10 Classic Problem Solving Heuristics
Michael Starbird and Ed Burger have written a nice textbook called The Heart of Mathematics aimed at liberal arts types and non-math majors generally—almost a philosophy book, really, but the authors are far smarter and far more mathematically sophisticated than the average navel-gazing philosopher. They also have good senses of humor and wear their erudition lightly.
Starbird and Burger describe ten basic heuristics, some of which I think come from Polya (another well known gem worth adding to your library if it’s not there already). These heuristics apply to almost every domain and are worth learning:
1. Just do it. If you’re faced with a problem and you don’t know how to solve it, begin by taking some action.
2. Make mistakes and fail but never give up. Mathematicians are supremely gifted at making mistakes. The key is to use the insight from your mistakes to identify the features of a correct solution to your problem.
3. Keep an open mind. If we are never willing to consider new ideas, then we can never hope to increase our understanding of the world around us.
4. Explore the consequences of new ideas. This strategy pushes us to see where an idea leads and in this way to discover new ideas and insights.
5. Seek the essential. One of the biggest obstacles in solving real-world problems is the noise and clutter of irrelevant issues that surround them.
6. Understand the issue. Identifying and clarifying the problem to be solved in a situation is often a significant step in reaching a solution.
7. Understand simple things deeply. We can never understand unknown situations without an intense focus on those aspects of the unknown that are familiar. The familiar, in other words, serves as the best guide to the unfamiliar.
8. Break a difficult problem into easier ones. This strategy is fundamental to mathematics and, indeed, applicable in everyday life.
9. Examine issues from several points of view. We can, for example, gain new insights by looking at the construction of an object, rather than the object itself.
10. Look for patterns. Similarities among situations and objects that are different on the surface should be viewed as flashing lights urging us to look for explanations. Patterns help us to structure our understanding of the world, and similarities are what we use to bring order and meaning to chaos.
(Note: the book’s not written to be read straight through, though. If you want something more readable, their Coincidences, Chaos, and All that Math Jazz is probably a better choice. The information density is lower but the writing quality is higher).
[composed and posted with ecto]