The real value of problem solving, he concludes, is in the ideas it produces. Thinking abstractly about ideas increases the flexibility of one s thinking capacity. Teachers should choose problems that are just beyond the solver s skill level. In fact, students who haven t yet mastered computations should be allowed to do open-ended problem solving. Nonroutine problems can provide ample opportunity to build computation skills while engaging the student in more challenging mathematics and higher-order thinking.
All students deserve the opportunity to develop their problem-solving ability.
Working collaboratively is a key feature of open-ended problem solving, though students will also work individually. To solve a problem, students draw on their previous knowledge and experience with related problems. Students will discuss the problem, identifying its features, considering possible solution methods, conjecturing, and explaining their thinking. Students who reflect on what they do and communicate with others about it are in the best position to build useful connections in mathematics. Through it all, the solver constructs a mathematical understanding of the problem that is both deep and flexible.