500

Independent study of a set of related topics agreed to by the student and the instructor.

A survey of software or technology used in teaching mathematics and of current issues in the use of technology in the teaching and learning of mathematics.

The course explores topics in secondary school mathematics from an advanced perspective to better foster deep conceptual understanding in core content areas. Foundational curricular topics include systems of equations, probability, statistical methods, parent functions, complex numbers, and elementary models including their applications to physical and life sciences. Candidates enrolled in the course should have teaching experience in middle/adolescence mathematics or STEM-related disciplines.

Study of the design and evaluation of various models of mathematics curricula including the latest National Council Teachers of Mathematics (NCTM) materials.

A critical study of the literature on current methods of teaching mathematics.

Overview of the development of mathematics education as a discipline, the major influence from other fields and society, trends in recent decades, and the impact on school mathematics.

Graduate level mathematics education course emphasizing connections between mathematics and other STEM disciplines. Mathematical topics include: Vector calculus; Jacobian matrices and their determinants; differentiation and integration of differential forms and applications to physics; generalizations of the fundamental theorem of calculus, including Green's theorem, the divergence theorem, Gauss's theorem, and Stokes' theorem; potential theory. Successful undergraduate completion of Linear Algebra and Calculus III recommended.

Introductory course with emphasis on applications of calculus. Topics include: first order equations, higher order linear equations, undetermined coefficients, variation of parameters, linear systems, bifurcation analysis, series solutions, Laplace transforms, numeric and qualitative techniques, mathematical modeling and applications, and computational technology. Additional topics selected from linear and nonlinear systems as time permits. Prerequisites: MATH 122, 123, 223, 231 or permission from the instructor.

Topology of the real line, measure and integration on the real line, function spaces and their topologies.

Geometric interpretation of the complex plane, analytic functions and power series representations, introduction to Riemann Surfaces, the Cauchy Integral Theorem and Intergral Formula, calculation of integrals by residues, application to potential theory.

An introduction to rings and fields: rings; integral domains; fields; charactersistic of a ring; subrings; ideals; factor rings; ring homomorphisms; polynomial rings; introduction to Galois theory as time permits.

Study of selected applications of abstract algebra. Topics selected from: identification numbers, symmetry, applications to combinatorics, coding theory, cryptography.

This course focuses on a problem with a rich mathematical history, that of solving polynomial equations. Topics selected from: review of rings and fields; construction of the real numbers from the rationals; algebraic and complex numbers; fundamental theorem of algebra; finite fields and integral domains; polynomial rings, including the division algorithm and Euclidean algorithm for polynomials; counting and locating real roots; cubic and quartic equations; introduction to Galois theory.

Topics selected from: congruence relations; arithmetic functions; primitive roots; quadratic reciprocity; sums of squares; continued fractions; Farey sequences; factoring and primality testing; analytic and additive number theory. The course relates directly to the following N.Y.S. Learning Standards for Mathematics, Science, and Technology: mathematical reasoning, numbers and numeration, and patterns/functions.

The addition, multiplication and pigeonhole principles. Permutations and combinations, partitions and distributions; the binomial and multinomial theorems. Generating functions; recurrence relations; principle of inclusion-exclusion; combinatorial algorithms or designs as time permits. The course relates directly to the following N.Y.S. Learning Standards for Mathematics, Science, and Technology: mathematical reasoning, modeling/multiple representation, and pattern/functions.

Graph models; graph structure and representation; trees; connectivity; graph traversals; graph colorings; other topics as time permits. The course relates directly to the N.Y.S. Learning Standards for Mathematics, Science, and Technology: mathematical reasoning, modeling/multiple representation, and pattern/functions.

Neutral geometry; Euclidean geometry; comparison of various models of Euclidean geometry, such as that of Hilbert and the SMSG; comparison of Euclidean geometryto various finite geometries and non-Euclidean geometries; geometry of circles and triangles; other topics such as analytic and transformational geometry as time permits.

Neutralgeometry; hyperbolic geometry; elliptic geometry; various models of hyperbolic and elliptic geometry; comparison of results to those of Euclidean geometry.

A study of varieties and morphisms between them; foundational material from commutative ring theory; the Zariski topology; dimension and degree of varieties; select problems in enumerative geometry.

Detailed study of topological spaces and their invariants. The emphasis will be on the geometric aspects of topology. Topics may include Sperner's lemma and Brouwer Fixed Point Theorem, classification of compact surfaces, application to systems of differential equations.

Given the emphasis on statistics in the mathematics 5 - 12 grades curriculum, and the increased need for teachers competent to teach AP Statistics, mathematics educators need to understand statistical concepts and methods to a greater depth than what is acquired in an undergraduate level statistics course or sequence. In this course, students will learn to understand and explain why certain statistical techniques are used in certain situations and why they work, why certain methods don't work well sometimes, and also common conceptual errors that users and consumers of statistics often make. Students in the course will investigate some current resources that exist to help their students understand statistics, as well as teaching methods and activities that increase student understanding. Prerequisite: a university-level statistics course.

Regression, hypothesis testing and confidence intervals in regression, nonparametric statistics, chi-square tests, sampling, use of activities in probability and statistics.

This course concerns the conic sections: the parabola, ellipse, and hyperbola. It covers their basic definitions and equations, from both a synthetic and analytic perspective, including their equations in polar coordinates. It then moves on to consider the conic sections in light of Bezout's theorem, which concerns the number of points of intersection of two algebraic curves.

Formulation and solution of linear programming problems using graphical and algebraic methods and computer software, sensitivity analysis, and the dual problem; additional topics chosen from network analysis, game theory, and integer programming.

A study of selected topics in the history of mathematics and their applications to teaching. Permission of instructor required.

The week-long academy is dedicated to the development of instructional and curricular materials in K-12 mathematics. Special topics include proof and communication, the use of current technology and manipulatives, and preparation strategies for N.Y.S. Math Assessments.

Selected readings, discussions, reports and problem solutions on a topic of interest in mathematics, mathematics education or statistics.