An effective problem-solving lesson should include several stages. In the lessons you viewed in this session, you saw the teacher introduce the problem by making connections to students' previous mathematical experiences and providing a context for the problem. After the problem was presented to the class, students worked in groups. The teacher provided guidance as he or she interacted with students, asking questions, modeling thinking for the class, or drawing everyone's attention to key points. In the Staircase problem from Part A, for instance, Mr. Solomon advised students not to rush to find a rule, but rather to explore cases and gather data first. Ms. Cho, in the Fish Derby problem, brought the whole group together to work on interpreting the graph and the area of feasibility.

Both of these examples show that a key component of an effective lesson is the balance between individual/group work and the chance for students to share and compare their approaches, strategies, and solutions in presentations to the whole class. Orchestrating this transition, as well as bringing the class period to a close, summarizing the work that has been done, and setting up the next day's extensions, are also key tasks for the teacher.

Use the information learned in this session to plan a lesson for one of your classes. Remember, for true problem solving to occur, a problem of mathematical significance should be selected in which no solution method is readily apparent at the outset. At the same time, however, all students should have access to the problem. Students should build on their previous knowledge as they begin to solve the problem. A critical goal of the solution process is the deepening and extension of students' understanding of mathematical concepts, along with development of problem-solving expertise. The problems you select and the way you teach them should foster a positive problem-solving disposition in all students.