Many great ideas in mathematics have resulted from examining problems and solutions from different perspectives or directions, but one-dimensional thinking in solving a problem is all too familiar. A number of approaches to teaching and learning mathematics, especially those involving algorithms of some form, lend themselves to a reinforcement of one direction in solving problems. Mathematical insight gained by changing perspective is occasionally suggested in specific problems, but the “big picture” of mathematical exploration, conjecture, and proof requires a deeper commitment to this process. This article focuses on problems that can benefit from a change in perspective by looking forward and backward. In particular, the transition from basic mathematical manipulations to higher-order levels of reasoning requires an awareness of such processes.
cliff Sloyer is also a faculty fellow in the Mathematics and Science Educational Resource Center at the same institution. His interests include problem solving and teaching mathematical modeling.