Problem solving

[This is a rough draft of a paper containing my ideas on problem solving. It is a work in progress, so please be understanding of areas where it is incomplete or has ideas that are not fully formed.]

An essential part of mathematics is finding patterns, posing questions about those patterns, and then solving the problems posed by those questions. As those problems are investigated, further interesting questions may naturally arise and lead to beautiful and intriguing developments for further investigation. Problem solving is at the heart of doing mathematics.

This paper looks at the process of doing problem solving. To distinguish problems from exercises, a problem will refer to a task whose method of solution is not readily apparent to the solver. Starting around 1980, Alan Schoenfeld created a framework for problem solving built on Polya’s groundbreaking 1945 book How to Solve It. In his book Mathematical Problem Solving, Schoenfeld identifies four areas that make up a framework for problem solving: Resources (the collection of known math facts, procedures, skills), Problem-solving Strategies, Self-monitoring (watching over problem solving steps), and Beliefs (beliefs and attitudes about doing mathematics). Roughly speaking, each area of the framework attends to how the previous area is managed. This paper is made up of four chapters corresponding to Schoenfeld’s four areas.