Ph.D. Courses in Mathematics and Mathematics Education

7111: Seminar in Teaching. Seminar on individual study projects concerned with selected problems in the teaching of mathematics. Prerequisite:

7301: Studies in Mathematics. This course provides basic foundations in Mathematics for students entering the doctoral program in Mathematics or the doctoral program in Mathematics Education. This course may be repeated, but cannot apply to a student's degree plan. Prerequisite:

7187: Seminar. This course is repeatable for credit. Students are required to attend weekly research seminars in mathematics and to give at least one research presentation in the seminar during the semester.

7188: Seminar This course is repeatable for credit. Students are required to attend weekly research seminars in Mathematics Education and to give at least one research presentation in the seminar during the semester.

7302 History of Mathematics. A study of the development of mathematics and of the accomplishments of men and women who contributed to its progress. Prerequisite:

7303 Analysis I. This course covers foundations of modern analysis. Topics include: sequences, LimSup, LimInf, Sigma Algebras of sets that include open and closed sets, sequences of functions, pointwise and uniform convergence, lower and upper semi-continuity, Borel sets, outer measure and Lebesgue measure. Prerequisite: Math 4315.

7306 Current Research in Math Education. This course surveys the various current social, political, and economic trends in local, state, national and international settings that are related to research in Mathematics Education. Prerequisite:

7307 Algebra I. Applications of Algebra, topics in modern algebra. Topics covered include permutation groups, symmetry groups, Sylow theorems, and select topics from Ring Theory. Prerequisite: Math 4307

7309 Topology I. A course in point-set topology emphasizing topological spaces, continuous functions, connectedness, compactness, countability, separability, metrizability, CW-complexes, simplicial complexes, nerves and dimension theory. Prerequisite: Math 4330

7313 Analysis II. This course covers the theory of integration with special emphasis on Lebesgue integrals. Topics include: Lebesque integral, Bounded Convergence theorem, differentiation and integration, absolute continuity, Lp spaces. Prerequisite: Math 7303,

7317 Algebra II. A study of the important algebraic structures of rings and fields. Topics covered include rings, ideals, modules, poly­nomial rings, Euclidean algorithm, finite fields, and field extensions. Topics also include an introduction to Galois theory with an emphasis on the geometric applications. Prerequisite: Math 7307

7319 Topology II: Algebraic Topology. This course covers the fundamental concepts and tool of algebraic topology. Topics include the fundamental group, covering spaces, homotopy type, the higher homotopy groups, singular homology theory, the computation of homology groups via exact sequences and applications. Prerequisite: Math 7307 and Math 7309,

7321 Graph Theory. Topics in this course include: Trees, connectivity of graphs, Eulerian graphs, Hamiltonian graphs, planar graphs, graph coloring, matchings, factorizations, digraphs, networks, and network flow problems. Prerequisite: Math 3398

7324 Curriculum Design & Analysis. This course examines, analyzes, and evaluates the various concepts, topics, methods, and techniques that are related to curriculum design in Mathematics Education for grades P-16. Prerequisite:

7325 Statistics I. A study of the mathematical and probabilistic underpinnings of the techniques used in statistical inference. Topics covered include sampling, sampling distributions, confidence intervals and hypothesis testing with an emphasis on both simulations and derivations. Prerequisite: Math 2472, Math 3305

7328 Instructional Techniques & Assessments. This course examines, analyzes, and evaluates the various concepts, topics, methods, and techniques of instruction in Mathematics Education and the related assessment procedures for each for grade levels P-20. Prerequisite:

7331 Combinatorics. This course is a study of fundamental principles of combinatorics. Topics include: Permutations and combinations, the Pigeonhole principle, the principle of inclusion-exclusion, binomial and multinomial theorems, special counting sequences, partitions, posets, extremal set theory, generating functions, recurrence relations, and Polya theory of counting. Prerequisite: Math 3398

7335 Statistics II: Linear Modeling. A study of the formulation and statistical methodologies for fitting linear models. Topics include the general linear hypothesis, least-squares estimation, Gauss-Markov theorem, assessment of model fit, effects of departures from assumptions, model design, and criteria for selection of optimal regression models. Prerequisite: Math 7325, Math 3377

7346 Quantitative Research Analysis in Mathematics Education. (3-0) This course surveys the various research techniques used in quantitative analysis for mathematics education and covers topics such as experimental design, statistical analysis, and use of appropriate design methodologies to achieve the strongest possible evidence to support or refute a knowledge claim. Prerequisite: Math 7306 and Math 7325

ED 7352 Beginning Qualitative Design and Analysis. Introduces the qualitative paradigm. Includes distinctive features, alternative qualitative traditions, purposeful sampling, common data collection methods, inductive analysis, the role of the researcher, and evaluating qualitative research. Prerequisite:

7356 Advanced Topics in Research. This course encompasses investigation, development, and demonstration of competence, design, and execution for Mathematics Education problems.
A: Advanced Quantitative Research. This course encompasses investigation, development, and demonstration of competence, design, and execution for mathematics education problems in quantitative research. Prerequisite: 7346
B: Advanced Qualitative Research. This course encompasses investigation, development, and demonstration of competence, design, and execution for mathematics education problems in qualitative research. Prerequisite: ED 7352

7361  Seminar in Advanced Mathematics.  Material in course will vary with the interest of students and faculty.  A detailed study of subject matter may be chosen from advanced areas of analysis; algebra; topology and geometry; applied mathematics; and probability and statistics.  This course is repeatable for credit when subject matter varies.

