My Blog Post about Math For Love's Blog Post about Math Mamma's Blog Post about...

Math for Love just wrote a post that I really enjoyed about play.  Go there and check it out and then come back here for the post-game analysis.  I was originally just going to comment over there, but this got too long.  Anyway, I found that it was a really well articulated process of how I want students to work with math problems.  Since you're good at following directions you've already read this but let's recap.

1. Play (Expansion)
2. Ask questions (Reflection)
3. Choose a question that you really want to explore (Contraction)
4. Try to answer the question with what you already know
5. If that fails (i.e., if your question is deep enough), start exploring other ways to approach the question, with your own ideas and by looking at what others have done. A teacher or expert can be very useful here (“Oh, you want to figure out how far apart those two dots are? Have you ever heard of the Pythagorean Theorem? It works like this…”)
6. Refine your question
7. Repeat Steps 4-6 until you solve your question
8. Explain your work to people
9. Realize that you can’t really explain it, and refine your solution until you can actually write down an explanation.
10. If in a class or group, share your work with the group, or have the group come to an agreement on how things really are.
11. Satisfaction.

I would stop at #11 (and although this is hard, I would also aim to have students feel #11 after every step).

12. Tweak the original "rules" in some way and return to #1.

In terms of addressing the floundering that can occur in progressive models, I think the key (not that this is easy) is teaching kids how to do #1 through #11 (well, imho the cycle of  #1 through #12).  Even "playing" can be taught. Consider the development kids go through when playing boardgames.  They start with games like Chutes & Ladders which has absolutely no strategy and children simply enjoy the unknown of what you're going to roll next and where you'll end up.  This eventually evolves into greater sophisticated game play where students have to make decisions (whether or not to buy that property in Monopoly, for example).  This evolves into more sophisticated strategies such as thinking ahead (Chess), thinking about your opponent's thinking (Settlers of Catan), making inferences based on past play (Poker), etc.

I believe the game metaphor can be extended to math problems, even traditional ones.  Just as an example, consider how the "rules" of subtraction evolve throughout elementary school:
1. Single digit whole #'s with the first being larger than the second
2. Multiple digit whole #'s with the first being larger than the second
3. Multiple digit whole #'s with the first being larger than the second, but requiring borrowing
4. Whole #'s with the second being larger than the first
5. Fractions
6. Subtraction of integers
7. Subtraction of rationals
etc.

We change (usually relaxing) the rules as we progress through the curriculum.  It's nice that this also mirrors what mathematicians often do, relaxing or restricting conditions and exploring the ramifications.