Puzzles and Problems

"I have a puzzle for you."

"The problem I have to deal with is how to...."

What is the difference between a puzzle and a problem?

I'd say that one distinguishing feature is that problems are plausible in that they purport to present a situation that one might conceivably find oneself in and need to deal with. Puzzles, on the other hand tend to be contrived. The implication is that they are intended primarily for amusement purposes.

Also, I think, that there are more or less recognized ways of dealing with most problems, and that one can get better at dealing with problems by acquiring relevant knowledge and practising.

Puzzles, on the other hand, tend to be approached in trial and error fashion. Crossword puzzles, and cryptograms are examples.

For a long time it was thought that learning transfer occurred even when there was only a tenuous relationship between the activity being practiced and the activity being prepared for. For example, Latin was taught almost universally on the theory that it prepared you for learning languages in general. Military Academies in some countries encouraged playing chess to develop the mindset for battle strategies.

The same sort of thinking was behind the very heavy commitment that School Mathematics used to make to Euclidean Geometry. It was important because you learned to think logically - and this was beneficial not only for the study of other aspects of Mathematics; but also because it spilled over to life outside the classroom.

I think that most of us recognize these claimed benefits were somewhat exaggerated.

On the other hand, I would say that if there is close similarity between problems that one learns about and practices finding solutions for, there is likely to be benefit.

Many of the exercises given in textbooks to practice manipulations on algebraic symbols are of the puzzle type in that they are never encountered outside the classroom and are often far more complex (convoluted) than is needed for the student to gain or demonstrate understanding of the underlying principles and processes. I have never been able to understand what the point of this is.

It is the practice in School Mathematics to include a number of "word problems" at the end of each chapter. These are characterized by the fact that they have been constructed or chosen in such a way that one supposedly needs to use the mathematics just covered in the preceding chapter to solve them. In other words we select problems to solve on the basis of the mathematics we learn - as opposed to learning the mathematics needed to solve the problems we are likely to face.

This, it seems to me, is backwards.

WHAT WAS DESIGN CRITERIA FOR SCHOOL MATHEMATICS? WHAT DO YOU THINK?