Defining Problem Solving
"Successful problem solving requires knowledge of mathematical content, knowledge of problem-solving strategies, effective self-monitoring and a productive disposition to pose and solve problems."

## (NCTM, 2000, p. 341)

Math teachers enjoy solving problems. In fact, this aspect of teaching mathematics may be its most appealing aspect, a key part of what has drawn them into the profession. The NCTM Problem-Solving Standard hopes to inspire teachers, who are already expert problem solvers, to help their students improve their skills, to give them a "problem-solving disposition."

How can we do this? This section of the session discusses both overall themes and specific problem-solving techniques. It is not an exhaustive list. Mathematical problem-solving is an inexhaustible sea! But this context and these techniques will give you a stronger background for addressing the goals of the Problem-Solving standard in your classroom.

Understanding Context

When a teacher is working with math problems in his or her class preparation, in the classroom, and in evaluating students, there are several factors to keep in mind. Choosing and sequencing problems for students is an important part of the planning job. Thought should be given to how problems build on previous knowledge or how they assist in helping students or a class with subject matter that has been particularly difficult.

Problems are posed to teach content, but they are also a means to mathematical discovery. Even when students are doing iterative problems to practice a technique, it is important to remember that they are "doing real mathematics," not simply turning a crank to get a preordained result that is checked in the back of the book.

This point leads to an overarching concern of the standard: preparing students to solve problems they have not encountered before. This is an exciting and challenging part of learning for teacher and student, and it divides into at least three distinct points: 1. learning to solve problems that use familiar techniques in new contexts; 2. solving new problems that require new techniques and insights (such as the staircase example presented in Part A of this session); and 3. solving problems and bringing a problem-solving disposition to tasks in related fields such as the sciences, and social sciences. These can be explicitly mathematical problems or more general issues -- for instance those that require step-by-step thinking and the application of logic.

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