Goal: To engage high school students in the problem-solving process used in this set.

Need: To develop a strategy for implementing this process within the restrictions of the classroom setting.

1. The Problem-Solving Cycle:
1. Stage 1: Familiarization with problem; Discover patterns; Attempt various solutions.
2. Stage 2: Find an intermediate solution, i.e. numerical value, summation or recursive formula.
3. Stage 3: Formalize solution, i.e. Closed formula.
4. Stage 4: Analyze solution; Proofs, Reflection, Connections, Communication.
5. Stage 5: Apply generalization to new situation.
2. Classroom Challenges:
1. Time limitations
2. Student attenuation
3. Pacing among students
3. Challenge Helpers
1. Timing
1. In order to control the amount of time involved in doing a classroom problem-solving exercise, the use of time must be very tightly structured. The greatest drawback here is that students work at very different pacing in coming to an understanding or realization of a problem and its solution. To begin with, students should be given part of a problem, or a very close variation of the problem the night before to ponder over and become familiar with. To insure that students do take the time to familiarize themselves, class should begin with a brief quiz asking: "What did you discover about your problem last night?"
2. Secondly, we must set reasonable expectations on what can be accomplished during the class hour. This may be unrealistic, but I believe that students should be able to complete stages A & B, and potentially, A through C, during one class day. A second hour should be devoted to C through E or D & E, depending on the pace of the class. Stage E should not take much time at all, but somewhat be a review of what has been learned. Stages C and D should receive the most attention.
3. Setting the goal of completing A-C in 50 minutes requires allowing 10-15 minutes for each stage. To accomplish C-E on day 2, allow 20 minutes for C&D, 10 for E.
4. A Proposed Design
1. Divide students into groups of four. Ideally, students have already been taught how to properly function in groups and are familiar with playing different roles, etc.
2. Each group desk could have a flagpole of sorts which would be used to communicate their status to the teacher. For example the flag has levels to mark Stage 1, Stage 2, ..., and a place for HELP! The teacher could also use this pole to communicate expectations to the group without disrupting their conversations. For example, letting a group know if they are going to be asked to present their findings to the class soon.
3. While students are working, the teacher needs a prepared list of questions which would reveal to her the students' understanding at various stages of the process. With this information she can assess whether she needs to intervene with the class or whether a particularly group would do well in explaining material to the rest.
4. Within the time structure for Day 1, 10 minutes could be allowed for group work on a stage. At the end of the stage, 5 minutes would be designated for sharing results and asking questions. This would keep all groups closer to keeping a good pace, and would perhaps foster a bit of competition among the groups to work as quickly and focused as possible. It wouldn't really "give away" anything, since only little bits and pieces would be shared at a time.
5. Evalution
1. Ideally, evaluation would take the form of an individual, written communication where the student expresses in prose and symbol the process and results of the two day activity.