Teaching+through+problems

Lessons:

 * Counting the fish -- Sample lesson by Vivian Shell and Diane Crim
 * Geometric radicals -- Shamelessly borrowed from Henri Picciotto's Geometry Labs - activities for grades 8-11. His website has a lot of great stuff: @http://www.mathedpage.org/
 * [|Cantor's carousel.pdf] -- Troy, Brian and Carrie's worksheet

Working definition:
Teaching through problems means posing a problem whose solution will necessitate development of mathematics new to the student. The work of solving the problem is done by the student, so that the mathematics is developed by the learner.

Questions:

 * What does it mean to teach through problems?
 * What are the goals we have for teaching through problems?
 * In what ways can we support our students so that they are willing to engage in the activities we prepared for them?
 * What kinds of questions should we prepare and prepare for as we are building the activities?
 * What is this "continuum" of teaching with problems?
 * How do "teaching through problems" and "teaching problem solving" differ, if at all?
 * What are the objectives of teaching through problems?
 * Are there structures a teacher can put in place so that the "reluctant" learner engages in the problems posed?
 * What kinds of questions does the teacher pose to ensure the learning: types, order, level?