First Idea – Mathematics of Order and Chaos

Fall Semester. The Mathematics of Order

As humans, we all seem to have a basic tendency to seek order and pattern in the natural world and to make use of order and pattern in the things we create. But exactly what constitutes a pattern, and how can we describe order? In this course, our major goal will be to explore unifying mathematical concepts known as group theory and symmetry. These provide a framework for understanding many of the patterns and kinds of order things can have. To illustrate the mathematics, we will study the natural origins of these ideas and the ways that humans in different cultures have incorporated order and symmetry into their crafts, art, architecture, and music.

Spring Semester. The Mathematics of Chaos

To what extent can we hope to use science and mathematics to understand the natural world? By the end of the 1700s humans had apparently succeeded in explaining and predicting the motions of the planets and it became possible to see the universe as a sort of clockwork mechanism – the ultimate in mathematical orderliness. More recently, however, it has become clear that nature is more subtle than that. In many real-world systems, small changes in starting conditions can lead to very large differences over time. There is often an inherent effective unpredictability and apparent chaos in the long-run behavior. We will explore some simple systems whose chaotic properties can be studied using only high school mathematics and think about their implications.

Second Idea – Math Across Time and Culture

Fall Semester. Mathematics Over Time

What is distinctive about mathematics as a field of human study? Where did the mathematics you learned in high school come from? Why were those topics taught and not others? In this course we will look at some of the development of arithmetic, geometry, and algebra over time. We will take a close look at some of the most important sources, such as the Elements of Euclid, and trace their influence on how mathematics is understood and presented today. Another key focus will be the idea of mathematical proof and its role in the teaching and learning of mathematics.

Spring Semester. Mathematics Across Cultures

To what extent are mathematical ideas common to all human cultures? In this course, we will examine some examples of ways of thinking about the world, crafts, works of art, games, and so forth, from a collection of African, Asian, and Native American cultures. These will share many features with parts of Western mathematics and can be explained and studied using some of that mathematics. But were the people who developed these ideas and objects doing mathematics in the same sense that we do now? What are the implications for how mathematics should be taught and learned in a multicultural world?