The present study reveals the importance of students imagining a situation and engaging in activities to conceptualize, vary, and covary the values of related quantities as they attempt to solve word problems. (p. 58)
Moore, K. C., & Carlson, M. P. (2012). Students’ images of problem contexts when solving applied problems. The Journal of Mathematical Behavior, 31(1), 48-59.
Gaining insights into students' ways of reasoning as they worked to understand a dynamic phenomenon including their use of graphs, tables, formulas, etc.
Methods and Participants
Semi-structured clinical interviews with nine undergraduate precalculus level students.
Matching the length of this study guide, the major takeaway of this empirical report is short. A student's image of a problem context dictates their use of graphs, tables, formulas, etc. For example, in the case of Travis, he developed a formula that most teachers and researchers characterize as "incorrect." That characterization is only in the eye of the beholder. From Travis's (and our) perspective(s), his formula accurately captured his image of the problem context; he envisioned a box with a fixed base and varying height, and his formula captured that. Similarly, in constructing graphs, the covariational relationships they developed simultaneously afforded and constrained their actions. Said another way, their re-presentational activity was constrained and supported by the covariational relationships they sought to re-present. It's a somewhat obvious takeaway in retrospect, yet an important research finding.
Support the images of problem contexts your students develop, and always consider their solution activity with respect to those images. It's likely their solutions are viable reflections of those images, and thus "correct" with respect to how they envision the problem context.