What is Problem Solving?

Today in one of our PCMI sessions we discussed problem solving.

According to the Principles and Standards for School Mathematics (NCTM document), 'problem solving means engaging in a task for which the solution method is not known in advance.' It also says that 'problem solving is an integral part of all mathematics learning, and so it should not be an isolated part of the mathematics program.'

I have enjoyed working on problems, but I don't know that I have given my students problems that are true problem solving problems. The problems I give my students are story problems that have a specific method for solving. I am usually hoping the students will see that method and use it to get an answer. I need to do a little more digging and find some problems that can help students get the math they are supposed to get.

I read a little on formative assessment last week, and read that formative assessment is assessment for learning, not assessment of learning. Problem solving would then be problems for learning new mathematical techniques, not problems that practice learned techniques.