#8 Exercise VS Problem and Mental Toughness

Excerpts from 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 that being able to, say, invert a matrix. A computer can do the later, 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.

Likewise, a good math problem, one that is interesting and worth solving, will not solve itself. You must expend effort to discover the combination of the right mathematical tactics with the proper strategies. “Strategy” is often non-mathematical. Some problem solving strategies will work on many kinds of problems, not just mathematical ones.

The moral of the story, of course, is that a good problem solver doesn’t give up. However, she doesn’t just stupidly keep banging her head against a wall (or cage!), but instead varies each attempt. But this is too simplistic. If people never gave up on problems, the world would be a very strange and unpleasant place. Sometimes you just cannot solve a problem. You will have to give up, at least temporarily. All good problem solvers occasionally admit defeat. An important part of the problem solver’s art is knowing when to give up.

But most beginners give up too soon, because they lack the mental toughness attributes of confidence and concentration. It is hard to work on a problem if you don’t believe that you can solve it, and it is impossible to keep working past your “frustration threshold.” The novice must improve her mental toughness in tandem with her mathematical skills in order to make significant progress. Questions:

1)Have you encountered any “problems” yet in this class, or are they all exercises?

2)Have you ever had to work “problems” in other classes, or have you only ever been assigned exercises?

3)Have you ever solved a problem that you thought was too difficult for you on the onset?

4)Where is your “frustration threshold”

5)What is your confidence level when it comes to solving “problems”. Do you think that working more “problems” with a strategy will help your confidence?

6)Include any other thought you have about yourself as a problem solver or on this article.