Part I. PROBLEM SOLVING
2. Problem Solving as a Process
Problem Solving as a Process
What is a problem?
Let’s establish this right away: a problem is an intellectual challenge. Solving a problem is then a process of undertaking and overcoming the challenge.
As we have indicated elsewhere, authentic problems are often poorly-structured and vague. They are not carefully-crafted to yield whole-number answers with a few minutes of symbolic manipulation, like those you typically encounter in school. Authentic problems could take hours, days, or even longer to solve, and you may not always know with confidence that you have succeeded because the answers aren’t in the back of the book. Problems don’t require only the application of a recently-learned method or algorithm; indeed, you may not know at the outset which methods are appropriate for solving the problem. You may not be given all the information needed to achieve a complete solution, or information you are given may be uncertain or incomplete. In short, authentic problems are hard, and that can be frustrating.
But wait! Don’t close your book (or laptop) and walk away just yet! Having just explained the difficulties, consider the flip side of the problem-solving coin: problems that are authentic are also inherently interesting, particularly when they address contemporary issues or puzzles in your chosen area of study. Solutions and solution methods for such problems are therefore not just an academic dead-end, but can lend themselves to practical applications in the real world. The rewards of achieving a clever and well-justified solution to a practical and interesting problem should outweigh by far the moments of uncertainty, frustration, or disappointment encountered along the way.
Most of the problems discussed in this book are designed to mimic authentic problems, and in many cases are drawn from or inspired by encounters with local researchers and practitioners. As we endeavor to address these problems, we’ll often find it helpful to explore simpler problems and exercises to help with sense-making. Thus, our time will be spent moving back and forth from focus problems to auxiliary problems and exercises. Read on through the end of this chapter to understand how this approach can lead to more successful problem-solving.
Problem solving cannot be reduced to a simple recipe or fast and easy method, but in the past 70 years, much has been learned about how successful construction of solutions differs from unsuccessful attempts. A key component is the use of heuristics, or habits of mind that are useful in solving problems. The modern idea of heuristics has its origin in the work of Hungarian mathematician George Pólya in the mid 20th century. Heuristics help guide us in decisions about how to approach a problem. With the help of heuristics and the benefit of experience, we may develop problem-solving strategies that lead to successful solutions. We’ll begin our study of problem-solving with a br