization Of Bailey's 6Ψ6 Summation Formula, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

Lucy Slater used Bailey's 6ψ6 summation formula to derive the Bailey pairs she used to construct her famous list of 130 identities of the Rogers-Ramanujan type.

In the present paper we apply the same techniques to Chu's 10ψ10 generalization of Bailey's formula to produce quite general Bailey pairs. Slater's Bailey pairs are then recovered as special limiting cases of these more general pairs.

In re-examining Slater's work, we find that her Bailey pairs are, for the most part, special cases of more general Bailey pairs containing one or more free parameters. Further, we also find new …

Homological Dimensions And Regular Rings, Alina Iacob, Srikanth B. Iyengar

Mathematical Sciences: Faculty Publications

A question of Avramov and Foxby concerning injective dimension of complexes is settled in the affirmative for the class of noetherian rings. A key step in the proof is to recast the problem on hand into one about the homotopy category of complexes of injective modules. Analogous results for flat dimension and projective dimension are also established.

Interior-Point Algorithms For A Class Of Convex Optimization Problems, Goran Lesaja, Verlynda Slaughter

Mathematical Sciences: Faculty Publications

In this paper we consider interior-point methods (IPM) for the nonlinear, convex optimization problem where the objective function is a weighted sum of reciprocals of variables subject to linear constraints (SOR). This problem appears often in various applications such as statistical stratified sampling and entropy problems, to mention just few examples. The SOR is solved using two IPMs. First, a homogeneous IPM is used to solve the Karush-Kuhn-Tucker conditions of the problem which is a standard approach. Second, a homogeneous conic quadratic IPM is used to solve the SOR as a reformulated conic quadratic problem. As far as we are …

Lifting Bailey Pairs To Wp-Bailey Pairs, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

A pair of sequences (α_n(a,k,q),β_n(a,k,q)) such that α₀(a,k,q)=1 and β_n(a,k,q)=∑ⁿ_j=0 (k/a;q)_n−j (k;q)_n+j / (q;q)_n−j (aq;q)_n+j α_j (a,k,q) is termed a WP-Bailey Pair. Upon setting k=0 in such a pair we obtain a Bailey pair.

In the present paper we consider the problem of “lifting” a Bailey pair to a WP-Bailey pair, and use some …

Rogers-Ramanujan Computer Searches, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

We describe three computer searches (in PARI/GP, Maple, and Mathematica, respectively) which led to the discovery of a number of identities of Rogers–Ramanujan type and identities of false theta functions.

Math Island At Brewer Elementary School: A Learner-Centered Model For Education, Vikki K. Collins, H. Marguerite Yates

Georgia Educational Researcher

This paper examines a learner-centered model for the teaching of elementary mathematics established at an urban Georgia school that serves a highly mobile, diverse population of approximately 475 students. The school was on its state’s list of schools needing improvement because students continued to perform poorly on standardized tests. Math Island, a support center for students designed to facilitate the conceptual understanding of mathematics, was created, and professional development programs were implemented to assist teachers in developing their pedagogical and content expertise in the teaching of mathematics. The school made substantial progress after these initiatives were employed and was removed …

Some More Identities Of Rogers-Ramanujan Type, Douglas Bowman, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

In this we paper we prove several new identities of the Rogers-Ramanujan-Slater type. These identities were found as the result of computer searches. The proofs involve a variety of techniques, including series-series identities, Bailey pairs, a theorem of Watson on basic hypergeometric series, generating functions and miscellaneous methods.

Interpolation Theorems For Self-Adjoint Operators, Shijun Zheng

Mathematical Sciences: Faculty Publications

We prove a complex and a real interpolation theorems on Besov spaces and Triebel-Lizorkin spaces associated with a self adjoint operator L, without assuming the gradient estimate for its spectral kernel. The result applies to the cases where L is a uniformly elliptic operator or a Schrödinger operator with electro-magnetic potential.

