CHARACTERISTICS OF PROBLEM REPRESENTATION INDICATIVE OF UNDERSTANDING IN MATHEMATICS PROBLEM SOLVING
Abstract
The purpose of this study was to develop a means for assessing understanding in mathematics problem solving. In phase one, a list of characteristics of problem representation that are potentially indicative of understanding was developed. Their relationship to understanding was examined in phase two. Information processing and cognitive psychology research in problem solving and representation served as the theoretical basis for developing the characterization. This basis was extended through analysis of mathematics problem solving protocols. In phase one, sixteen subjects, age 16-25, were interviewed individually or in pairs in two-hour, audio-taped sessions as they solved as many as ten nonroutine mathematics problems while thinking aloud. The experimenter's written record of observations, the audio-tape, and the subjects' written work were analyzed using an initial list of anticipated observable events which was modified as the analysis progressed. The list of characteristics of problem representation contained three broad categories. The first category, which describes the content of the representation, includes accuracy, completeness, and generalizability. The second category is external code which focuses on the solver's written and verbal expression. The third category is processes descriptive of how the solver established the representation. Specific factors within each of these categories have been identified. In phase two thirty-six university undergraduates were interviewed individually in two-hour sessions as they solved four nonroutine problems and for each completed four follow-up tasks assumed to indicate understanding, namely, two recollection tasks, a similar problem task and a problem creation task. The subject's representations for each of the four problems were assessed using the instrument developed in phase one. Kendall's tau correlation coefficients were computed for the representation variables with success on the follow-up tasks. Except for the recollection tasks, the results showed significant positive correlations between success on the follow-up tasks and the representation variables not concentrated on a single value. The main result of this study is that characteristics indicative of understanding in mathematics problem solving have been identified and verified.
Degree
Ph.D.
Subject Area
Mathematics education
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