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Data Analysis, Stat, Prob
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The PCMI 2005 Summer Session has three strands:
 Developing Mathematics: Doing it with Differences
 (2 hours per day, 5 days per week)
 Focused on learning mathematics by working problems together, this course explores the fundamental mathematics on a topic that has its roots in the secondary level, and is related to the mathematical theme of the Institute. Careful work on this topic allows teachers (and students) to understand exactly how elementary and more advanced procedures in the specific content area are derived and generalized.
The course is structured so that each participant can work at his/her own level. Those who are more mathematically advanced may be asked to help those with less preparation. The course is conducted by teacher leaders from the PROMYS program at Boston University. The focus of this strand is entirely on mathematics, although opportunity is provided within the course for reflection on the approach used by the instructors and to consider the implications of such an approach for teaching in secondary classrooms.
The topic for the summer is: Many situations, from monthly payments on a car loan to the absorption of drugs into the bloodstream, to Fibonacci numbers, to models of population growth can be described with "difference equations," equations that relate one term in a sequence to previous terms in a regular way. This course will develop several generalpurpose methods for dealing with difference equations, including methods that use algebra, matrix algebra, combinatorics, and the theory of equations. Little by way of background is assumed, but we promise new and beautiful results by the end of week three.
 Reflecting on Practice: Connections to Research
 (1 hour per day, 5 days per week, plus opportunities for informal sessions in late afternoon and evenings)
 After considering research related to teaching and learning mathematics, participants will reflect on the implications of this research for what takes place in classrooms. The discussion will be grounded in the development of lessons, student work, and videos of classroom practice. Participants will conduct small research investigations around some topic of interest in the secondary curriculum.

 Working Groups
 (2 hours, 4 days a week)
 As part of their summer activities, each participant selected for the 2005 Secondary School Teachers Program will be assigned to a small subjectspecific working group, which will prepare an activity or resource for the profession (with the associated mathematics) for piloting during the following year. The working groups are:

 Data Analysis, Statistics and Probability
 Discrete Mathematics
 Geometry
 Learning from Teaching Labs
 Lesson Study
 Visualizing Functions
