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A structure-mapping model for word problems is used to interpret the results of four experiments. In the first two experiments students rated the potential usefulness of solutions for pairs of problems, mixture problems in Experiment 1 and work problems in Experiment 2. The problems were either equivalent (same story, same procedure), similar (same story, different procedure), isomorphic (different story, same procedure), or unrelated (different story, different procedure). Students in Experiment 3 used an example solution for a work problem and a mixture problem to generate equations to related test problems that differed in their mappings from the example. In Experiment 4 students matched concepts in the test problems to corresponding concepts in the examples to provide a direct measure of their ability to construct mappings across different problems. In both Experiments 3 and 4, students performed significantly better on isomorphic problems than on similar problems, and significantly better on work isomorphs than on mixture isomorphs. The results suggest that a structure-mapping model that emphasizes the transparency and structure of the mapping can be used to predict the usefulness of a solution.

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We studied how successfully students could use examples and procedures to construct equations for work problems. According to the proposed theory, the procedures indicate how to generate values that differ in structure from the example. The fLrst experiment compared 3 groups of students who received a simple example, a set of procedures, or both. A mathematical model with 3 parameters (the probability of generating a correct value by matching the example, following a procedure, or using general knowledge) accounted for 94% of the variance for how the 3 instructional groups performed over 4 levels of transformation. A second experiment extended the predictions of the model to include either a complex example, a complex example and procedures, or a complex example and a simple example. Two alternative approaches for instructing people about a task are to present either a detailed example or a set of procedures. Each method has its advantages and disadvantages. The advantage of an example is that it illustrates how the procedures are applied to a particular situation. For example, students in a college algebra class could be given a detailed solution to the following problem: Ann can type a manuscript in 10 hr, and Florence can type it in 5 hr. How long will it take them if they both work together?. The disadvantage of an example is that it may not be very helpful for solving problems that are slightly different. Students often have difficulty in solving variations of the exampies, such as a problem in which one person worked more hours than the other (Reed, Dempster, & Ettinger, 1985). The advantage of procedures or rules is that they can specify the component steps for solving a variety of problems. One rule might specify what to do when one person works longer than another, and another rule might specify what to do when rate rather than time is the unknown. The disadvantage of procedures is that they can be rather abstract and isolated, resulting in minimal understanding of the task as a whole. Thus learning to operate a device can be facilitated if a set of procedures is supplemented with additional material (functional, structural, or diagrammatic information) that enables students to better understand and integrate the procedures

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