Mathematics

This course reviews arithmetic and covers algebraic expressions and equations; their manipulation and use in problem solving; word problems; and an introduction to inequalities, absolute values, and graphing. Prerequisite: satisfactory placement test score. (4 credits)

This is a complete course in first-semester calculus. Topics include the meaning, use, and interpretation of the derivative; techniques of differentiation; applications to curve sketching and optimization in a variety of disciplines; the definite integral and some applications; and the Fundamental Theorem of Calculus. Students enrolling for graduate credit participate in weekly pedagogical seminars designed for current and future K-12 teachers. Prerequisite: MATH E-10, or the equivalent, or satisfactory placement test score. (4 credits)

This course covers the following topics: calculus of functions of several variables; vectors and vector-valued functions; parameterized curves and surfaces; vector fields; partial derivatives and gradients; optimization; method of Lagrange multipliers; integration over regions in R² and R³; integration over curves and surfaces; Green's theorem, Stokes's theorem, divergence theorem. Graduate-credit students may be asked to complete additional projects consistent with the course material and their backgrounds. Prerequisites: MATH E-16, or the equivalent; placement test is recommended. (4 credits)

This course is an integrated treatment of linear algebra and multivariable differential calculus, with an introduction to manifolds. Students are required to learn 22 important proofs. Prerequisite: a grade of A in MATH E-16, or the equivalent. (4 credits)

This course is an introduction to the beauty and power of number theory. Ideal for inservice teachers and those aspiring to take discrete mathematics or abstract algebra, it emphasizes the various proof techniques and counterexamples, and presents many unsolved problems that have been studied for centuries. Once thought to be the purest of pure mathematics, number theory has a new dimenson since the advent of modern technology: constant practical applications, including computer security checks, such as check digits in UPCs, zip codes, ISBNs, ISSNs, EANs, VINs, credit card numbers, and library book numbers, to name a few. Topics include recursion, Pascal's triangle, binomial theorem, figurate numbers, Fibonacci and Lucas numbers, Fermat numbers, division algorithm, prime and composite numbers, Euclidean algorithm, gcd, lcm, linear diophantine equations, congruences, Pollard Rho factoring method, divisibility tests, perpetual calendar, Chinese remainder theorem, Wilson's theorem, Fermat's little theorem, Euler's theorem, perfect numbers, and Mersenne numbers. Graduate-credit students are expected to write programs in a structured language such as Java. Prerequisites: precalculus and mathematical maturity; knowledge of a structured language such as Java for graduate-credit students; placement test is recommended. (4 credits)

This course develops the theory of convex sets, normed infinite-dimensional vector spaces, and convex functionals and applies it as a unifying principle to a variety of optimization problems such as resource allocation, production planning, and optimal control. Topics include Hilbert space, dual spaces, the Hahn-Banach theorem, the Riesz representation theorem, calculus of variations, and Fenchel duality. Students are expected to understand and invent proofs of theorems in real and functional analysis. Prerequisites: MATH E-21a and MATH E-21b, or the equivalent, plus at least one other more advanced course in mathematics. (4 credits)

This course examines the mathematical underpinnings behind what is taught in secondary level algebra courses. It considers what, why, and how we teach what we teach, and investigates different strands of algebraic competence with particular emphasis on how we assess the students' proficiency in these various strands. Prerequisite: familiarity with K-12 mathematics. (4 credits)

This is a course in both linear algebra and group theory that develops the general structure and formulation of geometries over sets in terms of their transformation groups (Klein Erlangen Programme). By introducing elementary notions of Galois theory and symmetric groups, we also prove the impossibility of solving by radicals the quintic or higher order polynomial equations (in a single variable). Prerequisites: familiarity with K-12 mathematics, including calculus and proof methodology. (4 credits)

The difficulty of both learning and teaching math is evident in its history. The struggle of early research mathematicians who developed and formalized a topic parallels the struggle of students and teachers in the modern classroom. Students learning about the concept of limits and series undergo a similar process as the pioneers of calculus did when they developed the subject. Archimedes, Zeno, Cavalieri, Newton, Leibniz, and Cauchy had to find or invent structure. This struggle goes on today, as new flavors of calculus are developed and studied. Each week, this course considers a different math subject and pinpoints a moment when something interesting happened. This moment is condensed into a specific and concrete mathematical problem. Each story is tied to a particular mathematician. Prerequisite: single variable calculus is helpful. (4 credits)