I have just completed observing a CPM Algebra Connections classroom. The particular lesson that I thought was amazing occurred at the end of Chapter 4 (4.2.4). It was a summary of slope, intercepts, and graphing done through contextual problems. The students worked in well-managed, self-directed study teams facilitated by an instructor using superior questioning techniques. The critical thinking skills these students showed were incredible. During their large group share-out, I heard two to three alternative approaches for each problem that the students explained to their peers. As volunteers spoke, the rest of the class interspersed comments like: "That was sure a more efficient strategy than mine." "How can you justify that your answer really makes sense?" "My strategy was similar to yours except when I got to this point." I never once heard, "We haven't seen a problem like this. How would you start it?"
This seems like problem-solving to the "nth" degree to me. The alternative solutions show deep conceptual understanding. The math discourse, intelligent analysis and critiquing the students were doing via their comments demonstrates the deep thinking taking place during class. This classroom is certainly an example of student perseverance that we have long lacked in math classrooms.