Mathematical misunderstandings : qualitative reasoning about quantitative problems

Summary In this chapter, we summarize a research program on students' misunderstandings of mathematical story problems. A mathematical misunderstanding occurs when the problem solver constructs a mental model of the problem situation that conflicts with the information in the problem statement. In particular, we examine difficulties that students have in reasoning about problems containing relational statements, such as “Gas at Chevron is 5 cents more per gallon than gas at ARCO.” We report a series of studies in which relational statements cause systematic errors in the recall of story problems, in the pattern of solution errors, in the pattern of solution times, in students' eye fixations, and in the effectiveness of remediation training. This research supports the idea that instruction in mathematical problem solving should focus on the development of qualitative reasoning skills–such as how to build a mental model of the problem situation–in addition to quantitative reasoning skills–such as executing computational procedures.

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