Reflection on "What If Not" Strategy

The "What If Not" strategy is perhaps what distinguishes a mathematician most from any other type of scholar. By asking students questions constructed from an original problem and therefore exploring variations of the same theme students can be encouraged to engage in the same type of thinking that is so important in the study of mathematics. Because the nature of mathematics allows us to negate any physical restraints (for example, there surely is only one reasonable law of physics, if any), we can therefore allow students' minds to 'run wild'. The limitation of course is our own lack of creativity. While it gives us some tools students are still limited by their teachers' limited creativity. On the other hand, teachers might run wild with the idea and create problems that drift too far from the intended topic. For example it is not hard to get from plane geometry to constructing Darboux functions by asking a series of 'what if not' questions.