Defining Problem Solving
 Introduction | Posing Problems in a Variety of Contexts | Mathematical Stories | Assessing Student Thinking | Teacher's Role | Your Journal
 "The decisions that teachers make about problem-solving opportunities influence the depth and breadth of students' mathematical learning. Teachers must be clear about the mathematics they want their students to accomplish as they structure situations that are both [challenging] and attainable for a wide range of students." (NCTM, 2000, p. 119) Planning The teacher's role begins with selecting rich problem-solving tasks that focus on the mathematics the teacher wants his or her students to explore. Problem solving should be embedded throughout the mathematics content curriculum. When problem solving becomes the platform for developing mathematical ideas in a way that makes sense to students, they also begin to recognize the usefulness of different problem-solving strategies. Students may find several different ways to solve a problem. When this happens, teachers have an excellent opportunity to prompt students to discuss and compare their approaches. What is similar? What is different? How did this variety of problem-solving strategies lead to the solution? It is important to realize that the process is just as important, if not more important, than arriving at a solution, for it is in the solution process that students uncover the mathematics. Arriving at an answer isn't the end of the process. Looking at the approaches used to solve the problem provides additional learning experiences. Studying the strategy used for one problem helps students become more comfortable with using that strategy in a variety of other situations. Problem-solving lessons should provide ample time for students to explore and choose the materials and strategies they will use to solve a problem. A rich problem-solving task may fill one or more mathematics classes. Problem Solving in the Classroom Effective teachers model good problem-solving habits for their students. Their questions are designed to help children use a variety of strategies and materials to solve problems. Young children often want to begin without a plan in mind. Through appropriate questions, the teacher gives students some structure for beginning the task without telling them exactly what to do. In the Wheel Problem video, Ms. Gregory carefully constructed questions to help her students focus on the important information in the problem and the question being asked. Her review of the five-step problem-solving model (read the problem, decide what is being asked, decide on a problem-solving strategy, establish materials to use and a way to record work, solve the problem, explain how the problem was solved.) reminded the students of what they needed to do as they began to work independently or with partners. These steps might give students a way to make sense of a problem and to organize their thinking and their approach to solving it. The first steps, "read the problem" and "decide what is being asked," get students to focus on what the problem is about and what they are looking for. The next three steps are the crux of the mathematics; this is where students begin the actual problem-solving process. Teachers should emphasize these steps and make sure that students spend most of their time here. Within these steps, a key (and sometimes overlooked) point is to teach a variety of strategies. For students to become successful problem solvers, they must have a repertoire of such strategies and be able to find at least one that applies in a given situation. As teachers observe how students go about tackling a problem they must use their own judgment to decide how to help students who are having difficulty getting started, when to probe student thinking, when to give feedback on student work, and when to withhold comments. Students become more reflective about their work and their understanding when they have opportunities to listen to and share their thinking with others. Sharing of ideas occurs in the small-group setting, as well as when the entire class reconvenes to explain their different approaches and solutions.
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