The PCMI 2006 Summer Session has three strands:

Developing Mathematics: Some Applications of Geometric Thinking

(2 hours per day, 5 days per week)

While geometry is a wonderful field in its own right, geometric ideas are also useful tools for studying and understanding all kinds of phenomena inside and outside mathematics. Teachers in this course will look at some basic geometric habits of mind like studying continuous change and looking for things that don't change, and they'll apply these habits to a wide variety of situations. For example, we may look at

• Geometric optimization: what are some general-purpose tools for minimizing paths or maximizing area?

• Geometric invariants: what attributes of a figure stay the same when the figure is changed in some regular way?

• The fundamental theorem of algebra: what does it say, why is it fundamental, and why is it true?

• Fixed points: If you pick a positive number, take its square root, take the square root of that, and keep up the good work, what happens and why?

• What geometric thinking is common to all these and similar questions?

No background with any of these questions will be assumed. One of the tools we may use in the course is a dynamic geometry environment (like The Geometer's Sketchpad® or Cabri). Here, too, we'll start from scratch. By the end of week three, we promise some beautiful and exciting applications of geometry.

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. Participants will work collaboratively to develop teaching and learning resources in order to implement ideas from their discussion. The focus will be on teaching strategies that use problem-solving and open-ended approaches.

 

Working Groups

(2 hours, 4 days a week)

As part of their summer activities, each participant selected for the 2006 Secondary School Teacher Summer Program will be assigned to a small subject-specific 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 and Data and Chance
• Exploring Discrete Mathematics
• Investigating Geometry
• Learning from Teaching Cases
• Implementing Lesson Study
• Visualizing Functions
• Advanced Geometry
• Teacher Professional Continuum