Components of Understanding in Proportional Reasoning: A Fuzzy Set Representation of Developmental Progressions

Moore, Colleen F.; Dixon, James A.; Haines, Beth A.

doi:10.2307/1131122

Cited by 35 publications

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“…Because this task also required an understanding of 'random draw' and probability, children's difficulties may not have arisen because of lack of proportional knowledge. However, similarly low performance in children younger than 11 years was reported in subsequent studies using different procedures that did not involve probability judgments, for example tasks based on mixing juice and water (Fujimura, 2001;Noelting, 1980) or liquids of different temperature (Moore, Dixon, & Haines, 1991).…”

Section: Such Findings Documenting Children's Difficulties With Fractmentioning

confidence: 51%

Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

Möhring

Newcombe

Levine

et al. 2015

Journal of Cognition and Development

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Section: Such Findings Documenting Children's Difficulties With Fractmentioning

confidence: 51%

Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

Möhring

Newcombe

Levine

et al. 2015

Journal of Cognition and Development

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“…One important factor in generating mathematical solutions is the person's conceptual or intuitive representation of the problem. The intuitive representation of the problem is a qualitative, nonformal representation of the relationships between the objects in the problem (e.g., Moore, Dixon, & Haines, 1991). Three major lines of research have demonstrated a strong relationship between people's intuitive representations and their mathematical solutions.…”

mentioning

confidence: 99%

The representations of the arithmetic operations include functional relationships

2001

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Current theories of mathematical problem solving propose that people select a mathematical operation as the solution to a problem on the basis of a structure mapping between their problem representation and the representation of the mathematical operations. The structure-mapping hypothesis requires that the problem and the mathematical representations contain analogous relations. Past research has demonstrated that the problem representation consists of functional relationships, or principles. The present study tested whether people represent analogous principles for each arithmetic operation (i.e., addition, subtraction, multiplication, and division). For each operation, college (Experiments 1 and 2) and 8th grade (Experiment 2) participants were asked to rate the degree to which a set of completed problems was a good attempt at the operation. The pattern of presented answers either violated one of four principles or did not violate any principles. The distance of the presented answers from the correct answers was independently manipulated. Consistent with the hypothesis that people represent the principles, (1) violations of the principles were rated as poorer attempts at the operation, (2) operations that are learned first (e.g., addition) had more extensive principle representations than did operations learned later (multiplication), and (3) principles that are more frequently in evidence developed more quickly. In Experiment 3, college participants rated the degree to which statements were indicative of each operation. The statements were either consistent or inconsistent with one of two principles. The participants' ratings showed that operations with longer developmental histories had strong principle representations. The implications for a structure-mapping approach to mathematical problem solving are discussed.

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“…Two lines of evidence support the idea that subjects represent these principles. First, developmental differences in understanding mixture tasks are consistent with the hypothesis that children are acquiring the principles that govern the tasks (Ahl, Moore, & Dixon, 1992;Moore, Dixon, & Haines, 1991). Second, when performing mixture tasks, subjects sometimes mention principles, both spontaneously and when asked to explain their judgments Reed & Evans, 1987;Strauss & Stavy, 1982).…”

Section: Methodsmentioning

confidence: 53%

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

Dixon

Moore

1997

Memory & Cognition

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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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Components of Understanding in Proportional Reasoning: A Fuzzy Set Representation of Developmental Progressions

Cited by 35 publications

References 0 publications

Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

Spatial Proportional Reasoning Is Associated With Formal Knowledge About Fractions

The representations of the arithmetic operations include functional relationships

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

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