Just thought I would add a few things to the mix on the standards and problem solving. How about the paper by Thomas Schroeder and Frank Lester called "Developing Understanding in Mathematics via Problem Solving"(from New Directions for Elementary School Mathematics). In it there is discribed three different approaches to problem solving. 1. Teaching about problem solving. This is pretty much looking at Polya's model of problem solving(understanding the problem, devising a plan, carrying out the plan and looking back). Students are taught to understand and recognize the phases they go through when solving a problem(as expert problem solvers use according to polya). Typically they are taught "heuristics" or "strategies" from which they can choose when they solve math problems. Some of the strategies typically taught include looking for patterns, solving simpler problems and working backward. Sometimes teaching about problem solving actually includes actual problems, but it always involves a great deal of explicit discussion and teaching about how problems are solved. When we say problem solving this is usually what people are thinking of. A sort of loose algorithmic guide to follow. Is this what the standards are talking about? How many of us have used this when teaching word problems in algebra?
Scott Powell University of Hawaii University Lab school Honolulu, HI. 96822 firstname.lastname@example.org