The schemas of arithmetic story problems

The approach described here derives from the idea that in any problem-solving domain, only a limited set of basic situations will occur. The number of basic situations is presumed to be small, but many complex situations can arise because the basic ones can be logically combined in many ways. Given that there is but a small collection of basic situations, the objective is to delineate one and only one schema for each of them. Just as basic situations combine to create complex situations, the corresponding basic schemas are combined in solving complex problems. 

In many domains, a person may develop a basis set of schemas quite naturally. In a number of educational domains, however, we have evidence that students are not developing powerful schemas 

The issue of how to identify an effective basis set of schemas for use in instruction is a thorny one for a number of reasons. At first glance, it might seem that one need only ask experts how they solve arithmetic problems and then extract or abstract their schemas from the responses. This approach has at least three limitations. The first limitation is endemic to all schema research People are not very good at telling us how and what they think. Almost every researcher who studies expertise faces this problem. As Nisbett and Wilson (1977) illustrated in their research, individuals 

The second limitation also pertains to most schema research. Experts may not share a common set of schemas. To be sure, we would expect some consistent threads among their different configurations. But, for instruction and for cognitive models, it is necessary to identify a single set of schemas upon which to build. The problem is how to select an appropriate one given several alternatives. Which expert s perspective do we embrace  

One is tempted to turn instead to young experts, i.e., children who have mastered simple arithmetic story problems but who have not yet studied other areas of mathematics. I initially attempted this approach but abandoned it after interviewing all sixth-grade children in two elementary schools. The difficulty with young experts is that they have not necessarily developed appro- 

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