I just subscribed this newsgroup and read Cindy's argument that "problem solving should not be limited to an agreed upon list of set strategies." I agree with Cindy and in my experience find that many teachers misunderstand the standards call to teach through problem solving. This confusion stems from the belief that math knowledge is a set of rules that should be transmitted too students primarily through teacher directed lessons where the teacher tells the students how to solve a specific type of problem, the children practice the isolated skills in their workbook and then apply these skills to a selection of word problems after the teacher has "covered" the prerequisite computational skills. What is called problem solving is merely a means to practice an operation. In addition, the children are often instructed to find clue words to assist them. This approach not only denies children the opportunity to reason but it fragments mathematical activity rather than approaching it as an interrelated whole. It promotes the view that math is doing computation to get the right answer rather than reasoning and thinking critically to solve problems. It focuses on knowing how and not why.
The same thing occurs when students are explicitly taught the strategies used fo problem solving (typically Polya's phases and strategies - making a diagram, looking for a patten etc.) Although teaching these strrategies is not inherently opposed to teaching through problem solving,they become limiting if the child is directly taught a strategy and then provided a series of problems that can be solved using that particular strategy.
To teach problem solving effectively teachers must place emphasis on developing the child's abiltiy to reason and solve problems. Teaching problem solving requires teachers to abandon the linear progresssion of most math lessons from skill instruction to "problem soving" and create an environment where there is interplay among reason, problem solving and skill development. Teachers must create lessons where children can learn a variety of strategies - not through explicit instruction but through real problem solving, guidance and discourse.