Story Problem Solver

Chapter 5 shows how schema theory can be practically implemented. It gives a detailed description of one example of schema-based instruction, the Story Problem Solver (SPS). SPS is a computer-based system of instruction constructed around schema theory, using the basis set of schemas developed in chapter 3. It provides fundamental instruction enabling students to build schemas. A second computer program, the Problem Solving Environment (PSE), is also described in chapter 5. PSE provides an exploratory environment in which students can practice and utilize their schema knowledge about story problems. 

The Story Problem Solver and The Problem Solving Environment... 

The Story Problem Solver (SPS) is a computer-implemented program of instruction about arithmetic story problems.1 My research group and I developed SPS as an explicit instructional test of schema theory.2 Its companion, the Problem Solving Environment (PSE), is also a computer-based system, one that provides no additional instruction but that serves instead as a practice arena in which we can evaluate students acquisition of schema knowledge. Both SPS and PSE are written in Lisp and run on Xerox 1186 computer workstations equipped with 19-inch display monitors and three-button optical mice. In this chapter, I first describe SPS and its instructional objectives and then explain the contributions of PSE. 

A number of technical reports have been issued about the Story Problem Solver, e.g., Marshall, Pribe, Smith (1987) Marshall, Barthuli, Brewer, Rose (1989) Marshall (1991). 

Marshall, S. P., Barthuli, K. E., Brewer, M. A., Rose, F. E. (1989). STORY PROBLEM SOLVER A schema-based system of instruction (Tech. Rep. 89-01 ONR Contract N00014-85-K-0661). San Diego San Diego State University, Center for Research in Mathematics and Science Education. 

One surmises that the would-be problem solvers have failed to recognize the relevant relationships described by stories. They apparently lack the requisite knowledge for making decisions based on such recognition. Moreover, they rarely display evidence of creating problem-solving plans that use important relational information gleaned from the stories. For the most part, as students I interviewed often insisted, a number of the problems just do not make sense to them (Marshall, 1981). Moreover, the perceived senselessness of the problems does not seem to bother the students they do not expect otherwise. 

Surface features are perhaps the most obvious differences among story problems. However, they appear to be a poor basis for differentiating situations, as a number of novice-expert studies in mathematics and other domains have pointed out. What these studies have found is that experts rarely attempt to use surface features to understand problem situations. Such behavior is observed more often in unsuccessful solvers than in successful ones. This has been observed in domains such as physics (Chi, Feltovich, Glaser, 1981) and eighth grade mathematics (Silver, 1979). 

What do we lose by failing to categorize the 332 items classified as other in Tables 3.3-3.6 Very little. Most of them are not story problems at all but have been erroneously placed with the story problems in the curricular materials. Instead of requiring the student to use his or her own prior knowledge about situations, they depend upon the student s ability to retrieve specific formulas (e.g., area, perimeter, volume) or to understand the mathematical definitions of specific terms (e.g., median, mode, probability). These terms are always used explicitly in the question directed to the student. It is clear from the wording of such problems that the situations (i.e., the stories) are relatively unimportant. On the other hand, for all cases in which the situation was important -because a problem solver must extract the meaning of the problem from the story situation and use that information to carry out necessary computations - the five situations served well. 

Most of these studies begin with the assumption that students have a body of knowledge about arithmetic operations that they can then use to understand a story problem. My point, in contrast, is that this is not the memory organization that we desire and that it will not lead to competent problem solvers. The difficulty is that the operational approach assumes that the student already knows which operation to use and then determines whether the current problem matches any of the stored categories that fit the selected operation. How does the student know which operation applies Typically, it is the one most recently under discussion, or it is the one specified by the teacher or text (e.g., Use addition to solve the following problems. ). This approach misses an important point, namely, that the operation itself is frequently difficult for students to identify and must usually be determined from elements in the story situation. 

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