7366 Topics in Teaching. This course examines how to develop and teach specialized student-groups.

• A: Teaching Post-Secondary Students. (Developmental Math, Service Courses, and Majors) This course examines how to develop and teach post-secondary students. The course references the recommendations of government agencies and professional organizations and allows for the investigation of research-based models. Prerequisite: Math 7306
• B: Teaching K-12 Students. (Elementary, Middle School, and High School) This course examines how to develop and teach K-12 students. The course references the recommendations of government agencies and professional organizations and allows for the investigation of research-based models. Prerequisite: Math 7306
• C: Teaching Teachers. (In-Service, Pre-Service) This course examines how to prepare teachers of mathematics. The course references the recommendations of government agencies and professional organizations and allows for the investigation of research-based models. Prerequisite: Math 7306
• D: Teaching Specialized Content. This course will be an in-depth study of a specialized content area in mathematics with an emphasis on teaching. The specific content area will vary by instructor. Examples include Euclidean simplex geometry and Discrete Probability Spaces with Implications for Public School Curriculum.

7371 Topics in Discrete Mathematics. In depth study of advanced topics in discrete mathematics. Potential topics include: advanced graph theory, advanced combinatorics, combinatorial number theory, discrete optimization, algorithms and complexity and probabilistic methods.

• A: Advanced Graph Theory. Topics in this course include: Turan's problems, Ramsey theory, random graph theory, extremal graph theory, algebraic graph theory, domination of graphs, distance problems, and applications. Prerequisite: Math7321.
• B: Advanced Combinatorics. Topics in this course include: Block designs, Latin squares, combinatorial optimization problems, coding theory, matroids, difference sets, and finite geometry. Prerequisite: Math 7331.
• C: Combinatorial Number Theory. A study of fundamental techniques in combinatorial number theory. Instructor may choose from the following different topics: Waring's problem, additive number theory, and probabilistic methods in number theory. Prerequisite: Math 7331.
• D: Discrete Optimization. A study of some fundamental techniques in discrete optimization. Topics include: Discrete optimization, linear programming, integer programming, integer nonlinear programming, dynamic programming, location problem, scheduling problem, transportation problem, postman problem, traveling salesman problem, matroids, and NP-completeness. Prerequisite: Math 7321 and Math7331.
• E: Algorithms and Complexity. A study of some fundamental concepts of computability and complexity. Topics include: polynomially bounded problems, NP-complete problems, exponentially hard problems, undecidable problems, and reducibility. Prerequisite: Math7331.
• F: Probabilistic Methods in Discrete Mathematics. A study of some fundamental probabilistic techniques used to solve problems in graph theory, combinatorics, combinatorial number theory, combinatorial geometry, and algorithm. Topics include: linearity of expectation, alterations, second moment, local lemma, correlation inequalities, martingales, Poisson paradigm, and pseudo-randomness. Prerequisite: Math 7321 and Math 7331.

7373 Topics in Applied Mathematics. In depth study of advanced topics in applied mathematics. Potential topics include: systems of differential equations and partial differential equations.

• A: Systems of Differential Equations. Theory and applications of systems of differential equations, including the study of algebraic and qualitative properties of solutions, and numerical methods associated with computation of solutions. Prerequisites: Math 3323, Math 3373, and Math 4315
• B: Partial Differential Equations. Theory and application of partial differential equations; typical equations of mathematical physics; Cauchy problem for equations of the first order; classification of second-order equations; Cauchy problem for second-order hyperbolic equations; Duhamel's principle; potential theory and elliptic equations; maximum principle and parabolic equations. Prerequisites: Math 3323, Math 3373, and Math 3380

7378 Topics in Standards. This course examines the basic principles involved in Mathematics Education. Fundamental themes will be reviewed, researched, and discussed.

• A: Problem Solving, Reasoning, and Proof. A study of the fundamental concepts of problem solving, logic, set theory, and mathematical proof and applications of these concepts in mathematics curriculum for grades P-20. Prerequisite: Math 7306
• B: Connecting and Communicating Math. This course examines one of the basic principles involved in Mathematics Education: Connecting and Communicating Mathematics. This fundamental theme will be reviewed, researched, and discussed. Prerequisite: Math 7306
• C: Representing Fundamental Math Ideas (Function, Data Analysis, and Enumeration). This course examines the basic principles involved in Mathematics Education. The process of representing fundamental mathematics ideas will be reviewed, researched, and discussed. Prerequisite: Math 7306
• D: Math Technologies. This course examines the basic principles involved in Mathematics Education: Technology. This fundamental theme will be reviewed, researched, and discussed. Prerequisite: Math 7306

7385: Independent Study. Student will work directly with a faculty member and develop in-depth knowledge in a specific topic area of mathematics. Topics vary according to student's needs and demands.

7386: Independent Study. Student will work directly with a faculty member and develop in-depth knowledge in a specific topic area of Mathematics Education. Topics vary according to student's needs and demands.

7389 Internship.

7398A: Dissertation. This course represents a Mathematics Education student's first year dissertation enrollment.

7398B: Dissertation. This course represents a Mathematics Education student's dissertation enrollments. The course can be repeated as necessary. The dissertation credit (18 hours) will not be awarded until the dissertation is submitted for binding. Prerequisite: completion of the core and required concentration courses, or approval of student's dissertation advisor.