Content Knowledge And Pedagogical Content Knowledge Of Algebra Teachers And Changes In Both Types Of Knowledge As A Result Of Professional Development, Joy W. Black

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

In seeking to improve the mathematics education of all students, it is important to understand the connection between the content knowledge and pedagogical content knowledge of mathematics and how professionals can influence growth in both of these types of knowledge. We do not have an answer about the interplay of content knowledge and pedagogical content knowledge in successful instructional practices in the mathematics classroom. This study involves assessing the content knowledge and pedagogical content knowledge of secondary teachers of Algebra I. In addition, how are these types of knowledge expressed in instructional practices? Last, how do content knowledge, pedagogical content …

Authentic Discovery Projects In Statistics, Dianna J. Spence, Robb Sinn

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

We report the activities and findings of a 3-year project, ―Authentic, Career-Specific, Discovery Learning Projects in Introductory Statistics,‖ funded by the National Science Foundation. The project scope includes: 1) development of teaching materials for using discovery learning projects to teach statistics; 2) training secondary teachers to use the materials developed; 3) evaluation of student outcomes, in both content knowledge and attitudes toward statistics; and 4) extending and refining teacher training. With input from an interdisciplinary team of instructors, materials were developed to assist the teacher in facilitating collaborative discovery projects using linear regression techniques and comparison techniques with appropriate t-tests. …

Establishment Of Weak Conditions For Darboux-Goursat-Beudon Theorem, Mekki Terbeche, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

This paper is devoted to the establishment of weak conditions for Darboux-Goursat-Beudon (DGB) theorem in order to improve analogous results in [1, 7]. By adapting a technique proposed by [7] in another setting [8] via majorant method, we obtain the generalization of DGB theorem.

Introducing Interior-Point Methods For Introductory Operations Research Courses And/Or Linear Programming Courses, Goran Lesaja

Mathematical Sciences: Faculty Publications

In recent years the introduction and development of Interior-Point Methods has had a profound impact on optimization theory as well as practice, influencing the field of Operations Research and related areas. Development of these methods has quickly led to the design of new and efficient optimization codes particularly for Linear Programming. Consequently, there has been an increasing need to introduce theory and methods of this new area in optimization into the appropriate undergraduate and first year graduate courses such as introductory Operations Research and/or Linear Programming courses, Industrial Engineering courses and Math Modeling courses. The objective of this paper is …

Combinatorics Of Ramanujan-Slater Type Identities, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We provide the missing member of a family of four q-series identities related to the modulus 36, the other members having been found by Ramanujan and Slater. We examine combinatorial implications of the identities in this family, and of some of the identities we considered in Identities of the Ramanujan-Slater type related to the moduli 18 and 24.

On Fourier Parabolic And Wave Equation, Mekki Terbeche, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

This paper is devoted to a diffusion (heat) and wave equations for the function of two independent variables. We establish a criterion for existence and uniqueness of the solution of Fourier parabolic equation using Taylor series. A formal solution for a wave equation is investigated.

Proceedings Of The Third Annual Meeting Of The Georgia Association Of Mathematics Teacher Educators Front Matter

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Contents of 3rd Annual GAMTE Proceedings Front Matter:

• Proceedings Committee
• Officers of GAMTE
• Purposes and Goals of GAMTE
• Table of Contents
• Letter from President

Analysis Of Achievement For Understanding Geometry, Annita W. Hunt

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

The purpose of this study was to investigate the effectiveness of a mathematics professional development course. More specifically, in this study we examine whether geometric experiences have an impact on level of performance in mathematics. The van Hiele (Fuys, D., Geddes, D., & Tischler, R., 1988) model of geometric understanding provided a research framework from which to view geometric understanding. This model suggests five levels of understanding that should be taken into consideration when examining levels of geometric thinking: Visual, Descriptive/Analytic, Abstract/Relational, Formal Deduction/Proof, and Rigor. The sample under study was three cohorts of practicing elementary teachers and mathematics coaches …

Gorenstein Flat Dimension Of Complexes, Alina Iacob

Mathematical Sciences: Faculty Publications

We define a notion of Gorenstein flat dimension for unbounded complexes over left GF-closed rings. Over Gorenstein rings we introduce a notion of Gorenstein cohomology for complexes; we also define a generalized Tate cohomology for complexes over Gorenstein rings, and we show that there is a close connection between the absolute, the Gorenstein and the generalized Tate cohomology.

