How Can We Teach Problem Solving?
We continue to plan our co-op courses for next fall. Some of the classes I had hoped for will not happen, and my children are going to have to make some tough choices between the remaining topics. Unfortunately, they have not yet mastered the ability to be in two classrooms at once.
I have three math courses to plan, and I think I will focus as much as I can on teaching math through problems, even at the elementary level. These are once-a-week enrichment classes for homeschooled students, so I assume they have a “normal” math program at home. I want to introduce a few topics they might not otherwise see, to deepen their understanding of the topics they have studied, and to give them a taste of that “Aha!” feeling that comes from conquering a challenging problem. Has anybody done something like this, and can you recommend some good resources?
Meanwhile, I am in the mood for some inspriation:
There is a distinction between what may be called a problem and what may be considered an exercise. The latter serves to drill a student in some technique or procedure, and requires little, if any, original thought. In contrast to an exercise, a problem, if it is a good one for its level, should require thought on the part of the student. It is impossible to overstate the importance of problems in mathematics. It is by means of problems that mathematics develops and actually lifts itself by its own bootstraps. Every new discovery in mathematics results from an attempt to solve some problem.
— Howard Eves
quoted by Rosemary Schmalz, Out of the Mouths of Mathematicians
The first and foremost duty of the high school in teaching mathematics is to emphasize methodical work in problem solving…The teacher who wishes to serve equally all his students, future users and nonusers of mathematics, should teach problem solving so that it is about one-third mathematics and two-thirds common sense.
Mathematical Discovery, Volume II
Solving problems is a practical skill like, let us say, swimming. We acquire any practical skill by imitation and practice. Trying to swim, you imitate what other people do with their hands and feet to keep their heads above water, and, finally, you learn to swim by practicing swimming. Trying to solve problems, you have to observe and to imitate what other people do when solving problems, and, finally, you learn to do problems by doing them.
How to Solve It