"What-if-Not" reflection

At the risk of sounding somewhat informal, I think that the ‘what-if-not’ strategy outlined in The Art of Problem Posing is really cool. From my initial reading of it, the strategy basically involves analyzing an existing problem, definition or object, making a list of its characteristics, and then omitting certain of them to see what might happen. I think this is a fabulous way to stimulate creative thinking among math students, and I already have some ideas about how I can apply it to the micro-teaching exercise next week. The idea I had for the presentation was to demonstrate a transition from arithmetic sequences to arithmetic series. I plan to do this by posing the problem of how to calculate the sum of the first 100 natural numbers, extending that to the sum of the first n natural numbers, and then applying the results to find the sum of a sequence of the form t(n) = a + (n-1)d. Basically, my plan is to ‘what-if-not’ two characteristics of the sum 1+2+3+…+ 100. Namely, ‘what if the sum does not end at 100, but continues to an arbitrary number,’ and ‘what if the initial term and difference are not 1.’ I think that ‘what-if-not’-ing is an excellent way to promote original and creative thinking. Particularly, I think it provides a great opportunity for teachers and students to collaborate in ways that could foster an atmosphere of discovery in a math class. From a teacher’s point of view, I think that the ‘what-if-not’ strategy is an effective method for generating multiple problems from a single mathematical object. Most importantly, the strategy allows students to experience discovery, which might lead them to be more curious about mathematics. One potential pitfall of using this strategy is that it seems like it might require quite a bit of effort on the part of the student. This is fine for more capable students, but if a student has a poor relationship with mathematics, implementation of the ‘what-if-not’ strategy may require a great deal of teacher intervention. Aside from this concern, I think ‘what-if-not’-ing is a strategy which I could use to great effect.