On Some Differential Equations, Mekki Terbeche, Broderick O. Oluyede

Mathematical Sciences: Faculty Publications

This paper investigates Cauchy and Goursat problems for partial differential operators. Successive approximation techniques for partial differential equations and the estimated results are employed to obtain the existence and the uniqueness of the solutions of such problems. An extended Darboux-Goursat-Beudon problem is studied.

Ramanujan–Slater Type Identities Related To The Moduli 18 And 24, James Mclaughlin, Andrew Sills

Mathematical Sciences: Faculty Publications

We present several new families of Rogers–Ramanujan type identities related to the moduli 18 and 24. A few of the identities were found by either Ramanujan, Slater, or Dyson, but most are believed to be new. For one of these families, we discuss possible connections with Lie algebras. We also present two families of related false theta function identities.

Rogers-Ramanujan-Slater Type Identities, James Mclaughlin, Andrew Sills, Peter Zimmer

Mathematical Sciences: Faculty Publications

In this survey article, we present an expanded version of Lucy Slater's famous list of identities of the Rogers-Ramanujan type, including identities of similar type, which were discovered after the publication of Slater's papers, and older identities (such as those in Ramanujan's lost notebook) which were not included in Slater's papers. We attempt to supply the earliest known reference for each identity. Also included are identities of false theta functions, along with their relationship to Rogers-Ramanujan type identities. We also describe several ways in which pairs/larger sets of identities may be related, as well as dependence relationships between identities.

Virtual Vs. Concrete Manipulatives In Mathematics Teacher Education: A Call For Research, Annita W. Hunt

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Are virtual manipulatives as effective as concrete (hands-on) manipulatives to build conceptual understanding of number concepts and relationships in pre-service middle grades teachers? In the past, the use of concrete manipulatives in mathematics courses for Clayton State University’s preservice middle grades teachers has proven to be a very effective way to build conceptual understanding of a variety of mathematical topics. This paper presents an argument for the need for research into the usefulness of virtual manipulatives for enhancing mathematics teacher education and their potential to supplement (or replace?) concrete manipulatives.

Actively Engaging Pre-Service Teachers In Geometry And Measurement, Kadian M. Callahan

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

Current work in mathematics education suggests that the learning experiences in which teachers engage during undergraduate study influences their knowledge of and beliefs about mathematics and the ways in which they will teach (Allen, et. al., 2008; CBMS, 2001; Hill, Rowan, & Ball, 2005; National Research Council, 2001). However, very little is known about pre-service teachers’ learning experiences and how those experiences influence their thinking about mathematics teaching and learning. The classroom excerpt described here attempts to illuminate how pre-service, elementary teachers’ active engagement in the learning of geometry and measurement influences their mathematical power: a positive disposition toward mathematics, …

The Paaps Strategy For Teaching Mathematics Content, Greg Harrell

Proceedings of the Annual Meeting of the Georgia Association of Mathematics Teacher Educators

In the fall of 2005, I started teaching the mathematics content course Algebra and Geometry for Teachers. The majority of students in the course are pre-service middle school teachers. Instead of teaching the course by demonstrating rigorous proofs, I wanted to use teaching strategies that would build the students’ content knowledge and connect to their roles as future mathematics

teachers. I chose to make problem solving a focal process standard by having students problem- solve for a majority of classroom time. In addition, the students complete a major project entitled

“Provide, Attempt, and Assess Problem Solving” or PAAPS. For PAAPS, …

A Note On Extreme Bernoulli And Dependent Families Of Bivariate Distributions, Broderick O. Oluyede, Marvis Pararai

Mathematical Sciences: Faculty Publications

The objective and purpose of this note is to generate bivariate distributions via extreme Bernoulli distributions and obtain results on positively and negatively dependent families of bivariate binomial distributions generated by extreme Bernoulli distributions. Some distributional properties and results are presented. The factorial moment generating functions, correlation functions, conditional distributions and the regression functions are given.

A Partition Bijection Related To The Rogers-Selberg Identities And Gordon's Theorem, Andrew Sills

Mathematical Sciences: Faculty Publications

We provide a bijective map from the partitions enumerated by the series side of the Rogers–Selberg mod 7 identities onto partitions associated with a special case of Basil Gordon's combinatorial generalization of the Rogers–Ramanujan identities. The implications of applying the same map to a special case of David Bressoud's even modulus analog of Gordon's theorem are also explored.