> I'm also very interested in whether or how K-12 teachers use the various > problem-solving descriptors (e.g. "guess and check"). Let me be very > upfront about this and say that as a practicing mathematician I am very > suspicious of these check-lists, since they seem (a) vague, (b) susceptible > of being turned into an algorithm, and (c) have little to do with what > really happens when we solve mathematical problems. > > Having thrown down the gauntlet I'll sit back and listen to the responses.
This has bothered me quite a bit from my earliest years of teaching beginning algebra (to both high schoolers and college). The way we used to try to teach word problems and now try to teach Problem Solving is not really the way we solve problems. I feel like it is to a great extent a necessary evil, however. It seems the way we solve problems is by (1) insight from smarts, (2) insight developed from experience, (3) being willing to play with it, (4) being willing to persist, (5) etc. etc.
My take on things like the check-lists mentioned is that they are a way to encourage students to stick with it long enough to actually gain some experience and confidence. Often, the students who have the most difficulty with problem solving like to have things turned into algorithms. Whatever it takes to get them to spend time working at problems and really *thinking*. Other than genetic-engineering more smarts in future students, this is likely to be about the most successful approach we can take.
Above all, though, if we really have our hearts in the Standards with regard to critical thinking/problem solving, we absolutely have to allow lots and lots of time for the necessary process. (This returns us to the discussion last week about coverage vs. understanding.)