Problem Solving as an Instructional Strategy - Polya (1945) suggests that problem solving consists of four phases: understanding the problem, devising a plan, carrying out the plan, and looking back. Lajoie (1992) defines mathematical problem solving as: “modeling the problem and formulating and verifying hypotheses by collecting and interpreting data, using pattern analysis, graphing, or computers and calculators.” This definition focuses on the processes of formulation, investigation, and verification, but it does not encompass the important elements inherent in Polya’s looking back phase, which involve evaluating and interpreting methods and results. The looking back phase includes such activities as:
- Verifying the result
- Checking for alternative methods of solution
- Determining the validity of an argument
- Applying the result or method of solution to other problems
- Interpreting the result
- Generalizing the solution
- Generating new problems to be solved
Looking back may be the most important aspect of teaching problem solving because it provides students the opportunity to learn about problem-solving processes and how a problem is related to other problems. Schoenfeld (1985) and others have shown that the principal traits that separate expert from novice problem solvers are their ability to see past the surface features of problems to their common underlying structures, and their ability to self-monitor and recognize when an approach or tactic is not being productive.
Although teachers and researchers report that it is difficult to develop a willingness in students to continue past finding the correct answer to a problem, the development of self-awareness and reflection are critical for improving problem-solving ability.
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