Defining Problem Solving
 Introduction | Introducing New Content Via Problems | A Positive Problem-Solving Disposition | Problem-Solving Techniques and Examples | Working Backwards | Technology | The Teachers' Role | Your Journal
 How can the teacher put problem solving front and center in the classroom? As the Standards point out: "Teaching is itself a problem-solving activity. Effective teachers of problem solving must themselves have the knowledge and disposition of effective problem solvers." (NCTM, p. 341) This means that a first priority is modeling the skill you want to encourage in your students. Let them see firsthand your pleasure, resourcefulness, and understanding in the way you approach problems. In your preparation, attend to the role each problem is playing in the content development of your course. How does the sequence of problems serve the content goals and also the goals of establishing general skill and strategy development in problem solving? Don't put this idea on hold when you are preparing tests and other assessments. Problems selected for assessments offer a chance to evaluate students' content mastery but also their procedural fluency, not just in specific techniques associated with one concept, but in their overall disposition towards tackling mathematical problems. However, such assessments are only fair if students have gotten a steady diet of problem-solving opportunities in class, and thus the chance to improve skill in this area. Finally, be courageous! By definition, problems in mathematics involve an unknown, and at times, that will be something that is unknown even to the teacher. This means that surprise is part of every mathematics teacher's classroom life from time to time. Perhaps a student gets an unanticipated result when she works on a variant of a problem on a graphing calculator and calls over a teacher to say, "What's this graph about?" Or a student proposes a solution path that no one in the class has considered and asks the teacher, in front of everyone, why it works or fails to. These are challenges, to be sure, but they may also be rare opportunities to engage in real problem solving on the fly with students. Helping students with ways of looking at such problems will be rewarding for all involved. Even if the excursion turns out to be a dead end or beyond the technical means and scope of the class, students will know that their original ideas and problems are valid subjects for inquiry and will gain confidence. Watch the video segment (duration 0:15) at left to hear a reflection from Harriette Davis, a high school mathematics teacher.
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