The following passage is from the opening pages of The Art and Craft of Problem Solving, by Paul Zeitz
An exercise is a question that tests the student’s mastery of a narrowly focused technique, usually one that was recently “covered.” Exercises may be hard or easy, but they are never puzzling, for it is always immediately clear how to proceed. Getting the solution may involve hairy technical work, but the path towards solution is always apparent. In contrast, a problem is a question that cannot be answered immediately. Problems are often open-ended, paradoxical, and sometimes unsolvable, and require investigation before one can come close to a solution. Problems and problem solving are at the heart of mathematics. Research mathematicians do nothing but open-ended problem solving. In industry, being able to solve a poorly defined problem is much more important to an employer than being able to, say, invert a matrix. A computer can do the latter, but not the former.
A good problem solver is not just more employable. Someone who learns how to solve mathematical problems enters the mainstream culture of mathematics; he or she develops great confidence and can inspire others. Best of all, problem solvers have fun; the adept problem solver knows how to play with mathematics, and understands and appreciates beautiful mathematics.
An analogy: The average (non-problem-solver) math student is like someone who goes to a gym three times a week to do lots of repetitions with low weights on various exercise machines. In contrast, the problem solver goes on a long, hard backpacking trip. Both people get stronger. The problem solver gets hot, cold, wet, tired, and hungry. The problem solver gets lost, and has to find his or her way. The problem solver gets blisters. The problem solver climbs to the top of mountains, sees hitherto undreamed of vistas. The problem solver arrives at places of amazing beauty, and experiences ecstasy that is amplified by the effort expended to get there. When the problem solver returns home, he or she is energized by the adventure, and cannot stop gushing about the wonderful experience. Meanwhile, the gym rat has gotten steadily stronger, but has not had much fun, and has little to share with others (Zeitz 3).
Zeitz, Paul. The Art and Craft of Problem Solving. New York: John Wiley & Sons, 1999. Print.
Write a comment on this post that answers at least one of the following questions:
- In your opinion, what is the purpose of mathematics: to do exercises or to solve problems?
- How do you feel when you’re solving problems?
- How does this reading apply to life outside of a high school mathematics class?