Characterizing the intuitive representation in problem solving: Evidence from evaluating mathematical strategies

Abstract

Two experiments were conducted to investigate the nature of the intuitive problem representation used in evaluating mathematical strategies. The first experiment tested between two representations: a representation composed of principles and an integrated representation. Subjects judged the correctness of unseen math strategies based only on the answers they produced for a set of temperature mixture problems. The distance of the given answers from the correct answers and whether the answers violated one of the principles of temperature mixture were manipulated. The results supported the principle representation hypothesis. In the second experiment we manipulated subjects' understanding of an acid mixture task with a brief paragraph of instruction on one of the principles. Subjects then completed an estimation task intended to measure their understanding of the problem domain. The evaluation task from the first experiment was then presented, but with acid mixture instead of temperature mixture. The results showed that intuitive understanding of the domain mediates the effect of instruction on evaluating problems. Additionally, the results supported the hypothesis that subjects perform a mapping process between their intuitive understanding and math strategies.Problem solving often requires applying some formal strategy or algorithm to a problem. How these formal strategies are selected and evaluated is a central issue in theories ofproblem solving. Many theories ofproblem solving have suggested that the person's qualitative or intuitive representation of a problem is an important factor in arriving at an appropriate formal strategy. For example, Larkin (1983) proposed that when people are solving physics problems with mathematics, they construct a representation of the problem and use this representation to select appropriate math strategies. Similar frameworks have been proposed for the domain of children's counting (Briars & Siegler, 1984;Greeno, Riley,& Gelman, 1984), word problems (Briars & Larkin, 1984;Kintsch & Greeno, 1985), and mixture problems Reed & Evans, 1987). According to these models, what the person understands about the problem domain affects the type of strategies they select to solve the problem.Most models of problem solving also agree that the intuitive representation of the problem domain specifies how the variables in the task relate to one another. A number of computer simulations have demonstrated that information of this type is sufficient to select a formal strategy (see, e.g., de Kleer, 1975Kleer, , 1977Kintsch & Greeno, 1985;Larkin, 1983 There is agreement on the importance of the representation of the problem domain as well as agreement that the representation must specify the relationships between variables in the domain, but it is not clear how the representation is used in the selection and evaluation offormal strategies. In past work, we have proposed that people engage in a mapping process between their intuitive representation and a representation of the formal strategies in o...

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