AN EXPLORATORY STUDY OF THE PROCESSES THAT COLLEGE MATHEMATICS STUDENTS USE TO SOLVE REAL-WORLD PROBLEMS.

摘要

The goal of this exploratory study was to determine how effectively a sample of undergraduate mathematics majors solve real-world problems in which mathematics can be applied. To help define the processes involved in the complex task of real-world problem solving, a model was developed by extrapolation from operations research literature. This model enumerated the following steps: perception of a need, formulation of the problem, collection of data, construction of a model, solution of the problem as formulated by the model, testing the solution and model, and implementing and evaluating the solution in the real-world.;Thirty volunteers participated in a one-hour clinical interview in which they thought out loud while solving problems relating to various steps in the real-world problem solving model. The problems were divided into four sections: (1) estimation, (2) non-standard word problems, (3) model construction, and (4) open-ended questions. A tape recording of their thought processes was later transcribed and analyzed to determine the strategies which led to successful products and the difficulties encountered in each section.;Contrary to the investigator's expectation, students constructed more arithmetic than algebraic models. In addition, each student constructed only one model for a particular problem, although the entire group generated many different correct models for the same problem. A somewhat surprising number of students tested their solutions, but testing, by itself, did not ensure that the error was corrected. Only those students who were able to examine their work critically and learn from their errors benefited from testing their solutions.;The results of the investigation were consistent with previous research findings which found that students have trouble estimating physical quantities and determining appropriate data in the presence of extraneous data. In addition, students made various errors in arithmetic reasoning which indicated conceptual difficulties with percentages. Other difficulties included calculation errors and the inability to relate the given values in the problems. Moreover, students generally did a superficial job of listing factors prerequisite to making a decision in the given real-world situations. Generally, students seemed to have trouble transferring mathematical concepts, learned in an academic environment, to real-world problems.;To understand the difficulties in estimating physical quantities, a model which described the processes students employed in making estimates was developed. Two methods for obtaining estimates were described: a direct method in which recall, intuition, or the formulation of a one-to-one correspondence was used, and an indirect method in which the estimate was broken into subquantities each of which was estimated by the direct method and then combined by appropriate arithmetic operations.;On the basis of the analysis, several suggestions for improving student performance on real-world problem solving tasks were given. Students might become better estimators by acquiring more reference values through experience in measuring various physical quantities and by comparing the values obtained by a variety of indirect strategies. To encourage students to construct the model with which they feel most comfortable, it might be appropriate for textbooks to present a diversity of models for the same problem. Instruction and exercises on error detections and methods to help bridge the gap between classroom theory and real-world problems were also suggested. Further research should be undertaken to determine if these suggestions are effective measures for making students better problem solvers in real-world situations.