MATHEMATICS SAMPLE COURSE DESCRIPTION

GS-WEST

The mathematics program at GSW addresses several fundamental questions:

How does one "know" mathematics?
What components comprise a mathematical structure or system?
How is the beauty and abstraction of mathematics balanced with the world's need (desire) to apply it?

The instructional goals that address these questions are:

To select topics that will develop the mathematical maturity of each student.
To expose students to some of the many different areas of modern mathematical thought and to go beyond those areas typically covered in a standard high school curriculum.
To make sure each student understands the nature of mathematical systems: undefined terms, definitions, postulates, and theorems.
To introduce students to various forms of proof in different mathematical contexts and to impress upon them the centrality of the idea of proof as the fundamental way of knowing mathematics.
To expose students to carefully selected outside speakers, videos, and activities which support the other goals.

Each student will take four different courses, each lasting 3 weeks. A sample of recent courses are:

Group Theory

This course is a general look at the idea of proof in mathematics in the process of which the mathematical structure known as a group will be defined. After looking at many examples of groups, we will use the definition to discover and try to prove properties and structures that are common to any group. We will move on to a more complex structure known as a ring and spend time proving some of its properties.

Knot Theory

This course will introduce students to basic topological concepts via knot theory. A mathematical knot can be thought of as a piece of string knotted with the ends glued together. Primary questions include: Is a given knot the unknot (a circle)? Can a given knot with a certain number of crossings be simplified so that it has fewer crossings? Are two knots topologically the same? How many knots with a given number of crossings are there?

Non-Euclidean Geometry

When is the sum of the measures of the angles of a triangle not equal to 180°? When is a bounded line of infinite length? When is a boundless line of finite length? When is a circle a square? What is the true geometry of physical space? What is mathematical truth? These questions and many others are considered in studying the exciting development of some non-Euclidean geometries.

Discrete Dynamical Systems

A survey of recursively defined processes and their applications to economics, genetics, population, and mathematical chaos. Techniques from analysis, geometry, and linear algebra will be developed and employed to study and describe important parameters and behavior of these models.

Number Theory

Everyone knows that multiplying a number by 0 gives 0, but most people do not know why. In school we are told "that's the way it is". In this course we will return to first principles, the real nuts and bolts of how numbers behave. We will also explore how number theory, a field once considered to be a purely aesthetic pursuit, came to have many applications in the modern era, from plant biology to cryptography.