1) Problem solving involves exploration and discovery. While there is a need for some prior knowledge, it's not necessary to teach the skills to solve the problems first. Teaching occurs during the trials, mistakes, and final understandings after the explorations.
2) My personal feelings towards math will impact my teaching skills and the way my students view me as a competent, comfortable teacher. If it's clear in my actions or expressions that I'm not confident in my skills, this is easily passed on to my students. They in turn may become nervous, unconfident learners.
3) Students, given the proper guidance and problems to solve, will not have to reinvent the wheel. Much of the responsibility falls on the teacher's shoulders to keep the discussion moving at a decent pace, keep the discussion on topic, and keeping the students motivated. It can be done!
4) When a student has struggled with a concept for a long period of time, a good idea may be to have a peer offer to help, or to ask probing questions which might lead the student onto the right track. Sometimes a simple hint, leading question, or statement can guide someone to new insights and possibilities.
5) I read this article so long ago that I can't remember specific examples. I know that the researchers used modeling to assist students' learning; more specifically, they used manipulatives. They also allowed for student discussions in small groups during the problem solving activities.
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