Students’ images of problem contexts when solving applied problems - Moore, Kevin C.; Carlson, Marilyn P. (The Journal of mathematical behavior, 2012)
► We investigate precalculus students’ solutions to novel problems. ► The students’ mental images of the problems’ contexts influenced their solutions. ► Incorrect solutions were a result of incorrect images of the problem contexts. ► Solutions were corrected only after constructing correct quantitative structures. ► Curriculum must be attentive to students’ emergent quantitative structures. This article reports findings from an investigation of precalculus students’ approaches to solving novel problems. We characterize the images that students constructed during their solution attempts and describe the degree to which they were successful in imagining how the quantities in a problem's context change together. Our analyses revealed that students who mentally constructed a robust structure of the related quantities were able to produce meaningful and correct solutions. In contrast, students who provided incorrect solutions consistently constructed an image of the problem's context that was misaligned with the intent of the problem. We also observed that students who caught errors in their solutions did so by refining their image of how the quantities in a problem's context are related. These findings suggest that it is critical that students first engage in mental activity to visualize a situation and construct relevant quantitative relationships prior to determining formulas or graphs.