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{\it First Idea -- Mathematics of Order and Chaos}
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{\bf Fall Semester.  The Mathematics of Order}
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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.  
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{\bf Spring Semester.  The Mathematics of Chaos}
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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.  
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{\it Second Idea -- Math Across Time and Culture} 
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{\bf Fall Semester.  Mathematics Over Time}
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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 {\it 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.
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{\bf Spring Semester.  Mathematics Across Cultures}
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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?    
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