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Reasoning from Data and Chance
Exploring Discrete Math
Investigating Geometry
Learning from Teaching Cases
Implementing Lesson Study
Visualizing Functions
Algebraic Number Theory
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The PCMI 2009 Summer Session has three strands:
 Some Questions and Problems in Arithmetic
 (2 hours per day, 5 days per week)
 This course will investigate questions like these:
 In how many ways can an integer be written as the sum of two squares?
 In how many ways can an integer be written as the sum of four squares?
 What's the probability that an integer picked at random has no perfect square factor?
 What's the probability that two integers picked at random have no common factor?
 Which linear functions f(x) = ax + b (a and b integers) generate infinitely many prime numbers for integer values of x?
 What is the probability that an integer picked at random between 1 and 1020 is a prime number?
The real goal of the course is to answer the following question:
How are all of the above questions related?
In this three week course, you will investigate questions like those above and develop underlying structural similarities among them. In particular, you will see how to use the algebra of sequences and series as a general purpose tool for these investigations. No prior knowledge of number theory or sequences and series is assumed, and by the end of three weeks, you'll be astounded at what you've discovered about arithmetic and its applications to the 712 curriculum.
 Reflecting on Practice: Connections to Research
 (1 hour per day, 5 days per week, plus opportunities for informal sessions in late afternoon and evenings)
 Participants will consider research related to teaching and learning mathematics with a particular focus on student thinking and questions designed to provoke student thinking. The discussion will be grounded in the study of discourse in lessons and classroom practice in both the United States and other countries. Participants will work collaboratively to consider different approaches to questioning that better enable students to learn mathematics.
 download: Reading prior to Week 1 [username/password required]

 Working Groups
 (2 hours, 4 days a week)
 As part of their summer activities, each participant selected for the 2009 Secondary School Teacher Summer 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:

 Reasoning from Data and Chance
 Exploring Discrete Mathematics
 Investigating Geometry
 Learning from Teaching Cases
 Implementing Lesson Study
 Visualizing Functions
 Algebraic Number Theory
