"Letting go" and letting the students take charge. I don't know of too many teachers that actually like the sound of this. Why? Because it's the way we've been taught that the classroom is supposed to work; it is our understated idea of its mechanism. And heaven forbid we let a kid go slip through the cracks--please, no child should be left behind. Yet that's one major obstacle that's keeping our students from truly learning.
So, then, what happens when they are wildly off track? At this point, they usually beg--I don't just mean ask, but beg--the teacher to supply the answer. Hints are good here. Give them only enough information to shift their line of thinking, just enough so that they have a better chance of taking the problem from here. One way I like to do this is with questions. "Remember when we had a problem that looked like ___? How did we solve that one?" Eventually that gets them going. And it's very empowering when they realize that they solved the problem on their own, and all the teacher did was just ask them questions. That's really the whole purpose of this: student empowerment. Remember, we have to prepare them for the day when they won't have us around to help them, and that day is coming soon.
One of the features that I introduced to one of my math courses last year was the concept of a "Closing" session--essentially the same thing as the After. The key principle was that the students had to explain the solutions to each other, and if they got it wrong, they had to rely on each other to correct it, not me. For much of the course, this drove the students nuts. Why do we have to go over a problem that we already know the answer to? Why can't we just say whether the answer is right or not? But I didn't cave into this, because I knew that they would learn a lot more effectively if they had to not just explain their mistakes but why they were mistakes. That has a lot more power than just hearing the teacher give the correct answer.
Something I'm planning on doing new next year is this idea of teaching the idea from problems. Have them understand why you need to learn FOIL--or do they need to learn it at all? Have them figure out what makes the Distributive Property so powerful. Reinforce their findings with multiple examples, but by all means, avoid skill-and-drill as the primary locus of teaching.
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I love how you used the phrase "student empowerment". That is extremely important when dealing with student learning because as we have been discussing in all three of our classes, it gives the students a certain level of ownership to the solution. With a repeated uses of this process students may begin to view themselves as part of that solution. This can create confidence which can lead to self-motivation on behalf of the students.
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