The Art and Craft of Problem Solving

14 CHAPTER 2 STRATEGIES FOR INVESTIGATING PROBLEMS

2.1 Psychological Strategies

Effective problem solvers stand out from the crowd. Their brains seem to work differ­

ently. They are tougher, yet also more sensitive and flexible. Few people possess these

laudable attributes, but it is easy to begin acquiring them.

Mental Toughness: Learn from Polya's Mouse

We will summarize our ideas with a little story, "Mice and Men," told by George P6lya,

the great mathematician and teacher of problem solving ([33], p. 75).

The landlady hurried into the backyard, put the mousetrap on the

ground (it was an old-fashioned trap, a cage with a trapdoor) and called

to her daughter to fetch the cat. The mouse in the trap seemed to un­

derstand the gist of these proceedings; he raced frantically in his cage,

threw himself violently against the bars, now on this side and then on

the other, and in the last moment he succeeded in squeezing himself

through and disappeared in the neighbour's field. There must have

been on that side one slightly wider opening between the bars of the

mousetrap... I silently congratulated the mouse. He solved a great

problem, and gave a great example.

That is the way to solve problems. We must try and try again un­

til eventually we recognize the slight difference between the various

openings on which everything depends. We must vary our trials so that

we may explore all sides of the problem. Indeed, we cannot know in

advance on which side is the only practicable opening where we can

squeeze through.

The fundamental method of mice and men is the same: to try, try

again, and to vary the trials so that we do not miss the few favorable

possibilities. It is true that men are usually better in solving problems

than mice. A man need not throw himself bodily against the obstacle,

he can do so mentally; a man can vary his trials more and learn more

from the failure of his trials than a mouse.

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, becau se 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.