How Much Thinking?

This summer all the teachers in my school are reading some books which we will discuss in the fall.  One is Situated Learning by Lave and Wenger.  Another is How We Think by Dewey.  Situated Learning appears to be focusing on apprenticeships as a a possible learning style... it brings to mind a question that has been in the back of my mind for a while.  How important is it that students figure things out for themselves?  My teaching began with an approach that kept returning the questions back to the students...in fact, I think I tried to avoid answering a question directly at all if I could, instead returning questions to the student designed to help them come to their own conclusions.  But it was a string of logic that I constructed, because I chose the questions from my own established and ever-broadening understanding of the concept.  Sometimes the student could reconstruct it themselves, or came to an insight that stuck.  Sometimes not.  The former was always my goal, but early on I guess I was not very good at it, and I lacked some tools that make it more successful for me now.  So I began to try to explain things more clearly, illustrate the concepts with demonstrations and models, provide good mnemonics, and organize the lessons in a more expository sequence.  That didn't work either.  The students were clearly passive and not expecting much of themselves.  It was an environment in which "I can't do math" could easily propagate, fester and thrive.  So I came back to trying to find ways to enable and encourage the students to do as much of their own thinking as possible.  I believe that mathematical thinking itself is the skill we are teaching, as well as the ability to use mathematical tools, both literal and abstract.  So I want them to do and enjoy as much mathematical thinking as possible.

But it is not a simple, smooth path always, and it is a bit of a mystery what makes the path smooth and what makes it bumpy.  Visual models help, and can often turn an abstract, confusing idea into an engaging puzzle.  What does it mean to figure something out?  And how does one build understanding for oneself?  And then, of course, I want to know how to externally facilitate that internal process.  Through posing questions?  Through instructing them on the use of tools?  Through sharing my solutions and modeling my thinking processes?  How much struggle is necessary and/or useful? 

I wonder what is useful to have the students think about?  For example: the skill for finding a percentage of a number.  How useful or important is it for them to figure out that one way to find 17% of 95 is to use multiplication in some form?  The really important outcome is for them to be able to recognize when the skill is needed, and how to apply it appropriately, say in a problem that involves compound percentages.  Do they really have to have derived or "figured out" the skill themselves in order to go on to applying it?  How does that help them?  How does it hinder them if they don't?