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We confront problems all the time in our lives, and success in life is largely determined by how we confront and resolve these problems. Thus, it’s important to know how to approach problem solving.That’s a big issue that I don’t feel capable of tackling, so let’s consider a more specialized question: How does someone approach solving a mathematical problem? From answers to this question, we can hopefully extrapolate about the real world.
The most important skill in mathematical problem solving isn’t knowledge of theorems and facts; instead, it’s more important to have the right mindset for approaching a problem. Here’s some general advice:
Before beginning to approach a mathematical problem, the best problem solvers are psychologically prepared. They are confident in their own ability to solve problems, and they are not worried about failure. In the meantime, they are willing to try new and creative attacks. They are persistent and never give up, and they are also courageous — willing to try any technique that might seem to succeed.
When confronted with a problem, first try to understand it. There are a number of typical strategies for doing this:
Then, try to find a solution.
Once you’ve solved the problem, check your answer.
These problem solving heuristics do not consider any particular formula, theorem, or method; they apply to every problem. By being independent of specific techniques, they transcend such techniques and become universal; some of these heuristics (“Be brave and try it”, for example) could also apply to the real world.
For more information on problem solving heuristics, see George Pölya’s classic book How to Solve It. I leave you with a quotation from his book:
A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery.
— George Pölya, How to Solve It