MATH SAMPLE COURSE DESCRIPTION

GS- East

The purpose of the mathematics program is to encourage students to re-evaluate their conception of mathematics. Most of the students likely find school mathematics to be easy, but their experience may be limited to symbol manipulation and formulaic approaches to thinking about mathematics. The mission of the math department is to expose students to truly interesting mathematical ideas and processes and to inspire them to stretch their understanding of what mathematics encompasses. In math classes, students are encouraged and prodded to discover the reasons why statements are true and to explore the logic and connections across mathematical disciplines. Although the primary focus is on the 60 mathematics students during the math classes that they take, a secondary focus is to challenge the entire student body through activities and discussions designed to expand their view of mathematics.

Generally, the mathematics program offers eight courses that can be classified into two main fields: applied mathematics (with courses such as “Dynamic Systems Modeling,” “Game Theory,” “Problem Solving,” and “The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number”) and pure mathematics (“Advanced Euclidean Geometry,” “Number Theory,” “Survey of Geometries,” and “Infinity and Beyond”). Guest speakers address topics such as “Godel’s Incompleteness Theorems,” “Topology” and “Knot Theory, Chaos and Fractals.” Mathematics students have also collaborated with instrumental music students and dance students to create and perform original works based on various mathematical principles and patterns. Students also participate in department-wide team competitions as well as having the opportunity to participate in a campus-wide mathematics contest that spanned the entire six weeks of GSE.

GS-West

Instructional Goals:

1. To select topics which will develop the mathematical maturity of each student
2. To expose students to some of the many different areas of modern mathematical study, especially those areas not included in the standard high school curriculum
3. To share with each student the nature of a mathematical system: undefined terms, definitions, axioms, and theorems
4. To make various forms of proof a part of the experience of every student
5. To expose students to carefully selected outside speakers, videos, and other appropriate activities

Courses Offered

Each course is a 3-week course. Each student takes four of the following courses:

1. Group Theory
The properties of a group are defined, including group multiplication tables and graphs. Then, depending on the interests of the majority of the students, group theory is applied to describe all 32 crystallographic classifications of molecules or to discuss the Special Relativistic Lorentz group (or both).

2. Non-Euclidian Geometry
The notion of what constitutes a "flat” vs. a "curved" geometry is discussed. The class explores such questions as "When are the sum of the interior angles of a triangle not equal to 180 degrees?" and "Under what circumstances do parallel lines eventually cross?" The notion of curvature is then applied in a discussion of General Relativity and Black Holes. "

3. Number Theory
Number theory reveals the true beauty of mathematics by studying the properties of the integers. This course covers some of the widely known theorems, conjectures, unsolved problems, and proofs of number theory. Topics covered may include divisibility, prime numbers, perfect numbers, congruencies, diophantine equations, and arithmetic functions. Fermat’s Last Theorem is discussed as well as Andrew Wiles’ journey to prove the theorem.

4. Knot Theory
How can we determine if two knots are geometrically the same? When can a knot be untied? These are among the questions explored in knot theory, a branch of the mathematical field called topology. In topology, students have the freedom to stretch, rotate, shrink, and flip geometric figures thus allowing teacher and students to study and compare knots to one another.

5. Matrix Theory
This course introduces the concepts of matrices and their properties. The characterization of a wide variety of matrices gives rise to both its theory and structure. Close scrutiny is given to the study of linear systems (Ax = b), determinants, symmetric matrices, eigenvalues and eigenvectors, characteristic polynomials, diagonalization, Cayley’s Theorem, and LU- and QR- factorization techniques.

6. Probability
The course introduces the basic principles of the theory of probability and its applications. Topics include sample spaces, techniques of combinatorial analysis used in computing probabilities, the axioms of probability, conditional probability, independent events, discrete and continuous random variables, mathematical expectation, and a study of some of the classical probability distributions.